Geometry, Surfaces, Curves, Polyhedra

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)

Contouring Algorithm
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.

HyperSpace
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

Waterman Polyhedra
Cylinder intersections
Plexagons (Pleated hexagon)
Platonic solids
Verrill
We should make things as simple as possible, but not simpler. Albert Einstein
Time Star
Polar + Star sphere
Parallelohedron
SuperShape
80 Polyhedra
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
Superellipse
Build your own
Kuen's
Cube personalities
If triangles had a God, He'd have three sides. Old Yiddish proverb
Capsule
Rhombic Triacontahedron
Polyhedra data files
Kissing number
Implicit Surfaces
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
dForm
Spherical Harmonics
Inverse Truchet
Pseudosphere
Harmonograph
Borg
Bifolia
Twisted plane
Twisted Fano
Cross Hole
Klein Bottle
Fano planes
Tranguloid Trefoil
Chladni plates, 2D and 3D
Nose
Aesthetic delight lies somewhere between boredom and confusion. E. H. Gombrick
Witch hat
Slipper
Tangle
Strophoid
Chair
Horn
Cresent
Sea Shells
Decocube
Maeder's Owl
Truchet tiles
Cross Cap
Steiner
Hunt
Stiletto
Barth Decic
Twisted Triaxial
Mitre
Nodal cubic
Boy
A straight line may be the shortest distance between two points, but it is by no means the most interesting.
Klein Cycloid
Mobius strip
Heart surfaces
Pilz
Tear drop
"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
Prolatesphereoid
Jet
Spline curve/surface
Triaxial teardrop
Whitney umbrella
Lemniscape
Tractrix
Pseudocatenoid
Twisted heart
Piva surface
Calabi-Yau
Triaxial Tritorus
Torus & Supertoroid
Bohemian Dome
Double torus
Hexagonal Drum
OrthoCircle
Bow Tie
Triaxial Hexatorus
Rounded cube
Going to war over religion is basically killing each other to see who's got the better imaginary friend. Richard Jeni
Devil
Swallow
P1 atomic orbital
Ghost Plane
Bent Horns
Folium
Pretzel & Bretzel
Tubey
Catalan minimal
Henneburg minimal
Catenoid minimal
Helicoid minimal
Bour minimal
Ennepers minimal
Richmond minimal
Handkerchief
Scherk minimal
Kidney
Monkey saddle
Pillow shape
Cushion
Double Cone
Spring
Fish
Twisted pipe
There is a remote tribe that worships the number zero. Is nothing sacred? Les Dawson
Snail
The Blob
Kusner Schmitt
McMullen K3 model
Tri-Trumpet
Weird
Gerhard Miehlich
Kampyle of Eudoxus
Tetra Ellipse
Sextic s
Barth sextic
Cayley
Tooth
Wiffle cube
Horned Cube
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
Borromean rings
Mesh weave
Knots
Chainmail
Dini's
Bezier curves/surfaces
Apple shape
Gerono lemniscate
Cycloid
Conic sections
Butterfly curve
Chrysanthemum
Equiangular Spiral
Fermats Spiral
Hyperbolic Spiral
Lituus Spiral
Archimedes Spiral
Parabolic spiral
Sinusoidal Spiral
Square Archimedes Spiral
Cornu Spiral
Viviani
Tanh spiral
Coth spiral
Cayleys sextic
Helix
Freeths Nephroid
Cassini Oval
Spherical Nephroid
Epicycloid
Conchoid of Nicomedes
Cissoid of Diocles
Deltoid
Strophoid
Lemniscate Bernoulli
Astroid
Nephroid
Spherical Cardioid
Cardioid
Parabola
Hyperbola
Ellipse
2D Bow curve
Limacon
Snake Kolam
Reuleaux Triangle
Krishna Anklets
Mango Leaf
Bicorn curve
Glissette
Hypocycloid
Agnesi curve
Kappa curve
Bow curve
Trisectrix of Maclaurin
Tractrix
Diamond curve
Folium curve
Baseball seam
Blobbie
3D Chainmail
SGI logo

Other ...

Reflection of a ray
Direction Cosines
Rotation of a point about an arbitrary axis
Quadric equations in x and y of degree 2
Fowler angles: Comparing angles without trigonometry

New :   ECU Art Gallery, exhibition Oct 2014
Enhancing rock art recordings through hyperspectral photography
Mobius-Strip-Like Columnar Functional Connections are Revealed in ...
Repeatability of reconstructed models
Weld Range rock shelter: digital representations
Presenting :   Imaging and visualisation in heritage. Nov 2014
Workshop: Automated 3D model reconstruction from photographs. Dec 2014
Novel imaging of heritage objects and sites. Dec 2014
Novel physical representations for the visualisation of science data and mathematics. April 2015
Attending :   Siggraph Asia 2014 (3-6 Dec)
International Conference on Virtual Systems and MultiMedia (9-12 Dec)
8th Annual International Conference on Computer Games Multimedia & Allied Technologies. (13-14 April)

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