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)

Distance between a point, a line and a plane
The intersection of a line with another line (2D)
The closest line between two lines (3D)
The intersection of a line with a plane
Mathematics describing a plane
The intersection of two planes
The intersection of three planes
Polygon area and centroid calculation
Inside / outside polygon test
Reflection of a ray
Direction Cosines
Eulers number and closed surfaces
Determining whether a line segment intersects a facet
Coordinate transformations on the plane (2D)
Cartesian, Cylindrical, and Spherical
Euler angles and coordinate transformations
Converting between left and right coordinate systems

Clipping a line with a polygon
Clockwise test for polygons in 2D
Test for concave/convex polygon in 2D
Area of (planar) polygons in 3D
Spheres, equations and terminology
The intersection of a line and a sphere (or a circle)
Equation of the circle through 3 points
Equation of the sphere through 4 points
Intersecting area of circles on a plane
Rotation of a point about an arbitrary axis
Creating a plane/disk perpendicular to a line segment
Intersection of two circles on the plane
Circumference of an ellipse
Intersection of two spheres
Distributing Points on a Sphere
Quadric equations in x and y of degree 2
Fowler angles: Comparing angles without trigonometry

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 surface
Jet
Splines
Triaxial teardrop
Whitney umbrella
Lemniscape
Tractrix
Pseudocatenoid
Twisted heart
Piva surface
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
chainmail
Reuleaux Triangle

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360 video from the Endeavour
Random space filling tiling of the plane
Little planet images from the UWA campus
Update to the "Burning Ship" fractal

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