Object Files (.obj)

Minimal example: box.obj
Minimal textured example

Object files define the geometry and other properties for objects in
Wavefront's Advanced Visualizer. Object files can also be used to
transfer geometric data back and forth between the Advanced Visualizer
and other applications.

Object files can be in ASCII format (.obj) or binary format (.mod).
This appendix describes the ASCII format for object files. These files
must have the extension .obj.

In this release, the .obj file format supports both polygonal objects
and free-form objects. Polygonal geometry uses points, lines, and faces
to define objects while free-form geometry uses curves and surfaces.

About this section

The .obj appendix is for those who want to use the .obj format to
translate geometric data from other software applications to Wavefront
products. It also provides information for Advanced Visualizer users
who want detailed information on the Wavefront .obj file format.

If you are a 2.11 user and want to understand the significance of the
3.0 release and how it affects your existing files, you may be
especially interested in the section called "Superseded statements" at
the end of the appendix. The section, "Patches and free-form surfaces,"
gives examples of how 2.11 patches look in 3.0.

How this section is organized

Most of this appendix describes the different parts of an .obj file and
how those parts are arranged in the file. The three sections at the end
of the appendix provide background information on the 3.0 release of
the .obj format.

The .obj appendix includes the following sections:

o       File structure
o       General statement
o       Vertex data
o       Specifying free-form curves/surfaces
o       Free-form curve/surface attributes
o       Elements
o       Free-form curve/surface body statements
o       Connectivity between free-form surfaces
o       Grouping
o       Display/render attributes
o       Comments
o       Mathematics for free-form curves/surfaces
o       Superseded statements
o       Patches and free-form surfaces

    The curve and surface extensions to the .obj file format were
    developed in conjunction with mental images GmbH&Co.KG, Berlin,
    Germany, as part of a joint development project to incorporate
    free-form surfaces into Wavefront's Advanced Visualizer.

File structure

The following types of data may be included in an .obj file. In this
list, the keyword (in parentheses) follows the data type.

Vertex data

o       geometric vertices (v)
o       texture vertices (vt)
o       vertex normals (vn)
o       parameter space vertices (vp)
	   Free-form curve/surface attributes
o       rational or non-rational forms of curve or surface type:
	   basis matrix, Bezier, B-spline, Cardinal, Taylor (cstype)
o       degree (deg)
o       basis matrix (bmat)
o       step size (step)

Elements

o       point (p)
o       line (l)
o       face (f)
o       curve (curv)
o       2D curve (curv2)
o       surface (surf)

Free-form curve/surface body statements

o       parameter values (parm)
o       outer trimming loop (trim)
o       inner trimming loop (hole)
o       special curve (scrv)
o       special point (sp)
o       end statement (end)

Connectivity between free-form surfaces

o       connect (con)

Grouping

o       group name (g)
o       smoothing group (s)
o       merging group (mg)
o       object name (o)

Display/render attributes

o       bevel interpolation (bevel)
o       color interpolation (c_interp)
o       dissolve interpolation (d_interp)
o       level of detail (lod)
o       material name (usemtl)
o       material library (mtllib)
o       shadow casting (shadow_obj)
o       ray tracing (trace_obj)
o       curve approximation technique (ctech)
o       surface approximation technique (stech)

General statement

call  filename.ext arg1 arg2 . . .

    Reads the contents of the specified .obj or .mod file at this
    location.  The call statement can be inserted into .obj files using
    a text editor.

    filename.ext is the name of the .obj or .mod file to be read. You
    must include the extension with the filename.

    arg1  arg2 . . .  specifies a series of optional integer arguments
    that are passed to the called file. There is no limit to the number
    of nested calls that can be made.

    Arguments passed to the called file are substituted in the same way
    as in UNIX scripts; for example, $1 in the called file is replaced
    by arg1,  $2 in the called file is replaced by arg2, and so on.

    If the frame number is needed in the called file for variable
    substitution, "$1" must be used as the first argument in the call
    statement. For example:

	call filename.obj $1

    Then the statement in the called file,

	scmp filename.pv $1

    will work as expected. For more information on the scmp statement,
    see appendix C, Variable Substitution for more information.

    Another method to do the same thing is:

	scmp filename.pv $1

	call filename.obj

    Using this method, the scmp statement provides the .pv file for all
    subsequently called .obj or .mod files.

csh command

csh -command

    Executes the requested UNIX command. If the UNIX command returns an
    error, the parser flags an error during parsing.

    If a dash (-) precedes the UNIX command, the error is ignored.

    command is the UNIX command.

Vertex data

Vertex data provides coordinates for:

o        geometric vertices
o        texture vertices
o        vertex normals

For free-form objects, the vertex data also provides:

o        parameter space vertices

The vertex data is represented by four vertex lists; one for each type
of vertex coordinate. A right-hand coordinate system is used to specify
the coordinate locations.

The following sample is a portion of an .obj file that contains the
four types of vertex information.

    v      -5.000000       5.000000       0.000000
    v      -5.000000      -5.000000       0.000000
    v       5.000000      -5.000000       0.000000
    v       5.000000       5.000000       0.000000
    vt     -5.000000       5.000000       0.000000
    vt     -5.000000      -5.000000       0.000000
    vt      5.000000      -5.000000       0.000000
    vt      5.000000       5.000000       0.000000
    vn      0.000000       0.000000       1.000000
    vn      0.000000       0.000000       1.000000
    vn      0.000000       0.000000       1.000000
    vn      0.000000       0.000000       1.000000
    vp      0.210000       3.590000
    vp      0.000000       0.000000
    vp      1.000000       0.000000
    vp      0.500000       0.500000

When vertices are loaded into the Advanced Visualizer, they are
sequentially numbered, starting with 1. These reference numbers are
used in element statements.

Syntax

The following syntax statements are listed in order of complexity.

v x y z w

    Polygonal and free-form geometry statement.

    Specifies a geometric vertex and its x y z coordinates. Rational
    curves and surfaces require a fourth homogeneous coordinate, also
    called the weight.

    x y z are the x, y, and z coordinates for the vertex. These are
    floating point numbers that define the position of the vertex in
    three dimensions.

    w is the weight required for rational curves and surfaces. It is
    not required for non-rational curves and surfaces. If you do not
    specify a value for w, the default is 1.0.

    NOTE: A positive weight value is recommended. Using zero or
    negative values may result in an undefined point in a curve or
    surface.

vp u v w

    Free-form geometry statement.

    Specifies a point in the parameter space of a curve or surface.

    The usage determines how many coordinates are required. Special
    points for curves require a 1D control point (u only) in the
    parameter space of the curve. Special points for surfaces require a
    2D point (u and v) in the parameter space of the surface. Control
    points for non-rational trimming curves require u and v
    coordinates. Control points for rational trimming curves require u,
    v, and w (weight) coordinates.

    u is the point in the parameter space of a curve or the first
    coordinate in the parameter space of a surface.

    v is the second coordinate in the parameter space of a surface.

    w is the weight required for rational trimming curves. If you do
    not specify a value for w, it defaults to 1.0.

    NOTE: For additional information on parameter vertices, see the
    curv2 and sp statements

vn i j k

    Polygonal and free-form geometry statement.

    Specifies a normal vector with components i, j, and k.

    Vertex normals affect the smooth-shading and rendering of geometry.
    For polygons, vertex normals are used in place of the actual facet
    normals.  For surfaces, vertex normals are interpolated over the
    entire surface and replace the actual analytic surface normal.

    When vertex normals are present, they supersede smoothing groups.

    i j k are the i, j, and k coordinates for the vertex normal. They
    are floating point numbers.

vt u v w

    Vertex statement for both polygonal and free-form geometry.

    Specifies a texture vertex and its coordinates. A 1D texture
    requires only u texture coordinates, a 2D texture requires both u
    and v texture coordinates, and a 3D texture requires all three
    coordinates.

    u is the value for the horizontal direction of the texture.

    v is an optional argument.

    v is the value for the vertical direction of the texture. The
    default is 0.

    w is an optional argument.

    w is a value for the depth of the texture. The default is 0.

Specifying free-form curves/surfaces

There are three steps involved in specifying a free-form curve or
surface element.

o       Specify the type of curve or surface (basis matrix, Bezier,
	   B-spline, Cardinal, or Taylor) using free-form curve/surface
	   attributes.
o       Describe the curve or surface with element statements.
o       Supply additional information, using free-form curve/surface
	   body statements

The next three sections of this appendix provide detailed information
on each of these steps.

Data requirements for curves and surfaces

All curves and surfaces require a certain set of data. This consists of
the following:

Free-form curve/surface attributes

o       All curves and surfaces require type data, which is given with
	   the cstype statement.
o       All curves and surfaces require degree data, which is given
	   with the deg statement.
o       Basis matrix curves or surfaces require a bmat statement.
o       Basis matrix curves or surfaces also require a step size, which
	   is given with the step statement.

Elements

o       All curves and surfaces require control points, which are
	   referenced in the curv, curv2, or surf statements.
o       3D curves and surfaces require a parameter range, which is
	   given in the curv and surf statements, respectively.

Free-form curve/surface body statements

o       All curves and surfaces require a set of global parameters or a
	   knot vector, both of which are given with the parm statement.
o       All curves and surfaces body statements require an explicit end
	   statement.

Error checks

The above set of data starts out empty with no default values when
reading of an .obj file begins. While the file is being read,
statements are encountered, information is accumulated, and some errors
may be reported.

When the end statement is encountered, the following error checks,
which involve consistency between various statements, are performed:

o    All required information is present.
o    The number of control points, number of parameter values
	(knots), and degree are consistent with the curve or surface
	type. If the type is bmatrix, the step size is also consistent.
	(For more information, refer to the parameter vector equations
	in the section, "Mathematics of free-form curves/ surfaces" at
	the end of appendix B1.)
o    If the type is bmatrix and the degree is n, the size of the
	basis matrix is (n + 1) x (n + 1).

Note that any information given by the state-setting statements remains
in effect from one curve or surface to the next. Information given
within a curve or surface body is only effective for the curve or
surface it is given with.

Free-form curve/surface attributes

Five types of free-form geometry are available in the .obj file
format:

o       Bezier
o       basis matrix
o       B-spline
o       Cardinal
o       Taylor

You can apply these types only to curves and surfaces. Each of these
five types can be rational or non-rational.

In addition to specifying the type, you must define the degree for the
curve or surface. For basis matrix curve and surface elements, you must
also specify the basis matrix and step size.

All free-form curve and surface attribute statements are state-setting.
This means that once an attribute statement is set, it applies to all
elements that follow until it is reset to a different value.

Syntax

The following syntax statements are listed in order of use.

cstype rat type

    Free-form geometry statement.

    Specifies the type of curve or surface and indicates a rational or
    non-rational form.

    rat is an optional argument.

    rat specifies a rational form for the curve or surface type. If rat
    is not included, the curve or surface is non-rational

    type specifies the curve or surface type. Allowed types are:

	bmatrix		basis matrix
	bezier		Bezier
	bspline		B-spline
	cardinal        Cardinal
	taylor		Taylor

    There is no default. A value must be supplied.

deg degu degv

    Free-form geometry statement.

    Sets the polynomial degree for curves and surfaces.

    degu is the degree in the u direction. It is required for both
    curves and surfaces.

    degv is the degree in the v direction. It is required only for
    surfaces. For Bezier, B-spline, Taylor, and basis matrix, there is
    no default; a value must be supplied. For Cardinal, the degree is
    always 3. If some other value is given for Cardinal, it will be
    ignored.

bmat u matrix

bmat v matrix

    Free-form geometry statement.

    Sets the basis matrices used for basis matrix curves and surfaces.
    The u and v values must be specified in separate bmat statements.

    NOTE: The deg statement must be given before the bmat statements
    and the size of the matrix must be appropriate for the degree.

    u specifies that the basis matrix is applied in the u direction.
    v specifies that the basis matrix is applied in the v direction.

    matrix lists the contents of the basis matrix with column subscript
    j varying the fastest. If n is the degree in the given u or v
    direction, the matrix (i,j) should be of size (n + 1) x (n + 1).

    There is no default. A value must be supplied.

    NOTE: The arrangement of the matrix is different from that commonly
    found in other references. For more information, see the examples
    at the end of this section and also the section, "Mathematics for
    free-form curves and surfaces."

step stepu stepv

    Free-form geometry statement.

    Sets the step size for curves and surfaces that use a basis
    matrix.

    stepu is the step size in the u direction. It is required for both
    curves and surfaces that use a basis matrix.

    stepv is the step size in the v direction. It is required only for
    surfaces that use a basis matrix. There is no default. A value must
    be supplied.

    When a curve or surface is being evaluated and a transition from
    one segment or patch to the next occurs, the set of control points
    used is incremented by the step size. The appropriate step size
    depends on the representation type, which is expressed through the
    basis matrix, and on the degree.

    That is, suppose we are given a curve with k control points:
			{v , ... v }
			  1       k

    If the curve is of degree n, then n + 1 control points are needed
    for each polynomial segment. If the step size is given as s, then
    the ith polynomial segment, where i = 0 is the first segment, will
    use the control points:
			{v    ,...,v      }
			  is+1      is+n+1

    For example, for Bezier curves, s = n .

    For surfaces, the above description applies independently to each
    parametric direction.

    When you create a file which uses the basis matrix type, be sure to
    specify a step size appropriate for the current curve or surface
    representation.

Examples

1.      Cubic Bezier surface made with a basis matrix

    To create a cubic Bezier surface:

	cstype bmatrix
	deg 3 3
	step 3 3
	bmat u  1       -3      3       -1      \
		0       3       -6      3       \
		0       0       3       -3      \
		0       0       0       1
	bmat v  1       -3      3       -1      \
		0       3       -6      3       \
		0       0       3       -3      \
		0       0       0       1

2.      Hermite curve made with a basis matrix

    To create a Hermite curve:

	cstype bmatrix
	deg 3
	step 2
	bmat u  1     0     -3      2      0       0       3      -2 \
		0     1     -2      1      0       0      -1       1

3.      Bezier in u direction with B-spline in v direction;
	made with a basis matrix

    To create a surface with a cubic Bezier in the u direction and
    cubic uniform B-spline in the v direction:

	cstype bmatrix
	deg 3 3
	step 3 1
	bmat u  1      -3       3      -1 \
		0       3      -6       3 \
		0       0       3      -3 \
		0       0       0       1
	bmat v  0.16666 -0.50000  0.50000 -0.16666 \
		0.66666  0.00000 -1.00000  0.50000 \
		0.16666  0.50000  0.50000 -0.50000 \
		0.00000  0.00000  0.00000  0.16666

Elements

For polygonal geometry, the element types available in the .obj file
are:

o       points
o       lines
o       faces

For free-form geometry, the element types available in the .obj file
are:

o       curve
o       2D curve on a surface
o       surface

All elements can be freely intermixed in the file.

Referencing vertex data

For all elements, reference numbers are used to identify geometric
vertices, texture vertices, vertex normals, and parameter space
vertices.

Each of these types of vertices is numbered separately, starting with
1. This means that the first geometric vertex in the file is 1, the
second is 2, and so on. The first texture vertex in the file is 1, the
second is 2, and so on. The numbering continues sequentially throughout
the entire file. Frequently, files have multiple lists of vertex data.
This numbering sequence continues even when vertex data is separated by
other data.

In addition to counting vertices down from the top of the first list in
the file, you can also count vertices back up the list from an
element's position in the file. When you count up the list from an
element, the reference numbers are negative. A reference number of -1
indicates the vertex immediately above the element. A reference number
of -2 indicates two references above and so on.

Referencing groups of vertices

Some elements, such as faces and surfaces, may have a triplet of
numbers that reference vertex data.These numbers are the reference
numbers for a geometric vertex, a texture vertex, and a vertex normal.

Each triplet of numbers specifies a geometric vertex, texture vertex,
and vertex normal. The reference numbers must be in order and must
separated by slashes (/).

o       The first reference number is the geometric vertex.
o       The second reference number is the texture vertex. It follows
	   the first slash.
o       The third reference number is the vertex normal. It follows the
	   second slash.

There is no space between numbers and the slashes. There may be more
than one series of geometric vertex/texture vertex/vertex normal
numbers on a line.

The following is a portion of a sample file for a four-sided face
element:

    f 1/1/1 2/2/2 3/3/3 4/4/4

Using v, vt, and vn to represent geometric vertices, texture vertices,
and vertex normals, the statement would read:

    f v/vt/vn v/vt/vn v/vt/vn v/vt/vn

If there are only vertices and vertex normals for a face element (no
texture vertices), you would enter two slashes (//). For example, to
specify only the vertex and vertex normal reference numbers, you would
enter:

    f 1//1 2//2 3//3 4//4

When you are using a series of triplets, you must be consistent in the
way you reference the vertex data. For example, it is illegal to give
vertex normals for some vertices, but not all.

The following is an example of an illegal statement.

    f 1/1/1 2/2/2 3//3 4//4

Syntax

The following syntax statements are listed in order of complexity of
geometry.

p  v1 v2 v3 . . .

    Polygonal geometry statement.

    Specifies a point element and its vertex. You can specify multiple
    points with this statement. Although points cannot be shaded or
    rendered, they are used by other Advanced Visualizer programs.

    v is the vertex reference number for a point element. Each point
    element requires one vertex. Positive values indicate absolute
    vertex numbers. Negative values indicate relative vertex numbers.

l  v1/vt1   v2/vt2   v3/vt3 . . .

    Polygonal geometry statement.

    Specifies a line and its vertex reference numbers. You can
    optionally include the texture vertex reference numbers. Although
    lines cannot be shaded or rendered, they are used by other Advanced
    Visualizer programs.

    The reference numbers for the vertices and texture vertices must be
    separated by a slash (/). There is no space between the number and
    the slash.

    v is a reference number for a vertex on the line. A minimum of two
    vertex numbers are required. There is no limit on the maximum.
    Positive values indicate absolute vertex numbers. Negative values
    indicate relative vertex numbers.

    vt is an optional argument.

    vt is the reference number for a texture vertex in the line
    element. It must always follow the first slash.

f  v1/vt1/vn1   v2/vt2/vn2   v3/vt3/vn3 . . .

    Polygonal geometry statement.

    Specifies a face element and its vertex reference number. You can
    optionally include the texture vertex and vertex normal reference
    numbers.

    The reference numbers for the vertices, texture vertices, and
    vertex normals must be separated by slashes (/). There is no space
    between the number and the slash.

    v is the reference number for a vertex in the face element. A
    minimum of three vertices are required.

    vt is an optional argument.

    vt is the reference number for a texture vertex in the face
    element. It always follows the first slash.

    vn is an optional argument.

    vn is the reference number for a vertex normal in the face element.
    It must always follow the second slash.

    Face elements use surface normals to indicate their orientation. If
    vertices are ordered counterclockwise around the face, both the
    face and the normal will point toward the viewer. If the vertex
    ordering is clockwise, both will point away from the viewer. If
    vertex normals are assigned, they should point in the general
    direction of the surface normal, otherwise unpredictable results
    may occur.

    If a face has a texture map assigned to it and no texture vertices
    are assigned in the f statement, the texture map is ignored when
    the element is rendered.

    NOTE: Any references to fo (face outline) are no longer valid as of
    version 2.11. You can use f (face) to get the same results.
    References to fo in existing .obj files will still be read,
    however, they will be written out as f when the file is saved.

curv u0 u1 v1 v2 . . .

    Element statement for free-form geometry.

    Specifies a curve, its parameter range, and its control vertices.
    Although curves cannot be shaded or rendered, they are used by
    other Advanced Visualizer programs.

    u0 is the starting parameter value for the curve. This is a
    floating point number.

    u1 is the ending parameter value for the curve. This is a floating
    point number.

    v is the vertex reference number for a control point. You can
    specify multiple control points. A minimum of two control points
    are required for a curve.

    For a non-rational curve, the control points must be 3D. For a
    rational curve, the control points are 3D or 4D. The fourth
    coordinate (weight) defaults to 1.0 if omitted.

curv2  vp1  vp2   vp3. . .

    Free-form geometry statement.

    Specifies a 2D curve on a surface and its control points. A 2D
    curve is used as an outer or inner trimming curve, as a special
    curve, or for connectivity.

    vp is the parameter vertex reference number for the control point.
    You can specify multiple control points. A minimum of two control
    points is required for a 2D curve.

    The control points are parameter vertices because the curve must
    lie in the parameter space of some surface. For a non-rational
    curve, the control vertices can be 2D. For a rational curve, the
    control vertices can be 2D or 3D. The third coordinate (weight)
    defaults to 1.0 if omitted.

surf  s0  s1  t0  t1  v1/vt1/vn1   v2/vt2/vn2 . . .

    Element statement for free-form geometry.

    Specifies a surface, its parameter range, and its control vertices.
    The surface is evaluated within the global parameter range from s0
    to s1 in the u direction and t0 to t1 in the v direction.

    s0 is the starting parameter value for the surface in the u
    direction.

    s1 is the ending parameter value for the surface in the u
    direction.

    t0 is the starting parameter value for the surface in the v
    direction.

    t1 is the ending parameter value for the surface in the v
    direction.

    v is the reference number for a control vertex in the surface.

    vt is an optional argument.

    vt is the reference number for a texture vertex in the surface.  It
    must always follow the first slash.

    vn is an optional argument.

    vn is the reference number for a vertex normal in the surface.  It
    must always follow the second slash.

    For a non-rational surface, the control vertices are 3D.  For a
    rational surface the control vertices can be 3D or 4D.  The fourth
    coordinate (weight) defaults to 1.0 if ommitted.

    NOTE: For more information on the ordering of control points for
    survaces, refer to the section on surfaces and control points in
    "mathematics of free-form curves/surfaces" at the end of this
    appendix.

Examples

These are examples for polygonal geometry.

For examples using free-form geometry, see the examples at the end of
the next section, "Free-form curve/surface body statements."

1.	Square

This example shows a square that measures two units on each side and
faces in the positive direction (toward the camera).  Note that the
ordering of the vertices is counterclockwise. This ordering determines
that the square is facing forward.

    v 0.000000 2.000000 0.000000
    v 0.000000 0.000000 0.000000
    v 2.000000 0.000000 0.000000
    v 2.000000 2.000000 0.000000
    f 1 2 3 4

2.      Cube

This is a cube that measures two units on each side. Each vertex is
shared by three different faces.

    v 0.000000 2.000000 2.000000
    v 0.000000 0.000000 2.000000
    v 2.000000 0.000000 2.000000
    v 2.000000 2.000000 2.000000
    v 0.000000 2.000000 0.000000
    v 0.000000 0.000000 0.000000
    v 2.000000 0.000000 0.000000
    v 2.000000 2.000000 0.000000
    f 1 2 3 4
    f 8 7 6 5
    f 4 3 7 8
    f 5 1 4 8
    f 5 6 2 1
    f 2 6 7 3

3.      Cube with negative reference numbers

This is a cube with negative vertex reference numbers. Each element
references the vertices stored immediately above it in the file. Note
that vertices are not shared.

v 0.000000 2.000000 2.000000
v 0.000000 0.000000 2.000000
v 2.000000 0.000000 2.000000
v 2.000000 2.000000 2.000000
f -4 -3 -2 -1

v 2.000000 2.000000 0.000000
v 2.000000 0.000000 0.000000
v 0.000000 0.000000 0.000000
v 0.000000 2.000000 0.000000
f -4 -3 -2 -1

v 2.000000 2.000000 2.000000
v 2.000000 0.000000 2.000000
v 2.000000 0.000000 0.000000
v 2.000000 2.000000 0.000000
f -4 -3 -2 -1

v 0.000000 2.000000 0.000000
v 0.000000 2.000000 2.000000
v 2.000000 2.000000 2.000000
v 2.000000 2.000000 0.000000
f -4 -3 -2 -1

v 0.000000 2.000000 0.000000
v 0.000000 0.000000 0.000000
v 0.000000 0.000000 2.000000
v 0.000000 2.000000 2.000000
f -4 -3 -2 -1

v 0.000000 0.000000 2.000000
v 0.000000 0.000000 0.000000
v 2.000000 0.000000 0.000000
v 2.000000 0.000000 2.000000
f -4 -3 -2 -1

Free-form curve/surface body statements

You can specify additional information for free-form curve and surface
elements using a series of statements called body statements. The
series is concluded by an end statement.

Body statements are valid only when they appear between the free-form
element statement (curv, curv2, surf) and the end statement. If they
are anywhere else in the .obj file, they do not have any effect.

You can use body statements to specify the following values:

o       parameter
o       knot vector
o       trimming loop
o       hole
o       special curve
o       special point

You cannot use any other statements between the free-form curve or
surface statement and the end statement. Using any other of type of
statement may cause unpredictable results.

This portion of a sample file shows the knot vector values for a
rational B-spline surface with a trimming loop. Notice the end
statement to conclude the body statements.

    cstype rat bspline
    deg 2 2
    surf -1.0 2.5 -2.0 2.0 -9 -8 -7 -6 -5 -4 -3 -2 -1
    parm u -1.00 -1.00 -1.00 2.50 2.50 2.50
    parm v -2.00 -2.00 -2.00 -2.00 -2.00 -2.00
    trim 0.0 2.0 1
    end

Parameter values and knot vectors

All curve and surface elements require a set of parameter values.

For polynomial curves and surfaces, this specifies global parameter
values. For B-spline curves and surfaces, this specifies the knot
vectors.

For surfaces, the parameter values must be specified for both the u and
v directions. For curves, the parameter values must be specified for
only the u direction.

If multiple parameter value statements for the same parametric
direction are used inside a single curve or surface body, the last
statement is used.

Trimming loops and holes

The trimming loop statement builds a single outer trimming loop as a
sequence of curves which lie on a given surface.

The hole statement builds a single inner trimming loop as a sequence of
curves which lie on a given surface. The inner loop creates a hole.

The curves are referenced by number in the same way vertices are
referenced by face elements.

The individual curves must lie end-to-end to form a closed loop which
does not intersect itself and which lies within the parameter range
specified for the surface. The loop as a whole may be oriented in
either direction (clockwise or counterclockwise).

To cut one or more holes in a region, use a trim statement followed by
one or more hole statements. To introduce another trimmed region in the
same surface, use another trim statement followed by one or more hole
statements. The ordering that associates holes and the regions they cut
is important and must be maintained.

If the first trim statement in the sequence is omitted, the enclosing
outer trimming loop is taken to be the parameter range of the surface.
If no trim or hole statements are specified, then the surface is
trimmed at its parameter range.

This portion of a sample file shows a non-rational Bezier surface with
two regions, each with a single hole:

    cstype bezier
    deg 1 1
    surf 0.0 2.0 0.0 2.0 1 2 3 4
    parm u 0.00 2.00
    parm v 0.00 2.00
    trim 0.0 4.0 1
    hole 0.0 4.0 2
    trim 0.0 4.0 3
    hole 0.0 4.0 4
    end

Special curve

A special curve statement builds a single special curve as a sequence
of curves which lie on a given surface.

The curves are referenced by number in the same way vertices are
referenced by face elements.

A special curve is guaranteed to be included in any triangulation of
the surface. This means that the line formed by approximating the
special curve with a sequence of straight line segments will actually
appear as a sequence of triangle edges in the final triangulation.

Special point

A special point statement specifies that special geometric points are
to be associated with a curve or surface. For space curves and trimming
curves, the parameter vertices must be 1D. For surfaces, the parameter
vertices must be 2D.

These special points will be included in any linear approximation of
the curve or surface.

For space curves, this means that the point corresponding to the given
curve parameter is included as one of the vertices in an approximation
consisting of a sequence of line segments.

For surfaces, this means that the point corresponding to the given
surface parameters is included as a triangle vertex in the
triangulation.

For trimming curves, the treatment is slightly different: a special
point on a trimming curve is essentially the same as a special point on
the surface it trims.

The following portion of a sample files shows special points for a
rational Bezier 2D curve on a surface.

    vp -0.675  1.850  3.000
    vp  0.915  1.930
    vp  2.485  0.470  2.000
    vp  2.485 -1.030
    vp  1.605 -1.890 10.700
    vp -0.745 -0.654  0.500
    cstype rat bezier
    curv2 -6 -5 -4 -3 -2 -1 -6
    parm u 0.00 1.00 2.00
    sp 2 3
    end

Syntax

The following syntax statement are listed in order of normal use.

parm u p1 p2 p3. . .

parm v p1 p2 p3 . . .

    Body statement for free-form geometry.

    Specifies global parameter values. For B-spline curves and
    surfaces, this specifies the knot vectors.

    u is the u direction for the parameter values.

    v is the v direction for the parameter values.

    To set u and v values, use separate command lines.

    p is the global parameter or knot value. You can specify multiple
    values. A minimum of two parameter values are required. Parameter
    values must increase monotonically. The type of surface and the
    degree dictate the number of values required.

trim  u0  u1  curv2d  u0  u1  curv2d . . .

    Body statement for free-form geometry.

    Specifies a sequence of curves to build a single outer trimming
    loop.

    u0 is the starting parameter value for the trimming curve curv2d.

    u1 is the ending parameter value for the trimming curve curv2d.

    curv2d is the index of the trimming curve lying in the parameter
    space of the surface. This curve must have been previously defined
    with the curv2 statement.

hole  u0  u1  curv2d  u0  u1  curv2d . . .

    Body statement for free-form geometry.

    Specifies a sequence of curves to build a single inner trimming
    loop (hole).

    u0 is the starting parameter value for the trimming curve curv2d.

    u1 is the ending parameter value for the trimming curve curv2d.

    curv2d is the index of the trimming curve lying in the parameter
    space of the surface. This curve must have been previously defined
    with the curv2 statement.

scrv u0 u1 curv2d u0 u1 curv2d . . .

    Body statement for free-form geometry.

    Specifies a sequence of curves which lie on the given surface to
    build a single special curve.

    u0 is the starting parameter value for the special curve curv2d.

    u1 is the ending parameter value for the special curve curv2d.

    curv2d is the index of the special curve lying in the parameter
    space of the surface. This curve must have been previously defined
    with the curv2 statement.

sp vp1  vp. . .

    Body statement for free-form geometry.

    Specifies special geometric points to be associated with a curve or
    surface. For space curves and trimming curves, the parameter
    vertices must be 1D. For surfaces, the parameter vertices must be
    2D.

    vp is the reference number for the parameter vertex of a special
    point to be associated with the parameter space point of the curve
    or surface.

end

    Body statement for free-form geometry.

    Specifies the end of a curve or surface body begun by a curv,
    curv2, or surf statement.

Examples

1.      Taylor curve

    For creating a single-segment Taylor polynomial curve of the form:

                                   2         3         4
	x =  3.00 +  2.30t +  7.98t  +  8.30t  +  6.34t 

                                   2         3         4
	y =  1.00 - 10.10t +  5.40t  -  4.70t  +  2.03t 

                                   2         3         4
	z = -2.50 +  0.50t -  7.00t  + 18.10t  +  0.08t 

and evaluated between the global parameters 0.5 and 1.6:

	v       3.000    1.000   -2.500
	v       2.300  -10.100    0.500
	v       7.980    5.400   -7.000
	v       8.300   -4.700   18.100
	v       6.340    2.030    0.080
	cstype taylor
	deg 4
	curv 0.500 1.600 1 2 3 4 5
	parm u 0.000 2.000
	end

2.      Bezier curve

This example shows a non-rational Bezier curve with 13 control points.

    v -2.300000 1.950000 0.000000
    v -2.200000 0.790000 0.000000
    v -2.340000 -1.510000 0.000000
    v -1.530000 -1.490000 0.000000
    v -0.720000 -1.470000 0.000000
    v -0.780000 0.230000 0.000000
    v 0.070000 0.250000 0.000000
    v 0.920000 0.270000 0.000000
    v 0.800000 -1.610000 0.000000
    v 1.620000 -1.590000 0.000000
    v 2.440000 -1.570000 0.000000
    v 2.690000 0.670000 0.000000
    v 2.900000 1.980000 0.000000
    # 13 vertices

    cstype bezier
    ctech cparm 1.000000
    deg 3
    curv 0.000000 4.000000 1 2 3 4 5 6 7 8 9 10 \
    11 12 13
    parm u 0.000000 1.000000 2.000000 3.000000  \
    4.000000
    end
    # 1 element

3.      B-spline surface

This is an example of a cubic B-spline surface.

    g bspatch
    v -5.000000 -5.000000 -7.808327
    v -5.000000 -1.666667 -7.808327
    v -5.000000 1.666667 -7.808327
    v -5.000000 5.000000 -7.808327
    v -1.666667 -5.000000 -7.808327
    v -1.666667 -1.666667 11.977780
    v -1.666667 1.666667 11.977780
    v -1.666667 5.000000 -7.808327
    v 1.666667 -5.000000 -7.808327
    v 1.666667 -1.666667 11.977780
    v 1.666667 1.666667 11.977780
    v 1.666667 5.000000 -7.808327
    v 5.000000 -5.000000 -7.808327
    v 5.000000 -1.666667 -7.808327
    v 5.000000 1.666667 -7.808327
    v 5.000000 5.000000 -7.808327
    # 16 vertices

    cstype bspline
    stech curv 0.5 10.000000
    deg 3 3
    8surf 0.000000 1.000000 0.000000 1.000000 13 14 \ 15 16 9 10 11 12 5 6
    7 8 1 2 3 4
    parm u -3.000000 -2.000000 -1.000000 0.000000  \
    1.000000 2.000000 3.000000 4.000000
    parm v -3.000000 -2.000000 -1.000000 0.000000  \
    1.000000 2.000000 3.000000 4.000000
    end
    # 1 element

4.      Cardinal surface

This example shows a Cardinal surface.

    v -5.000000 -5.000000 0.000000
    v -5.000000 -1.666667 0.000000
    v -5.000000 1.666667 0.000000
    v -5.000000 5.000000 0.000000
    v -1.666667 -5.000000 0.000000
    v -1.666667 -1.666667 0.000000
    v -1.666667 1.666667 0.000000
    v -1.666667 5.000000 0.000000
    v 1.666667 -5.000000 0.000000
    v 1.666667 -1.666667 0.000000
    v 1.666667 1.666667 0.000000
    v 1.666667 5.000000 0.000000
    v 5.000000 -5.000000 0.000000
    v 5.000000 -1.666667 0.000000
    v 5.000000 1.666667 0.000000
    v 5.000000 5.000000 0.000000
    # 16 vertices

    cstype cardinal
    stech cparma 1.000000 1.000000
    deg 3 3
    surf 0.000000 1.000000 0.000000 1.000000 13 14 \
    15 16 9 10 11 12 5 6 7 8 1 2 3 4
    parm u 0.000000 1.000000
    parm v 0.000000 1.000000
    end
    # 1 element

5.      Rational B-spline surface

This example creates a second-degree, rational B-spline surface using
open, uniform knot vectors. A texture map is applied to the surface.

    v -1.3 -1.0  0.0
    v  0.1 -1.0  0.4  7.6
    v  1.4 -1.0  0.0  2.3
    v -1.4  0.0  0.2
    v  0.1  0.0  0.9  0.5
    v  1.3  0.0  0.4  1.5
    v -1.4  1.0  0.0  2.3
    v  0.1  1.0  0.3  6.1
    v  1.1  1.0  0.0  3.3
    vt 0.0  0.0
    vt 0.5  0.0
    vt 1.0  0.0
    vt 0.0  0.5
    vt 0.5  0.5
    vt 1.0  0.5
    vt 0.0  1.0
    vt 0.5  1.0
    vt 1.0  1.0
    cstype rat bspline
    deg 2 2
    surf 0.0 1.0 0.0 1.0 1/1 2/2 3/3 4/4 5/5 6/6 \
    7/7 8/8 9/9
    parm u 0.0 0.0 0.0 1.0 1.0 1.0
    parm v 0.0 0.0 0.0 1.0 1.0 1.0
    end

6.      Trimmed NURB surface

This is a complete example of a file containing a trimmed NURB surface
with negative reference numbers for vertices.

    # trimming curve
    vp -0.675  1.850  3.000
    vp  0.915  1.930
    vp  2.485  0.470  2.000
    vp  2.485 -1.030
    vp  1.605 -1.890 10.700
    vp -0.745 -0.654  0.500
    cstype rat bezier
    deg 3
    curv2 -6 -5 -4 -3 -2 -1 -6
    parm u 0.00 1.00 2.00
    end
    # surface
    v -1.350 -1.030 0.000
    v  0.130 -1.030 0.432 7.600
    v  1.480 -1.030 0.000 2.300
    v -1.460  0.060 0.201
    v  0.120  0.060 0.915 0.500
    v  1.380  0.060 0.454 1.500
    v -1.480  1.030 0.000 2.300
    v  0.120  1.030 0.394 6.100
    v  1.170  1.030 0.000 3.300
    cstype rat bspline
    deg 2 2
    surf -1.0 2.5 -2.0 2.0 -9 -8 -7 -6 -5 -4 -3 -2 -1
    parm u -1.00 -1.00 -1.00 2.50 2.50 2.50
    parm v -2.00 -2.00 -2.00 -2.00 -2.00 -2.00
    trim 0.0 2.0 1
    end

7.      Two trimming regions with a hole

This example shows a Bezier surface with two trimming regions, each
with a hole in them.

    # outer loop of first region
    deg 1
    cstype bezier
    vp 0.100 0.100
    vp 0.900 0.100
    vp 0.900 0.900
    vp 0.100 0.900
    curv2 1 2 3 4 1
    parm u 0.00 1.00 2.00 3.00 4.00
    end
    # hole in first region
    vp 0.300 0.300
    vp 0.700 0.300
    vp 0.700 0.700
    vp 0.300 0.700
    curv2 5 6 7 8 5
    parm u 0.00 1.00 2.00 3.00 4.00
    end
    # outer loop of second region
    vp 1.100 1.100
    vp 1.900 1.100
    vp 1.900 1.900
    vp 1.100 1.900
    curv2 9 10 11 12 9
    parm u 0.00 1.00 2.00 3.00 4.00
    end
    # hole in second region
    vp 1.300 1.300
    vp 1.700 1.300
    vp 1.700 1.700
    vp 1.300 1.700
    curv2 13 14 15 16 13
    parm u 0.00 1.00 2.00 3.00 4.00
    end
    # surface
    v 0.000 0.000 0.000
    v 1.000 0.000 0.000
    v 0.000 1.000 0.000
    v 1.000 1.000 0.000
    deg 1 1
    cstype bezier
    surf 0.0 2.0 0.0 2.0 1 2 3 4
    parm u 0.00 2.00
    parm v 0.00 2.00
    trim 0.0 4.0 1
    hole 0.0 4.0 2
    trim 0.0 4.0 3
    hole 0.0 4.0 4
    end

8.      Trimming with a special curve

This example is similar to the trimmed NURB surface example (6), except
there is a special curve on the surface. This example uses negative
vertex numbers.

    # trimming curve
    vp -0.675  1.850  3.000
    vp  0.915  1.930
    vp  2.485  0.470  2.000
    vp  2.485 -1.030
    vp  1.605 -1.890 10.700
    vp -0.745 -0.654  0.500
    cstype rat bezier
    deg 3
    curv2 -6 -5 -4 -3 -2 -1 -6
    parm u 0.00 1.00 2.00
    end
    # special curve
    vp -0.185  0.322
    vp  0.214  0.818
    vp  1.652  0.207
    vp  1.652 -0.455
    curv2 -4 -3 -2 -1
    parm u 2.00 10.00
    end
    # surface
    v -1.350 -1.030 0.000
    v  0.130 -1.030 0.432 7.600
    v  1.480 -1.030 0.000 2.300
    v -1.460  0.060 0.201
    v  0.120  0.060 0.915 0.500
    v  1.380  0.060 0.454 1.500
    v -1.480  1.030 0.000 2.300
    v  0.120  1.030 0.394 6.100
    v  1.170  1.030 0.000 3.300
    cstype rat bspline
    deg 2 2
    surf -1.0 2.5 -2.0 2.0 -9 -8 -7 -6 -5 -4 -3 -2 -1
    parm u -1.00 -1.00 -1.00 2.50 2.50 2.50
    parm v -2.00 -2.00 -2.00 2.00 2.00 2.00
    trim 0.0 2.0 1
    scrv 4.2 9.7 2
    end

9.      Trimming with special points

This example extends the trimmed NURB surface example (6) to include
special points on both the trimming curve and surface. A space curve
with a special point is also included. This example uses negative
vertex numbers.

    # special point and space curve data
    vp 0.500
    vp 0.700
    vp 1.100
    vp 0.200 0.950
    v  0.300 1.500 0.100
    v  0.000  0.000  0.000
    v  1.000  1.000  0.000
    v  2.000  1.000  0.000
    v  3.000  0.000  0.000
    cstype bezier
    deg 3
    curv 0.2 0.9 -4 -3 -2 -1
    sp 1
    parm u 0.00 1.00
    end
    # trimming curve
    vp -0.675  1.850  3.000
    vp  0.915  1.930
    vp  2.485  0.470  2.000
    vp  2.485 -1.030
    vp  1.605 -1.890 10.700
    vp -0.745 -0.654  0.500
    cstype rat bezier
    curv2 -6 -5 -4 -3 -2 -1 -6
    parm u 0.00 1.00 2.00
    sp 2 3
    end
    # surface
    v -1.350 -1.030 0.000
    v  0.130 -1.030 0.432 7.600
    v  1.480 -1.030 0.000 2.300
    v -1.460  0.060 0.201
    v  0.120  0.060 0.915 0.500
    v  1.380  0.060 0.454 1.500
    v -1.480  1.030 0.000 2.300
    v  0.120  1.030 0.394 6.100
    v  1.170  1.030 0.000 3.300
    cstype rat bspline
    deg 2 2
    surf -1.0 2.5 -2.0 2.0 -9 -8 -7 -6 -5 -4 -3 -2 -1
    parm u -1.00 -1.00 -1.00 2.50 2.50 2.50
    parm v -2.00 -2.00 -2.00 2.00 2.00 2.00
    trim 0.0 2.0 1
    sp 4
    end

Connectivity between free-form surfaces

Connectivity connects two surfaces along their trimming curves.

The con statement specifies the first surface with its trimming curve
and the second surface with its trimming curve. This information is
useful for edge merging. Without this surface and curve data,
connectivity must be determined numerically at greater expense and with
reduced accuracy using the mg statement.

Connectivity between surfaces in different merging groups is ignored.
Also, although connectivity which crosses points of C1discontinuity in
trimming curves is legal, it is not recommended. Instead, use two
connectivity statements which meet at the point of discontinuity.

The two curves and their starting and ending parameters should all map
to the same curve and starting and ending points in object space.

Syntax

con  surf_1  q0_1  q1_1   curv2d_1   surf_2  q0_2  q1_2  curv2d_2

    Free-form geometry statement.

    Specifies connectivity between two surfaces.

    surf_1 is the index of the first surface.

    q0_1 is the starting parameter for the curve referenced by
    curv2d_1.

    q1_1 is the ending parameter for the curve referenced by curv2d_1.

    curv2d_1 is the index of a curve on the first surface. This curve
    must have been previously defined with the curv2 statement.

    surf_2 is the index of the second surface.

    q0_2 is the starting parameter for the curve referenced by
    curv2d_2.

    q1_2 is the ending parameter for the curve referenced by curv2d_2.

    curv2d_2 is the index of a curve on the second surface. This curve
    must have been previously defined with the curv2 statement.

Example

1.      Connectivity between two surfaces

This example shows the connectivity between two surfaces with trimming
curves.

    cstype bezier
    deg 1 1

    v 0 0 0
    v 1 0 0
    v 0 1 0
    v 1 1 0

    vp 0 0
    vp 1 0
    vp 1 1
    vp 0 1

    curv2 1 2 3 4 1
    parm u 0.0 1.0 2.0 3.0 4.0
    end

    surf 0.0 1.0 0.0 1.0 1 2 3 4
    parm u 0.0 1.0
    parm v 0.0 1.0
    trim 0.0 4.0 1
    end

    v 1 0 0
    v 2 0 0
    v 1 1 0
    v 2 1 0

    surf 0.0 1.0 0.0 1.0 5 6 7 8
    parm u 0.0 1.0
    parm v 0.0 1.0
    trim 0.0 4.0 1
    end

    con 1 2.0 2.0 1 2 4.0 3.0 1

Grouping

There are four statements in the .obj file to help you manipulate groups
of elements:

o	Gropu name statements are used to organize collections of
	elements and simplify data manipulation for operations in
	Model.
o	Smoothing group statements let you identify elements over which
	normals are to be interpolated to give those elements a smooth,
	non-faceted appearance.  This is a quick way to specify vertex
	normals.
o	Merging group statements are used to ideneify free-form elements
	that should be inspected for adjacency detection.  You can also
	use merging groups to exclude surfaces which are close enough to
	be considered adjacent but should not be merged.
o	Object name statements let you assign a name to an entire object
	in a single file.

All grouping statements are state-setting.  This means that once a
group statement is set, it alpplies to all elements that follow
until the next group statement.

This portion of a sample file shows a single element which belongs to
three groups.  The smoothing group is turned off.

    g square thing all
    s off
    f 1 2 3 4

This example shows two surfaces in merging group 1 with a merge
resolution of 0.5.

    mg 1 .5
    surf 0.0 1.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
    surf 0.0 1.0 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

Syntax

g group_name1 group_name2 . . .

    Polygonal and free-form geometry statement.

    Specifies the group name for the elements that follow it. You can
    have multiple group names. If there are multiple groups on one
    line, the data that follows belong to all groups. Group information
    is optional.

    group_name is the name for the group. Letters, numbers, and
    combinations of letters and numbers are accepted for group names.
    The default group name is default.

s group_number

    Polygonal and free-form geometry statement.

    Sets the smoothing group for the elements that follow it. If you do
    not want to use a smoothing group, specify off or a value of 0.

    To display with smooth shading in Model and PreView, you must
    create vertex normals after you have assigned the smoothing groups.
    You can create vertex normals with the vn statement or with the
    Model program.

    To smooth polygonal geometry for rendering with Image, it is
    sufficient to put elements in some smoothing group. However, vertex
    normals override smoothing information for Image.

    group_number is the smoothing group number. To turn off smoothing
    groups, use a value of 0 or off. Polygonal elements use group
    numbers to put elements in different smoothing groups. For
    free-form surfaces, smoothing groups are either turned on or off;
    there is no difference between values greater than 0.

mg group_number res

    Free-form geometry statement.

    Sets the merging group and merge resolution for the free-form
    surfaces that follow it. If you do not want to use a merging group,
    specify off or a value of 0.

    Adjacency detection is performed only within groups, never between
    groups. Connectivity between surfaces in different merging groups
    is not allowed. Surfaces in the same merging group are merged
    together along edges that are within the distance res apart.

    NOTE: Adjacency detection is an expensive numerical comparison
    process.  It is best to restrict this process to as small a domain
    as possible by using small merging groups.

    group_number is the merging group number. To turn off adjacency
    detection, use a value of 0 or off.

    res is the maximum distance between two surfaces that will be
    merged together. The resolution must be a value greater than 0.
    This is a required argument only when using merging groups.

o object_name

    Polygonal and free-form geometry statement.

    Optional statement; it is not processed by any Wavefront programs.
    It specifies a user-defined object name for the elements defined
    after this statement.

    object_name is the user-defined object name. There is no default.

Examples

1.      Cube with group names

The following example is a cube with each of its faces placed in a
separate group. In addition, all elements belong to the group cube.

    v 0.000000 2.000000 2.000000
    v 0.000000 0.000000 2.000000
    v 2.000000 0.000000 2.000000
    v 2.000000 2.000000 2.000000
    v 0.000000 2.000000 0.000000
    v 0.000000 0.000000 0.000000
    v 2.000000 0.000000 0.000000
    v 2.000000 2.000000 0.000000
    # 8 vertices

    g front cube
    f 1 2 3 4
    g back cube
    f 8 7 6 5
    g right cube
    f 4 3 7 8
    g top cube
    f 5 1 4 8
    g left cube
    f 5 6 2 1
    g bottom cube
    f 2 6 7 3
    # 6 elements

2.      Two adjoining squares with a smoothing group

This example shows two adjoining squares that share a common edge. The
squares are placed in a smoothing group to ensure that their common
edge will be smoothed when rendered with Image.

    v 0.000000 2.000000 0.000000
    v 0.000000 0.000000 0.000000
    v 2.000000 0.000000 0.000000
    v 2.000000 2.000000 0.000000
    v 4.000000 0.000000 -1.255298
    v 4.000000 2.000000 -1.255298
    # 6 vertices

    g all
    s 1
    f 1 2 3 4
    f 4 3 5 6
    # 2 elements

3.      Two adjoining squares with vertex normals

This example also shows two squares that share a common edge. Vertex
normals have been added to the corners of each square to ensure that
their common edge will be smoothed during display in Model and PreView
and when rendered with Image.

    v 0.000000 2.000000 0.000000
    v 0.000000 0.000000 0.000000
    v 2.000000 0.000000 0.000000
    v 2.000000 2.000000 0.000000
    v 4.000000 0.000000 -1.255298
    v 4.000000 2.000000 -1.255298
    vn 0.000000 0.000000 1.000000
    vn 0.000000 0.000000 1.000000
    vn 0.276597 0.000000 0.960986
    vn 0.276597 0.000000 0.960986
    vn 0.531611 0.000000 0.846988
    vn 0.531611 0.000000 0.846988
    # 6 vertices

    # 6 normals

    g all
    s 1
    f 1//1 2//2 3//3 4//4
    f 4//4 3//3 5//5 6//6
    # 2 elements

4.      Merging group

This example shows two Bezier surfaces that meet at a common edge. They
have both been placed in the same merging group to ensure continuity at
the edge where they meet. This prevents "cracks" from appearing along
the seam between the two surfaces during rendering. Merging groups will
be ignored during flat-shading, smooth-shading, and material shading of
the surface.

    v -4.949854 -5.000000 0.000000
    v -4.949854 -1.666667 0.000000
    v -4.949854 1.666667 0.000000
    v -4.949854 5.000000 0.000000
    v -1.616521 -5.000000 0.000000
    v -1.616521 -1.666667 0.000000
    v -1.616521 1.666667 0.000000
    v -1.616521 5.000000 0.000000
    v 1.716813 -5.000000 0.000000
    v 1.716813 -1.666667 0.000000
    v 1.716813 1.666667 0.000000
    v 1.716813 5.000000 0.000000
    v 5.050146 -5.000000 0.000000
    v 5.050146 -1.666667 0.000000
    v 5.050146 1.666667 0.000000
    v 5.050146 5.000000 0.000000
    v -15.015566 -4.974991 0.000000
    v -15.015566 -1.641658 0.000000
    v -15.015566 1.691675 0.000000
    v -15.015566 5.025009 0.000000
    v -11.682233 -4.974991 0.000000
    v -11.682233 -1.641658 0.000000
    v -11.682233 1.691675 0.000000
    v -11.682233 5.025009 0.000000
    v -8.348900 -4.974991 0.000000
    v -8.348900 -1.641658 0.000000
    v -8.348900 1.691675 0.000000
    v -8.348900 5.025009 0.000000
    v -5.015566 -4.974991 0.000000
    v -5.015566 -1.641658 0.000000
    v -5.015566 1.691675 0.000000
    v -5.015566 5.025009 0.000000

    mg 1 0.500000

    cstype bezier
    deg 3 3
    surf 0.000000 1.000000 0.000000 1.000000 13 14 \
    15 16 9 10 11 12 5 6 7 8 1 2 3 4
    parm u 0.000000 1.000000
    parm v 0.000000 1.000000
    end
    surf 0.000000 1.000000 0.000000 1.000000 29 30 31 32 25 26 27 28 21 22 \
    23 24 17 18 19 20
    parm u 0.000000 1.000000
    parm v 0.000000 1.000000
    end

Display/render attributes

Display and render attributes describe how an object looks when
displayed in Model and PreView or when rendered with Image.

Some attributes apply to both free-form and polygonal geometry, such as
material name and library, ray tracing, and shadow casting.
Interpolation attributes apply only to polygonal geometry. Curve and
surface resolutions are used for only free-form geometry.

The following chart shows the display and render statements available
for polygonal and free-form geometry.

Table B1-1.     Display and render attributes

polygonal only		polygonal or free-form	free-form only
--------------		----------------------	--------------
bevel			lod			ctech
c_interp		usemtl			stech
d_interp		mtllib
			shadow_obj
			trace_obj

All display and render attribute statements are state-setting. This
means that once an attribute statement is set, it applies to all
elements that follow until it is reset to a different value.

The following sample shows rendering and display statements for a face
element.:

    s 1
    usemtl blue
    usemap marble
    f 1 2 3 4

Syntax

The following syntax statements are listed by the type of geometry.
First are statements for polygonal geometry. Second are statements for
both free-form and polygonal geometry. Third are statements for
free-form geometry only.

bevel on/off

    Polygonal geometry statement.

    Sets bevel interpolation on or off. It works only with beveled
    objects, that is, objects with sides separated by beveled faces.

    Bevel interpolation uses normal vector interpolation to give an
    illusion of roundness to a flat bevel. It does not affect the
    smoothing of non-bevelled faces.

    Bevel interpolation does not alter the geometry of the original
    object.

    on turns on bevel interpolation.

    off turns off bevel interpolation. The default is off.

    NOTE: Image cannot render bevel-interpolated elements that have
    vertex normals.

c_interp on/off

    Polygonal geometry statement.

    Sets color interpolation on or off.

    Color interpolation creates a blend across the surface of a polygon
    between the materials assigned to its vertices. This creates a
    blending of colors across a face element.

    To support color interpolation, materials must be assigned per
    vertex, not per element. The illumination models for all materials
    of vertices attached to the polygon must be the same. Color
    interpolation applies to the values for ambient (Ka), diffuse (Kd),
    specular (Ks), and specular highlight (Ns) material properties.

    on turns on color interpolation.

    off turns off color interpolation. The default is off.

d_interp on/off

    Polygonal geometry statement.

    Sets dissolve interpolation on or off.

    Dissolve interpolation creates an interpolation or blend across a
    polygon between the dissolve (d) values of the materials assigned
    to its vertices. This feature is used to create effects exhibiting
    varying degrees of apparent transparency, as in glass or clouds.

    To support dissolve interpolation, materials must be assigned per
    vertex, not per element. All the materials assigned to the vertices
    involved in the dissolve interpolation must contain a dissolve
    factor command to specify a dissolve.

    on turns on dissolve interpolation.

    off turns off dissolve interpolation. The default is off.

lod level

    Polygonal and free-form geometry statement.

    Sets the level of detail to be displayed in a PreView animation.
    The level of detail feature lets you control which elements of an
    object are displayed while working in PreView.

    level is the level of detail to be displayed. When you set the
    level of detail to 0 or omit the lod statement, all elements are
    displayed.  Specifying an integer between 1 and 100 sets the level
    of detail to be displayed when reading the .obj file.

maplib filename1 filename2 . . .

    This is a rendering identifier that specifies the map library file
    for the texture map definitions set with the usemap identifier. You
    can specify multiple filenames with maplib. If multiple filenames
    are specified, the first file listed is searched first for the map
    definition, the second file is searched next, and so on.

    When you assign a map library using the Model program, Model allows
    only one map library per .obj file. You can assign multiple
    libraries using a text editor.

    filename is the name of the library file where the texture maps are
    defined. There is no default.

usemap map_name/off

    This is a rendering identifier that specifies the texture map name
    for the element following it. To turn off texture mapping, specify
    off instead of the map name.

    If you specify texture mapping for a face without texture vertices,
    the texture map will be ignored.

    map_name is the name of the texture map.

    off turns off texture mapping. The default is off.

usemtl material_name

    Polygonal and free-form geometry statement.

    Specifies the material name for the element following it. Once a
    material is assigned, it cannot be turned off; it can only be
    changed.

    material_name is the name of the material. If a material name is
    not specified, a white material is used.

mtllib filename1 filename2 . . .

    Polygonal and free-form geometry statement.

    Specifies the material library file for the material definitions
    set with the usemtl statement. You can specify multiple filenames
    with mtllib. If multiple filenames are specified, the first file
    listed is searched first for the material definition, the second
    file is searched next, and so on.

    When you assign a material library using the Model program, only
    one map library per .obj file is allowed. You can assign multiple
    libraries using a text editor.

    filename is the name of the library file that defines the
    materials.  There is no default.

shadow_obj filename

    Polygonal and free-form geometry statement.

    Specifies the shadow object filename. This object is used to cast
    shadows for the current object. Shadows are only visible in a
    rendered image; they cannot be seen using hardware shading. The
    shadow object is invisible except for its shadow.

    An object will cast shadows only if it has a shadow object. You can
    use an object as its own shadow object. However, a simplified
    version of the original object is usually preferable for shadow
    objects, since shadow casting can greatly increase rendering time.

    filename is the filename for the shadow object. You can enter any
    valid object filename for the shadow object. The object file can be
    an .obj or .mod file. If a filename is given without an extension,
    an extension of .obj is assumed.

    Only one shadow object can be stored in a file. If more than one
    shadow object is specified, the last one specified will be used.

trace_obj filename

    Polygonal and free-form geometry statement.

    Specifies the ray tracing object filename. This object will be used
    in generating reflections of the current object on reflective
    surfaces.  Reflections are only visible in a rendered image; they
    cannot be seen using hardware shading.

    An object will appear in reflections only if it has a trace object.
    You can use an object as its own trace object. However, a
    simplified version of the original object is usually preferable for
    trace objects, since ray tracing can greatly increase rendering
    time.

    filename is the filename for the ray tracing object. You can enter
    any valid object filename for the trace object. You can enter any
    valid object filename for the shadow object. The object file can be
    an .obj or .mod file. If a filename is given without an extension,
    an extension of .obj is assumed.

    Only one trace object can be stored in a file. If more than one is
    specified, the last one is used.

ctech  technique  resolution

    Free-form geometry statement.

    Specifies a curve approximation technique. The arguments specify
    the technique and resolution for the curve.

    You must select from one of the following three techniques.

    ctech cparm res

	Specifies a curve with constant parametric subdivision using
	one resolution parameter. Each polynomial segment of the curve
	is subdivided n times in parameter space, where n is the
	resolution parameter multiplied by the degree of the curve.

	res is the resolution parameter. The larger the value, the
	finer the resolution. If res has a value of 0, each polynomial
	curve segment is represented by a single line segment.

    ctech cspace maxlength

	Specifies a curve with constant spatial subdivision. The curve
	is approximated by a series of line segments whose lengths in
	real space are less than or equal to the maxlength.

	maxlength is the maximum length of the line segments. The
	smaller the value, the finer the resolution.

    ctech curv maxdist maxangle

	Specifies curvature-dependent subdivision using separate
	resolution parameters for the maximum distance and the maximum
	angle.

	The curve is approximated by a series of line segments in which
	1) the distance in object space between a line segment and the
	actual curve must be less than the maxdist parameter and 2) the
	angle in degrees between tangent vectors at the ends of a line
	segment must be less than the maxangle parameter.

	maxdist is the distance in real space between a line segment
	and the actual curve.

	maxangle is the angle (in degrees) between tangent vectors at
	the ends of a line segment.

	The smaller the values for maxdist and maxangle, the finer the
	resolution.

    NOTE: Approximation information for trimming, hole, and special
    curves is stored in the corresponding surface. The ctech statement
    for the surface is used, not the ctech statement applied to the
    curv2 statement. Although untrimmed surfaces have no explicit
    trimming loop, a loop is constructed which bounds the legal
    parameter range. This implicit loop follows the same rules as any
    other loop and is approximated according to the ctech information
    for the surface.

stech  technique  resolution

    Free-form geometry statement.

    Specifies a surface approximation technique. The arguments specify
    the technique and resolution for the surface.

    You must select from one of the following techniques:

    stech cparma ures vres

	Specifies a surface with constant parametric subdivision using
	separate resolution parameters for the u and v directions. Each
	patch of the surface is subdivided n times in parameter space,
	where n is the resolution parameter multiplied by the degree of
	the surface.

	ures is the resolution parameter for the u direction.

	vres is the resolution parameter for the v direction.

	The larger the values for ures and vres, the finer the
	resolution.  If you enter a value of 0 for both ures and vres,
	each patch is approximated by two triangles.

    stech cparmb uvres

	Specifies a surface with constant parametric subdivision, with
	refinement using one resolution parameter for both the u and v
	directions.

	An initial triangulation is performed using only the points on
	the trimming curves. This triangulation is then refined until
	all edges are of an appropriate length. The resulting triangles
	are not oriented along isoparametric lines as they are in the
	cparma technique.

	uvres is the resolution parameter for both the u and v
	directions.  The larger the value, the finer the resolution.

    stech cspace maxlength

	Specifies a surface with constant spatial subdivision.

	The surface is subdivided in rectangular regions until the
	length in real space of any rectangle edge is less than the
	maxlength.  These rectangular regions are then triangulated.

	maxlength is the length in real space of any rectangle edge.
	The smaller the value, the finer the resolution.

    stech curv maxdist maxangle

	Specifies a surface with curvature-dependent subdivision using
	separate resolution parameters for the maximum distance and the
	maximum angle.

	The surface is subdivided in rectangular regions until 1) the
	distance in real space between the approximating rectangle and
	the actual surface is less than the maxdist (approximately) and
	2) the angle in degrees between surface normals at the corners
	of the rectangle is less than the maxangle. Following
	subdivision, the regions are triangulated.

	maxdist is the distance in real space between the approximating
	rectangle and the actual surface.

	maxangle is the angle in degrees between surface normals at the
	corners of the rectangle.

	The smaller the values for maxdist and maxangle, the finer the
	resolution.

Examples

1.      Cube with materials

This cube has a different material applied to each of its faces.

    mtllib master.mtl

    v 0.000000 2.000000 2.000000
    v 0.000000 0.000000 2.000000
    v 2.000000 0.000000 2.000000
    v 2.000000 2.000000 2.000000
    v 0.000000 2.000000 0.000000
    v 0.000000 0.000000 0.000000
    v 2.000000 0.000000 0.000000
    v 2.000000 2.000000 0.000000
    # 8 vertices

    g front
    usemtl red
    f 1 2 3 4
    g back
    usemtl blue
    f 8 7 6 5
    g right
    usemtl green
    f 4 3 7 8
    g top
    usemtl gold
    f 5 1 4 8
    g left
    usemtl orange
    f 5 6 2 1
    g bottom
    usemtl purple
    f 2 6 7 3
    # 6 elements

2.      Cube casting a shadow

In this example, the cube casts a shadow on the other objects when it
is rendered with Image. The cube, which is stored in the file cube.obj,
references itself as the shadow object.

    mtllib master.mtl
    shadow_obj cube.obj

    v 0.000000 2.000000 2.000000
    v 0.000000 0.000000 2.000000
    v 2.000000 0.000000 2.000000
    v 2.000000 2.000000 2.000000
    v 0.000000 2.000000 0.000000
    v 0.000000 0.000000 0.000000
    v 2.000000 0.000000 0.000000
    v 2.000000 2.000000 0.000000
    # 8 vertices

    g front
    usemtl red
    f 1 2 3 4
    g back
    usemtl blue
    f 8 7 6 5
    g right
    usemtl green
    f 4 3 7 8
    g top
    usemtl gold
    f 5 1 4 8
    g left
    usemtl orange
    f 5 6 2 1
    g bottom
    usemtl purple
    f 2 6 7 3
    # 6 elements

3.      Cube casting a reflection

This cube casts its reflection on any reflective objects when it is
rendered with Image. The cube, which is stored in the file cube.obj,
references itself as the trace object.

    mtllib master.mtl
    trace_obj cube.obj

    v 0.000000 2.000000 2.000000
    v 0.000000 0.000000 2.000000
    v 2.000000 0.000000 2.000000
    v 2.000000 2.000000 2.000000
    v 0.000000 2.000000 0.000000
    v 0.000000 0.000000 0.000000
    v 2.000000 0.000000 0.000000
    v 2.000000 2.000000 0.000000
    # 8 vertices

    g front
    usemtl red
    f 1 2 3 4
    g back
    usemtl blue
    f 8 7 6 5
    g right
    usemtl green
    f 4 3 7 8
    g top
    usemtl gold
    f 5 1 4 8
    g left
    usemtl orange
    f 5 6 2 1
    g bottom
    usemtl purple
    f 2 6 7 3
    # 6 elements

4.      Texture-mapped square

This example describes a 2 x 2 square. It is mapped with a 1 x 1 square
texture. The texture is stretched to fit the square exactly.

mtllib master.mtl

v 0.000000 2.000000 0.000000
v 0.000000 0.000000 0.000000
v 2.000000 0.000000 0.000000
v 2.000000 2.000000 0.000000
vt 0.000000 1.000000 0.000000
vt 0.000000 0.000000 0.000000
vt 1.000000 0.000000 0.000000
vt 1.000000 1.000000 0.000000
# 4 vertices

usemtl wood
f 1/1 2/2 3/3 4/4
# 1 element

5.      Approximation technique for a surface

This example shows a B-spline surface which will be approximated using
curvature-dependent subdivision specified by the stech command.

    g bspatch
    v -5.000000 -5.000000 -7.808327
    v -5.000000 -1.666667 -7.808327
    v -5.000000 1.666667 -7.808327
    v -5.000000 5.000000 -7.808327
    v -1.666667 -5.000000 -7.808327
    v -1.666667 -1.666667 11.977780
    v -1.666667 1.666667 11.977780
    v -1.666667 5.000000 -7.808327
    v 1.666667 -5.000000 -7.808327
    v 1.666667 -1.666667 11.977780
    v 1.666667 1.666667 11.977780
    v 1.666667 5.000000 -7.808327
    v 5.000000 -5.000000 -7.808327
    v 5.000000 -1.666667 -7.808327
    v 5.000000 1.666667 -7.808327
    v 5.000000 5.000000 -7.808327
    # 16 vertices

    g bspatch
    cstype bspline
    stech curv 0.5 10.000000
    deg 3 3
    surf 0.000000 1.000000 0.000000 1.000000 13 14 \ 15 16 9 10 11 12 5 6 7
    8 1 2 3 4
    parm u -3.000000 -2.000000 -1.000000 0.000000  \
    1.000000 2.000000 3.000000 4.000000
    parm v -3.000000 -2.000000 -1.000000 0.000000  \
    1.000000 2.000000 3.000000 4.000000
    end
    # 1 element

6.      Approximation technique for a curve

This example shows a Bezier curve which will be approximated using
constant parametric subdivision specified by the ctech command.

    v -2.300000 1.950000 0.000000
    v -2.200000 0.790000 0.000000
    v -2.340000 -1.510000 0.000000
    v -1.530000 -1.490000 0.000000
    v -0.720000 -1.470000 0.000000
    v -0.780000 0.230000 0.000000
    v 0.070000 0.250000 0.000000
    v 0.920000 0.270000 0.000000
    v 0.800000 -1.610000 0.000000
    v 1.620000 -1.590000 0.000000
    v 2.440000 -1.570000 0.000000
    v 2.690000 0.670000 0.000000
    v 2.900000 1.980000 0.000000
    # 13 vertices

    g default
    cstype bezier
    ctech cparm 1.000000
    deg 3
    curv 0.000000 4.000000 1 2 3 4 5 6 7 8 9 10 \
    11 12 13
    parm u 0.000000 1.000000 2.000000 3.000000  \
    4.000000
    end
    # 1 element

Comments

Comments can appear anywhere in an .obj file. They are used to annotate
the file; they are not processed.

Here is an example:

    # this is a comment

The Model program automatically inserts comments when it creates .obj
files. For example, it reports the number of geometric vertices,
texture vertices, and vertex normals in a file.

    # 4 vertices
    # 4 texture vertices
    # 4 normals

Mathematics for free-form curves/surfaces

[I apologize but this section will make absolutely no sense whatsoever
 without the equations and diagrams and there was just no easy way to
 include them in a pure ASCII document.  You should probably just skip
 ahead to the section "Superseded statements."  -Jim]

General forms

Rational and non-rational curves and surfaces

In general, any non-rational curve segment may be written as:

where

K + 1    is the number of control points
di       are the control points
n        is the degree of the curve
Ni,n(t)  are the degree n basis functions

Extending this to the bivariate case, any non-rational surface patch
may be written as:

where:

K1 + 1   is the number of control points in the u direction
K2 + 1   is the number of control points in the v direction
di,j     are the control points
m        is the degree of the surface in the u direction
n        is the degree of the surface in the v direction
Ni,m(u)  are the degree m basis functions in the u direction
Nj,n(v)  are the degree n basis functions in the v direction

NOTE: The front of the surface is defined as the side where the u
parameter increases to the right and the v parameter increases upward.

We may extend this curve to the rational case as:

where wi are the weights associated with the control points di.
Similarly, a rational surface may be expressed as:

where wi,j  are the weights associated with the control points di,j.

NOTE: If a curve or surface in an .obj file is rational, it must use
the rat option with the cstype statement and it requires some weight
values for each control point.

The weights for the rational form are given as a third control point
coordinate (for trimming curves) or fourth coordinate (for space curves
and surfaces). These weights are optional and default to 1.0 if not
given.

This default weight is only reasonable for curves and surfaces whose
basis functions sum to 1.0, such as Bezier, Cardinal, and NURB. It does
not make sense for Taylor and may or may not make sense for a
representation given in basis-matrix form.

For all forms other than B-spline, the final curve or surface is
constructed by piecing together the individual curve segments or
surface patches. A global parameter space is then defined over the
entire composite curve or surface using the parameter vector given with
the parm statement.

The parameter vector for a curve is a list of p global parameter values
{t1, . . . , tp}. If t1  t < ti+1 is a point in global parameter space,
then:

is the corresponding point in local parameter space for the ith
polynomial segment. It is this t which is used when evaluating a given
segment of the piecewise curve. For surfaces, this mapping from global
to local parameter space is applied independently in both the u and v
parametric directions.

B-splines require a knot vector rather than a parameter vector,
although this is also given with the parm statement. Refer to the
description of B-splines below.

The following discussion of each type is expressed in terms of the
above definitions.

NOTE: The maximum degree for all curve and surface types is currently
set at 20, which is high enough for most purposes.

Free-form curve and surface types

B-spline

Type bspline specifies arbitrary degree non-uniform B-splines which are
commonly referred to as NURBs in their rational form. The basis
functions are defined by the Cox-deBoor recursion formulas as:

and:

where, by convention, 0/0 = 0.

The xi  {x0, . . . ,xq} form a set known as the knot vector which is
given by the parm statement. It is required that

1.      xi  xi + 1,
2.      x0 < xn + 1,
3.      xq -n -1 < xq,
4.      xi < xi + n for 0 < i < q - n - 1,
5.      xn  t min < tmax  xK+ 1, where [tmin, tmax] is the parameter
over which the B-spline is to be evaluated, and
6.      K = q - n - 1.

A knot is said to be of multiplicity r if its value is repeated r times
in the knot vector. The second through fourth conditions above restrict
knots to be of at most multiplicity n + 1 at the ends of the vector and
at most n everywhere else.

The last condition requires that the number of control points is equal
to one less than the number of knots minus the degree. For surfaces,
all of the above conditions apply independently for the u and v
parametric directions.

Bezier

Type bezier specifies arbitrary degree Bezier curves and surfaces. This
basis function is defined as:

where:

When using type bezier, the number of global parameter values given
with the parm statement must be K/n + 1, where K is the number of
control points. For surfaces, this requirement applies independently
for the u and v parametric directions.

Cardinal

Type cardinal specifies a cubic, first derivative, continuous curve or
surface. For curves, this interpolates all but the first and last
control points. For surfaces, all but the first and last row and column
of control points are interpolated.

Cardinal splines, also known as Catmull-Rom splines, are best
understood by considering the conversion from Cardinal to Bezier
control points for a single curve segment:

Here, the ci variables are the Cardinal control points and the bi
variables are the Bezier control points. We see that the second and
third Cardinal points are the beginning and ending points for the
segment, respectively. Also, the beginning tangent lies along the
vector from the first to the third point, and the ending tangent along
the vector from the second to the last point.

If we let Bi(t) be the cubic Bezier basis functions (i.e. what was
given above for Bezier as Ni,n(t) with n = 3), then we may write the
Cardinal basis functions as:

Note that Cardinal splines are only defined for the cubic case.

When using type cardinal, the number of global parameter values given
with the parm statement must be K - n + 2, where K is the number of
control points. For surfaces, this requirement applies independently
for the u and v parametric directions.

Taylor

Type taylor specifies arbitrary degree Taylor polynomial curves and
surfaces. The basis function is simply:

NOTE: The control points in this case are the polynomial coefficients
and have no obvious geometric significance.

When using type taylor, the number of global parameter values given
with the parm statement must be (K + 1)/(n + 1) + 1, where K is the
number of control points. For surfaces, this requirement applies
independently for the u and v parametric directions.

Basis matrix

Type bmatrix specifies general, arbitrary-degree curves defined through
the use of a basis matrix rather than an explicit type such as Bezier.
The basis functions are defined as:

where the basis matrix is the bi,j. In order to make the matrix nature
of this more obvious, we may also write:

When constructing basis matrices, you should keep this definition in
mind, as different authors write this in different ways. A more common
matrix representation is:

To use such matrices in the .obj file, simply transpose the matrix and
reverse the column ordering.

When using type basis, the number of global parameter values given with
the parm statement must be (K - n)/s + 2, where K is the number of
control points and s is the step size given with the step statement.
For surfaces, this requirement applies independently for the u and v
parametric directions.

Surface vertex data

Control points

The control points for a surface consisting of a single patch are
listed in the order i = 0 to K1 for j = 0, followed by i = 0 to K1 for
j = 1, and so on until j = K2.

For surfaces made up of many patches, which is the usual case, the
control points are ordered as if the surface were a single large patch.
For example, the control points for a bicubic Bezier surface consisting
of four patches would be arranged as follows:

where (m, n) is the global parameter space of the surface and the
numbers indicate the ordering of the vertex indices in the surf
statement.

Texture vertices and texture mapping

When texture vertices are not supplied, the original surface
parameterization is used for texture mapping. However, if texture
vertices are supplied, they are interpreted as additional information
to be interpolated or approximated separately from, but using the same
interpolation functions as the control vertices.

That is, whereas the surface itself, in the non-rational case, was
given in the section "Rational and non-rational curves and surfaces"
as: the texture vertices are interpolated or approximated by:

where ti,j are the texture vertices and the basis functions are the
same as for S(u,v). It is T(u,v), rather than the surface
parameterization (u,v), which is used when a texture map is applied.

Vertex normals and normal mapping

Vertex normals are treated exactly like texture vertices. When vertex
normals are not supplied, the true surface normals are used. If vertex
normals are supplied, they are calculated as:

where qi,j are the vertex normals and the basis functions are the same
as for S(u,v) and T(u,v).

NOTE: Vertex normals do not affect the shape of the surface; they are
simply associated with the triangle vertices in the final
triangulation. As with faces, supplying vertex normals only affects
lighting calculations for the surface.

The treatment of both texture vertices and vertex normals in the case
of rational surfaces is identical. It is important to notice that even
when the surface S(u,v) is rational, the texture and normal surfaces,
T(u,v) and Q(u,v), are not rational. This is because the control points
(the texture vertices and vertex normals) are never rational.

Curve and surface operations

Special points

The following equations give a more precise description of special
points for space curves and discuss the extension to trimming curves
and surfaces.

Let C(t) be a space curve with the global parameter t. We can
approximate this curve by a set of k-1 line segments which connect the
points:

for some set of k global parameter values {t1,...,tk}

Given a special point ts in the parameter space of the curve
(referenced by vp), we guarantee that ts  {t1, . . . ,tk}. More
specifically, we approximate the curve by:

where, at the point i where ts is inserted, we have ti  ts < ti+1.

Special curves

The following equations give a more precise description of a special
curve.

Let T(t) be a special curve with the global parameter t. We have:

where (m,n) is a point in the global parameter space of a surface. We
can approximate this curve by a set of k-1 line segments which connect
the points:

for some set of k global parameter values.

Let S(m,n) be a surface with the global parameters m and n. We can
approximate this surface by a triangulation of a set of p points.

which lie on the surface. We further define E as the set of all edges
such that ei,j  E implies that S(mi,ni) and S(mj,nj) are connected in
the triangulation. Finally, we guarantee that there exists some subset
of E:

such that the points:

are connected in the triangulation.

Connectivity

Recall that the syntax of the con statement is:

con surf_1 q0_1 q1_1 curv2d_1 surf_2 q0_2 q1_2 curv2d_2

If we let:

T1(t1)  be the curve referenced by curv2d_1

S1(m1, n1)      be the surface referenced by surf1 on which T1(t1) lies

T2(t2)  be the curve referenced by curv2d_2

S2(m2, n2)      be the surface referenced by surf2 on which T2(t2) lies

then S1(T1(t1)), S2(T2(t2)) must be identical up to reparameterization.
Moreover, it must be the case that:

S1(T1(q0_1)) = S2(T2(q0_2))

and:

S1(T1(q1_1)) = S2(T2(q1_2))

It is along the curve S1(T1(t1)) between t1 = q0_1 and t1 = q1_1, and
the curve S2(T2(t2)) between t2 = q0_2 and t2 = q1_2 that the surface
S1(m1, n1) is connected to the surface S2(m2, n2).

Superseded statements

The new .obj file format has eliminated the need for several patch and
curve statements. These statements have been replaced by free-form
geometry statements.

In the 3.0 release, the following keywords have been superseded:

o       bsp
o       bzp
o       cdc
o       cdp
o       res

You can still read these statements in this version 3.0, however, the
system will no longer write files in this format.

This release is the last release that will read these statements. If
you want to save any data from this format, read in the file and write
it out. The system will convert the data to the new .obj format.

For more information on the new syntax statements, see "Specifying
free-form curves and surfaces."

Syntax

The following syntax statements are for the superseded keywords.

bsp v1 v2 . . . v16

    Specifies a B-spline patch. B-spline patches have sixteen control
    points, defined as vertices. Only four of the control points are
    distributed over the surface of the patch; the remainder are
    distributed around the perimeter of the patch.

    Patches must be tessellated in Model before they can be correctly
    shaded or rendered.

    v is the vertex number for a control point. Sixteen vertex numbers
    are required. Positive values indicate absolute vertex numbers.
    Negative values indicate relative vertex numbers.

bzp v1 v2 . . . v16

    Specifies a Bezier patch. Bezier patches have sixteen control
    points, defined as vertices. The control points are distributed
    uniformly over its surface.

    Patches must be tessellated in Model before they can be correctly
    shaded or rendered.

    v is the vertex number for a control point. Sixteen vertex numbers
    are required. Positive values indicate absolute vertex numbers.
    Negative values indicate relative vertex numbers.

cdc v1 v2 v3 v4 v5 . . .

    Specifies a Cardinal curve. Cardinal curves have a minimum of four
    control points, defined as vertices.

    Cardinal curves cannot be correctly shaded or rendered. They can be
    tessellated and then extruded in Model to create 3D shapes.

    v is the vertex number for a control point. A minimum of four
    vertex numbers are required. There is no limit on the maximum.
    Positive values indicate absolute vertex numbers. Negative values
    indicate relative vertex numbers.

cdp v1 v2 v3 . . . v16

    Specifies a Cardinal patch. Cardinal patches have sixteen control
    points, defined as vertices. Four of the control points are
    attached to the corners of the patch.

    Patches must be tessellated in Model before they can be correctly
    shaded or rendered.

    v is the vertex number for a control point. Sixteen vertex numbers
    are required. Positive values indicate absolute vertex numbers.
    Negative values indicate relative vertex numbers.

res useg vseg

    Reference and display statement.

    Sets the number of segments for Bezier, B-spline and Cardinal
    patches that follow it.

    useg is the number of segments in the u direction (horizontal or x
    direction). The minimum setting is 3 and the maximum setting is
    120.  The default is 4.

    vseg is the number of segments in the v direction (vertical or y
    direction). The minimum setting is 3 and the maximum setting is
    120.  The default is 4.

Comparison of 2.11 and 3.0 syntax

Cardinal curve

The following example shows the 2.11 syntax and the 3.0 syntax for the
same Cardinal curve.

2.11 Cardinal curve

    # 2.11 Cardinal Curve

    v 2.570000 1.280000 0.000000
    v 0.940000 1.340000 0.000000
    v -0.670000 0.820000 0.000000
    v -0.770000 -0.940000 0.000000
    v 1.030000 -1.350000 0.000000
    v 3.070000 -1.310000 0.000000
    # 6 vertices

    cdc 1 2 3 4 5 6

3.0 Cardinal curve

    # 3.0 Cardinal curve

    v 2.570000 1.280000 0.000000
    v 0.940000 1.340000 0.000000
    v -0.670000 0.820000 0.000000
    v -0.770000 -0.940000 0.000000
    v 1.030000 -1.350000 0.000000
    v 3.070000 -1.310000 0.000000
    # 6 vertices

    cstype cardinal
    deg 3
    curv 0.000000 3.000000 1 2 3 4 5 6
    parm u 0.000000 1.000000 2.000000 3.000000
    end
    # 1 element

Bezier patch

 The following example shows the 2.11 syntax and the 3.0 syntax for the
 same Bezier patch.

2.11 Bezier patch

    # 2.11 Bezier Patch
    v -5.000000 -5.000000 0.000000
    v -5.000000 -1.666667 0.000000
    v -5.000000 1.666667 0.000000
    v -5.000000 5.000000 0.000000
    v -1.666667 -5.000000 0.000000
    v -1.666667 -1.666667 0.000000
    v -1.666667 1.666667 0.000000
    v -1.666667 5.000000 0.000000
    v 1.666667 -5.000000 0.000000
    v 1.666667 -1.666667 0.000000
    v 1.666667 1.666667 0.000000
    v 1.666667 5.000000 0.000000
    v 5.000000 -5.000000 0.000000
    v 5.000000 -1.666667 0.000000
    v 5.000000 1.666667 0.000000
    v 5.000000 5.000000 0.000000
    # 16 vertices

    bzp 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
    # 1 element

3.0 Bezier patch

    #   3.0 Bezier patch

    v -5.000000 -5.000000 0.000000
    v -5.000000 -1.666667 0.000000
    v -5.000000 1.666667 0.000000
    v -5.000000 5.000000 0.000000
    v -1.666667 -5.000000 0.000000
    v -1.666667 -1.666667 0.000000
    v -1.666667 1.666667 0.000000
    v -1.666667 5.000000 0.000000
    v 1.666667 -5.000000 0.000000
    v 1.666667 -1.666667 0.000000
    v 1.666667 1.666667 0.000000
    v 1.666667 5.000000 0.000000
    v 5.000000 -5.000000 0.000000
    v 5.000000 -1.666667 0.000000
    v 5.000000 1.666667 0.000000
    v 5.000000 5.000000 0.000000
    # 16 vertices

    cstype bezier
    deg 3 3
    surf 0.000000 1.000000 0.000000 1.000000 13 14 \
    15 16 9 10 11 12 5 6 7 8 1 2 3 4
    parm u 0.000000 1.000000
    parm v 0.000000 1.000000
    end
    # 1 element