Geometry, Surfaces, Curves, Polyhedra

POV-Ray: A Tool for Creating Engaging Visualisation of Geometry

Various 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.

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.

Spherical 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)

Projection types
Classification of projections from 3D to 2D and specific examples of oblique projections.

A triangle was an improvement to the square wheel. It eliminated one bump. BC comics

Planar (stretching) distortion in the plane
Including Anamorphic projections and Mappings in the Complex Plane (Otherwise known as Conformal maps)

Polygonising a scalar field
Otherwise known as marching cubes and marching tetrahedrons.

HyperSpace (Historic)
A Macintosh 4 dimensional geometry viewer and manual.

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)	4D platonic solids	Polar Sphere
We should make things as simple as possible, but not simpler. Albert Einstein
Coordinates for 4D polyhedra	Polar + Star	Parallelohedron	Platonic solids	Time Star
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	Star Sphere	Build your own	80 Polyhedra	Cube personalities
If triangles had a God, He'd have three sides. Old Yiddish proverb
	Rhombic Triacontahedron	Polyhedra data files	Kissing number
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	3D SuperShape	Pseudosphere	Kuen's
Steinmetz solid	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
Steiner look-a-like	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.
Apple shape	The Blob	Klein Cycloid	Boy #2	Boy-Like
Pisod triaxial	Mobius strip	Heart #1	Heart #2	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
Dini's	Verrill	Pilz	Cassini Oval	Glob teardrop
Gerono lemniscate	Piriform	Superellipse	Cymbelloid	Prolatesphereoid
Bezier s	Jet	Splines	Triaxial teardrop	Whitney umbrella
Lemniscape	Tractrix	Pseudocatenoid	Twisted heart
Figure 8 Torus	Triaxial Tritorus	Torus/Supertoroid	Bohemian Dome	Double torus
Gumdrop Torus	Saddle Torus	Bow Tie	Triaxial Hexatorus
2-Torus	Limpet Torus	Elliptic Torus	Kinky torus
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	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		Kusner Schmitt	McMullen K3 model	Tri-Trumpet
Weird	Gerhard Miehlich	Kampyle of Eudoxus	Tetra Ellipse	Bretzel
Barth sextic	Cayley	Tooth	Wiffle cube	Horned Cube
Borg	OrthoCircle	Ribbed sphere	Sextic s	Calabi-Yau
Harmonograph	SuperShape	Truchet tiles	Inverse Truchet	Supercircle
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
Bezier curves	Piecewise cubic bezier	Spline curves	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
Klein Cycloid	Krishna Anklets	Mango Leaf	Bicorn curve	Glissette
Hypocycloid	Agnesi curve	Kappa curve	Piriform curve	Trisectrix of Maclaurin
Tractrix	Diamond curve	Folium curve	Baseball seam	SGI logo
Knots	Bow curve	Eight knot	Cinquefoil knot	Trefoil knot
Borromean rings	Mesh weave	Granny knot		Reuleaux Triangle