Notes on polygons and meshes
Includes Surface (polygon) simplification, Clipping a polygonal facet with an arbitrary plane, Surface Relaxation and Smoothing of polygonal data, Mesh crumpling, splitting polygons, two sided facets, polygon types, tests for clockwise and concavity, clipping line to polygons, area of a 3D polygon, area of general polygons, determining inside/outside test, intersection of a line and a facet, Eulers numbers.
Notes on points, lines and planes
Includes calculations for the distance between points, lines and planes. The intersection between 2 lines in 2D and 3D, the intersection of a line with a plane. The intersection of two and three planes.
The only thing that saves us from the bureaucracy is its inefficiency.
Notes on circles, cylinders and spheres
Includes equations and terminology. Equation of the circle through 3 points and sphere thought 4 points. The intersection of a line and a sphere (or a circle). Intersection of two circles on a plane and two spheres in 3D. Distributing Points on a Sphere. The area of multiple intersecting circles. Creating a plane/disk perpendicular to a line segment. Modelling with spheres and cylinders, including facet approximation to a sphere and cylinder, rounded boxes, pipes, and modelling with spheres.
Transformations and projections
Methods for mapping points on a spherical surface onto a plane, stereographic and cylindrical (including Mercator) projections. Includes Aitoff map projection: Conversion to/from longitude/latitude (spherical map). Transformations on the plane. Cartesian, Cylindrical, and Spherical coordinate systems. Euler angles and coordinate transformations. Converting between left and right coordinate systems. Classification of projections from 3D to 2D and specific examples of oblique projections. Planar (stretching) distortion in the plane. Anamorphic projections and Mappings in the Complex Plane (Otherwise known as Conformal maps). 3D projection: Transforming 3D world coordinates into 2D screen coordinates. Cubic to Cylindrical conversion and spherical projections. Convert spherical projection into a cylindrical projection.
Circumference of an ellipse
The circumference of an ellipse, one might think this was a "solved" problem, noting could be further from the truth.
Philosophy is written in this grand book - I mean universe - which stands continuously open to our gaze, but which cannot be understood unless one first learns to comprehend the language in which it is written. It is written in the language of mathematics, and its characters are triangles, circles and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth. Galileo (1623)
Description of an efficient contouring algorithm as it appeared in Byte magazine. (Byte Magazine, 1987) and a more general approach for arbitrary contour planes and polygonal meshes.
A triangle was an improvement to the square wheel. It eliminated one bump. BC comics
Polygonising a scalar field
Otherwise known as marching cubes and marching tetrahedrons.
Notes on 4 dimensional geometry, including an old Macintosh 4 dimensional geometry viewer and manual. List of 4D platonic solids and the coordinates for 4D polyhedra.
POV-Ray: A Tool for Creating Engaging Visualisation of Geometry
There are holes in the sky. Where the rain gets in. But they're ever so small. That's why the rain is thin. Spike Milligan
Plexagons (Pleated hexagon)||
We should make things as simple as possible, but not simpler. Albert Einstein
Polar + Star sphere||
To see a World in a Grain of Sand, And Heaven in a Wild Flower. Hold Infinity in the palm of your hand, And Eternity in an hour. William Blake
Photos by Gayla Chandler||
Build your own||
If triangles had a God, He'd have three sides. Old Yiddish proverb
Polyhedra data files||
Experience hath shewn, that even under the best forms of government those entrusted with power have, in time, and by slow operations, perverted it into tyranny. Thomas Jefferson
Chladni plates, 2D and 3D||
Aesthetic delight lies somewhere between boredom and confusion. E. H. Gombrick
A straight line may be the shortest distance between two points, but it is by no means the most interesting.
"I know what you're thinking about," said Tweedledum; "but it isn't so, nohow." "Contrariwise," continued Tweedledee, "if it was so, it might be, and if it were so, it would be, but as it isn't, it ain't. That's logic" Lewis Carroll
Torus & Supertoroid||
Going to war over religion is basically killing each other to see who's got the better imaginary friend. Richard Jeni
P1 atomic orbital||
Pretzel & Bretzel||
Catalan minimal ||
Henneburg minimal |
Catenoid minimal ||
Helicoid minimal ||
Bour minimal ||
Ennepers minimal ||
Richmond minimal |
Scherk minimal ||
There is a remote tribe that worships the number zero. Is nothing sacred? Les Dawson
McMullen K3 model||
Kampyle of Eudoxus||
I believe the geometric proportion served the creator as an idea when He introduced the continuous generation of similar objects from similar objects. Johannes Kepler
Square Archimedes Spiral|
Conchoid of Nicomedes||
Cissoid of Diocles||
2D Bow curve||
Trisectrix of Maclaurin|
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