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.
No amount of genius can overcome a preoccupation with detail. Law 8, Marion Levy Jr.
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.
Convert spherical projection into a cylindrical projection.
Tiling textures
An introduction to texture tiling using characteristics of the texture itself.
A general method is presented that converts any texture into one that tiles without seams.
Illustrates the most common texture mapping methods in use by rendering applications. The
mathematics of how to map a rectangular texture onto a sphere, creating a textured mesh in
OpenGL and how to correct for polar distortion of texture maps on spheres.
Texture library
I don't do drugs. If I want a rush then I get out of the chair when I'm not expecting it.
Dylan Moran
Tiling on the plane
Includes Truchet tiling in 2D and 3D, Regular pentagonal tiles,
block tessellation, weaving, and more.
Non periodic (aperiodic) tiling of the plane:
Methods of tiling that are never periodic, for example,
Penrose tiles, Danzer tiles, Chair tiles, Trilobite tiles, Pinwheel tiles.
Most of the tiles are presented accurately and large enough to be printed and cut out.
Hexagonal tiling and SHM Calculator
Including MacOSX and Linux software for experimenting with transformation in the Spiral Harmonic Mosaic.
Relationship between base 7 and base 10:
Exploration by Paolo Di Pasquale © 19882012
Tiling tricurves
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.
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.
POVRay: 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 shown, 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

CalabiYau

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

TriTrumpet

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

SGI logo

Reuleaux Triangle

Network

3D Chainmail

Bicorn curve

Glissette

Hypocycloid

Agnesi curve

Kappa curve

Bow curve

Trisectrix of Maclaurin

Tractrix

Diamond curve

Folium curve

Baseball seam

Blobbie

Sport Balls





Other ...
Solid angle of a pixel
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
Contents :
( Index )
( All pages )

Papers, Articles, Seminars, Workshops
Data Formats: 3D, Audio, Image
Geometry, Surfaces, Curves, Polyhedra
Fractals, Chaos, Self similarity

Domes, Planetariums, Fisheye, Spherical mirror
Stereographics, 3D projection
Miscellaneous: Projection, Modelling, Rendering
Fun, Puzzles, Travel

Search :


New/Updated :

What does it look like being on a mountain top engulfed by fire? 
Panasonic Lumix GH5 and Sigma 4.5mm fisheye 
Pilbara rock art: laser scanning, photogrammetry and 3D photographic reconstruction as heritage management tools 
Spatial errors in stereoscopic displays 
Visualising volumetric fractals 
Converting dual fisheye images into a spherical (equirectangular) projection

Offsite :

Sketchfab 
Vimeo 
Fotomerchant 
Shapeways 
Zazzle 
YouTube 
FaceBook 
360Cities
ResearchGate 
Orchid 
Loop 
Dark the movie

Copyright :

The contents of this web site are © Copyright Paul Bourke or
a third party contributor where indicated.
You may print or save an electronic copy of parts of this web site for your own personal use.
Permission must be sought for any other use.
Any source code found here may be freely used provided credits are given to the author.
Purchase of credit free licenses of material found on this site can be negotiated
with the author. The author can also quote for unique variations and/or higher
resolution versions of the images found on this site.

