Encyclopedia - Convex Polygon: Encyclopedia Ii - Convex Polygon - Concave Polygons
If a simple polygon is not convex, it is called concave. At least one internal angle of a concave polygon is larger than 180 degrees. A c...   » Read the article

Encyclopedia - Polygon: Encyclopedia Ii - Polygon - Taxonomic Classification
The taxonomic classification of polygons is illustrated by the following graph: Polygon ...   » Read the article

Encyclopedia - Fundamental Polygon: Encyclopedia Ii - Fundamental Polygon - Standard Fundamental Polygons
An orientable closed surface of genus n has the following standard fundamental polygon: A non-orientable closed surface of (non-orient...   » Read the article

Encyclopedia - Diagonal: Encyclopedia Ii - Diagonal - Polygons
As applied to a polygon, a diagonal is a line segment joining two vertices that are not adjacent. Therefore a quadrilateral has two diago...   » Read the article

Encyclopedia - Constructible Polygon: Encyclopedia - Constructible Polygon
In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regula...   » Read the article

Encyclopedia - Circumcircle: Encyclopedia - Circumcircle
In geometry, the circumcircle of a given two-dimensional geometric shape is a circle which contains the shape completely within it. For ...   » Read the article

Encyclopedia - Surface: Encyclopedia - Surface
In mathematics (topology), a surface is a two-dimensional manifold. Examples arise in three-dimensional space as the boundaries of three-...   » Read the article

Encyclopedia - 3d Studio Max: Encyclopedia - 3d Studio Max
3D Studio Max (sometimes called 3ds Max or just MAX) is a 3D computer graphics and animation program, written by Autodesk Media & En...   » Read the article

Encyclopedia - Zonohedron: Encyclopedia - Zonohedron
A zonohedron is a convex polyhedron where every face is a polygon with point symmetry or, equivalently, symmetry under rotations through ...   » Read the article

Encyclopedia - Chiliagon: Encyclopedia - Chiliagon
A chiliagon (pronounced /ˈkɪli.əˌgɑn/) is a polygon with 1000 sides. The measure of each angle in a regular chiliagon is 179.64°. S...   » Read the article

Encyclopedia - Constructibility: Encyclopedia - Constructibility
In mathematics, there are several notions of constructibility: a point in the Euclidean plane that can be constructed with unruled strai...   » Read the article

Encyclopedia - Concavity: Encyclopedia - Concavity
The word concave means curving in or hollowed inward. The term may rever to Concave lens — a lens with inward-curving (concave) surfac...   » Read the article

Encyclopedia - Archimedean Solid: Encyclopedia - Archimedean Solid
In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polyg...   » Read the article

Encyclopedia - Heptagram: Encyclopedia - Heptagram
A heptagram or septagram is a seven-pointed star drawn with seven straight strokes. There are two kinds of heptagram: Acute heptagram, t...   » Read the article

Encyclopedia - Wings 3d: Encyclopedia - Wings 3d
Wings 3D is a free and open source polygon mesh modeler inspired by Nendo and Mirai from Izware. Wings 3D is available for most platforms...   » Read the article

Encyclopedia - Cell Mathematics: Encyclopedia - Cell Mathematics
A cell is a three-dimensional object that is part of a higher-dimensional object, such as a polychoron. A cell is related to higher-dimen...   » Read the article

Encyclopedia - Containment Hierarchy: Encyclopedia - Containment Hierarchy
A containment hierarchy is a hierarchical collection of strictly nested sets. Each entry in the hierarchy designates a set such that the ...   » Read the article

Encyclopedia - Polygon: Encyclopedia Ii - Polygon - Taxonomic Classification
The taxonomic classification of polygons is illustrated by the following tree: Polygon ...   » Read the article

Encyclopedia - Polygon: Encyclopedia Ii - Polygon - Properties
We will assume Euclidean geometry throughout. An n-gon has 2n degrees of freedom, including 2 for position and 1 for rotational orientati...   » Read the article

Encyclopedia - Polygon: Encyclopedia Ii - Polygon - Special Cases
Some special cases are: Angle of 0° or 180° (degenerate case) Two non-adjacent sides are on the same line Equilateral polygon: a polyg...   » Read the article

Encyclopedia - Polygon: Encyclopedia Ii - Polygon - Special Cases
Some special cases are: angle of 0° or 180° (degenerate case) two non-adjacent sides are on the same line equilateral polygon: a polyg...   » Read the article

Encyclopedia - Polygon Triangulation: Encyclopedia Ii - Polygon Triangulation - Using Monotone Polygons
A monotone polygon is one with a boundary that consists of two parts, each of which consists of points that have incrementing coordinates...   » Read the article

Encyclopedia - Polygon: Encyclopedia Ii - Polygon - Names And Types
Polygons are named according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon, dode...   » Read the article

Encyclopedia - Polygon Triangulation: Encyclopedia Ii - Polygon Triangulation - Substracting Ears Method
One way to triangulate a simple polygon is by using the assertion that any simple polygon without holes has at least two so called 'ears'...   » Read the article

Encyclopedia - Constructible Polygon: Encyclopedia Ii - Constructible Polygon - Conditions For Constructibility
Some regular polygons are easy to construct with compass and straightedge; others are not. This led to the question being posed: is it po...   » Read the article

Encyclopedia - Constructible Polygon: Encyclopedia Ii - Constructible Polygon - General Theory
In the light of later work on Galois theory, the principles of these proofs have been clarified. It is straightforward to show from analy...   » Read the article

Encyclopedia - Point In Polygon: Encyclopedia Ii - Point In Polygon - Ray Casting Algorithms
One simple way of finding whether the point is inside or outside a simple polygon is to test how many times a ray starting from the point...   » Read the article

Encyclopedia - Constructible Polygon: Encyclopedia Ii - Constructible Polygon - Compass-and-straightedge Constructions
Compass-and-straightedge constructions are known for all constructible polygons. If n = p·q with p = 2 or p and q co...   » Read the article

Encyclopedia - Tiling By Regular Polygons: Encyclopedia Ii - Tiling By Regular Polygons - Combinations Of Regular Polygons That Can Meet At A Vertex
The internal angles of the polygons meeting at a vertex must add to 360 degrees. A regular n-gon has internal angle degrees. There are s...   » Read the article

Encyclopedia - Constructible Polygon: Encyclopedia Ii - Constructible Polygon - Detailed Results In Terms Of Fermat Primes
Only five Fermat primes are known: F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537 (sequence A019434 in OEIS). Thus an n-gon is const...   » Read the article

Encyclopedia - Tiling By Regular Polygons: Encyclopedia Ii - Tiling By Regular Polygons - Beyond The Plane
These tessellations are also related to regular and semiregular polyhedra and tessellations of the hyperbolic plane. Semiregular polyhedr...   » Read the article

Encyclopedia - Tiling By Regular Polygons: Encyclopedia Ii - Tiling By Regular Polygons - Other Edge-to-edge Tilings
Any number of non-uniform (sometimes called demiregular) edge-to-edge tilings by regular polygons may be drawn. Here are four examples: S...   » Read the article

Encyclopedia - Surface: Encyclopedia Ii - Surface - Fundamental Polygon
Each closed surface can be constructed from an even sided oriented polygon, called a fundamental polygon by pairwise identification of it...   » Read the article

Encyclopedia - Tiling By Regular Polygons: Encyclopedia Ii - Tiling By Regular Polygons - Archimedean Uniform Or Semiregular Tilings
Vertex-transitivity means that for every pair of vertices there is a symmetry operation mapping the first vertex to the second. If the re...   » Read the article

Encyclopedia - Schläfli Symbol: Encyclopedia Ii - Schläfli Symbol - Regular Polygons Plane
The Schläfli symbol of a polygon with n edges is {n}. For example {5} is a pentagon. There are nonconvex polygons which are considered r...   » Read the article

Encyclopedia - Pyramid Geometry: Encyclopedia Ii - Pyramid Geometry - Pyramids With Regular Polygon Faces
If all faces are regular polygons, the pyramid base can be a regular polygon of 3, 4 or 5 sided: The geometric center of a square-based p...   » Read the article

Encyclopedia - Fermat Number: Encyclopedia Ii - Fermat Number - Relationship To Constructible Polygons
An n-sided regular polygon can be constructed with ruler and compass if and only if n is a power of 2 or the product of a power of 2 and ...   » Read the article

Encyclopedia - Gunpowder Warfare: Encyclopedia Ii - Gunpowder Warfare - Beginning Of Polygonal Fortifications
A polygonal fort is a fortification in the style that evolved around the middle of the nineteenth century, in response to the development...   » Read the article

Encyclopedia - Ruler-and-compass Construction: Encyclopedia Ii - Ruler-and-compass Construction - Constructing Regular Polygons
Some regular polygons (e.g. a pentagon) are easy to construct with ruler and compass; others are not. This led to the question: Is it pos...   » Read the article

Encyclopedia - Tessellation: Encyclopedia Ii - Tessellation - Number Of Sides Of A Polygon Versus Number Of Sides At A Vertex
For an infinite tiling, let a be the average number of sides of a polygon, and b the average number of sides meeting at a vertex. Then ( ...   » Read the article

Encyclopedia - Diagonal: Encyclopedia Ii - Diagonal - Geometry
By analogy, the subset of the Cartesian product X×X of any set X with itself, consisting of all pairs (x,x), is called the diagonal. It ...   » Read the article

Encyclopedia - Tessellation: Encyclopedia Ii - Tessellation - Regular And Irregular Tessellations
A regular tessellation is a highly symmetric tessellation made up of congruent regular polygons. Only three regular tessellations exist: ...   » Read the article

Encyclopedia - Schläfli Symbol: Encyclopedia Ii - Schläfli Symbol - Regular Polychora 4-space
The Schläfli symbol of a regular polychoron is of the form {p,q,r}. It has {p} regular polygonal faces, {p,q} cells, {q,r} regular polyh...   » Read the article

Encyclopedia - Circumcircle: Encyclopedia Ii - Circumcircle - Circumcircles Of Triangles
The circumcircle of a triangle is the unique circle on which all its three vertices lie. (This is not the same as the first definition fo...   » Read the article

Encyclopedia - Regular Polytope: Encyclopedia Ii - Regular Polytope - Constructions
Regular polytope - Polygons. The traditional way to "construct" a regular polygon, or indeed any other figure on the plane, is by use o...   » Read the article

Encyclopedia - Tessellation: Encyclopedia Ii - Tessellation - Tessellations And Color
When discussing a tiling that is displayed in colors, to avoid ambiguity one needs to specify whether the colors are part of the tiling o...   » Read the article

Encyclopedia - Exact Trigonometric Constants: Encyclopedia Ii - Exact Trigonometric Constants - Table Of Constants
Values outside 0° ... 45° angle range are trivially extracted from circle axis reflection symmetry from these values. (See Trigonometri...   » Read the article

Encyclopedia - Exact Trigonometric Constants: Encyclopedia Ii - Exact Trigonometric Constants - Notes
Exact trigonometric constants - Uses for constants. As an example of the use of these constants, consider a dodecahedron with the follo...   » Read the article

Encyclopedia - Regular Polytope: Encyclopedia Ii - Regular Polytope - Polytopes In Nature
Regular polytope - Polygons. Numerous regular polygons may be seen in nature. In the world of minerals, crystals often have faces which...   » Read the article

Encyclopedia - Exact Trigonometric Constants: Encyclopedia Ii - Exact Trigonometric Constants - Plans For Simplifying
Exact trigonometric constants - Rationalize the denominator. If the denominator is a square root, multiply the numerator and denominat...   » Read the article

Encyclopedia - Schläfli Symbol: Encyclopedia Ii - Schläfli Symbol - Regular Polyhedra 3-space
The Schläfli symbol of a polyhedron is {p,q} if its faces are p-gons, and each vertex is surrounded by q faces. The Schläfli symbols of...   » Read the article

Encyclopedia - Computer Graphics: Encyclopedia Ii - Computer Graphics - Texturing
Polygon surfaces (the sequence of faces) can contain data corresponding to not only a color, but in more advanced software, can be a virt...   » Read the article

Encyclopedia - Computer Graphics: Encyclopedia Ii - Computer Graphics - Computer Graphics 3d
With the birth of the workstation computers (like LISP machines, paintbox computers and Silicon Graphics workstations) came the 3D comput...   » Read the article

Encyclopedia - Computer Graphics: Encyclopedia Ii - Computer Graphics - Computer Graphics, 3d
With the birth of the workstation computers (like LISP machines, paintbox computers and Silicon Graphics workstations) came the 3D comput...   » Read the article

Encyclopedia - Surface: Encyclopedia Ii - Surface - Some Models
To make some models of various surfaces, attach the sides of these squares (A with A, B with B) so that the directions of the arrows matc...   » Read the article

Encyclopedia - Computer Graphics: Encyclopedia Ii - Computer Graphics - Computer Graphics 3d
With the birth of the workstation computers (like LISP machines, paintbox computers and Silicon Graphics workstations) came the 3D comput...   » Read the article

Encyclopedia - 3d Studio Max: Encyclopedia Ii - 3d Studio Max - Modeling Methods
There are 5 basic modeling methods: Modeling with primitives NURMS (subdivision surfaces) Surface tool/Editable patch object NURBS Polyg...   » Read the article

Encyclopedia - 3d Studio Max: Encyclopedia Ii - 3d Studio Max - Modeling Methods
There are 5 basic modeling methods: Modeling with primitives NURMS (subdivision surfaces) Surface tool/Editable patch object NURBS Polyg...   » Read the article

Encyclopedia - Computer Graphics: Encyclopedia Ii - Computer Graphics - Texturing
Polygon surfaces (the sequence of faces) can contain data corresponding to not only a color, but in more advanced software, can be a virt...   » Read the article

Encyclopedia - Surface: Encyclopedia Ii - Surface - Definition
In what follows, all surfaces are considered to be second-countable 2-dimensional manifolds. More precisely: a topological surface (with ...   » Read the article

Encyclopedia - Surface: Encyclopedia Ii - Surface - Classification Of Closed Surfaces
There is a complete classification of closed (i.e compact without boundary) connected, surfaces up to homeomorphism. Any such surface fal...   » Read the article

Encyclopedia - Surface: Encyclopedia Ii - Surface - Connected Sum Of Surfaces
Given two surfaces M and M', their connected sum M # M' is obtained by removing a disk in each of them and gluing them along the newly fo...   » Read the article

Encyclopedia - Gunpowder Warfare: Encyclopedia Ii - Gunpowder Warfare - Nature Of War
For the most part the wars were not particularly deadly by later standards. Armies were slow moving in an era before good roads and canal...   » Read the article

Encyclopedia - Gunpowder Warfare: Encyclopedia Ii - Gunpowder Warfare - Cannons
Gunpowder weapons had been used in China centuries before cannon appeared in Europe. They appeared in Europe in the late Middle Ages; how...   » Read the article

Encyclopedia - Gunpowder Warfare: Encyclopedia Ii - Gunpowder Warfare - Outside Of Europe
The gunpowder era of warfare is largely confined to Europe. This was also the time of the beginning of European exploration and colonial ...   » Read the article

Encyclopedia - Ruler-and-compass Construction: Encyclopedia Ii - Ruler-and-compass Construction - Ruler And Compass
The "ruler" and "compass" of ruler-and-compass constructions is an idealization of rulers and compasses in the real world: The ruler is ...   » Read the article

Encyclopedia - Ruler-and-compass Construction: Encyclopedia Ii - Ruler-and-compass Construction - Constructible Points And Lengths
There are many different ways to prove something is impossible. In this particular problem we carefully demarcate the limit of the possib...   » Read the article

Encyclopedia - Ruler-and-compass Construction: Encyclopedia Ii - Ruler-and-compass Construction - Constructible Points And Lengths
How do you prove something impossible? There are many different ways, but this particular problem we carefully demarcate the limit of the...   » Read the article

Encyclopedia - Ruler-and-compass Construction: Encyclopedia Ii - Ruler-and-compass Construction - Impossible Constructions
Ruler-and-compass construction - Squaring the circle. The most famous of these problems, "squaring the circle", involves constructing a...   » Read the article

Encyclopedia - Gunpowder Warfare: Encyclopedia Ii - Gunpowder Warfare - Firearms
The power of aristocracies fell throughout Western Europe during this period. Their ancestral castles were no longer useful defences. The...   » Read the article

Encyclopedia - Fermat Number: Encyclopedia Ii - Fermat Number - Fermat's Little Theorem And Pseudoprimes
Fermat's little theorem ...Using Fermat numbers to generate infinitely many pseudoprimes... Fermat number - Other theorems about Fermat'...   » Read the article

Encyclopedia - 3d Studio Max: Encyclopedia Ii - 3d Studio Max - Overview
3ds Max is one of the most widely-used 3D animation software. Its has strong editing capabilities, an ubiquitous plugin architecture and ...   » Read the article

Encyclopedia - Fermat Number: Encyclopedia Ii - Fermat Number - Basic Properties
The Fermat numbers satisfy the following recurrence relations Fn = (Fn − 1 − 1)2 + 1 for n ≥ 2. Each of these relations can be...   » Read the article

Encyclopedia - 3d Studio Max: Encyclopedia Ii - 3d Studio Max - Overview
3ds Max is one of the most widely-used 3D animation software. It has strong editing capabilities, a ubiquitous plugin architecture and a ...   » Read the article

Encyclopedia - Fermat Number: Encyclopedia Ii - Fermat Number - Primality Of Fermat Numbers
Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who conjectured that all Fermat numbers are prime. Indeed, the f...   » Read the article

Encyclopedia - Fuchsian Model: Encyclopedia Ii - Fuchsian Model - A More Precise Definition
To be more precise, every Riemann surface has a universal covering map that is either the Riemann sphere, the complex plane or the upper ...   » Read the article

Encyclopedia - Watershed: Encyclopedia Ii - Watershed - Analyzing Watersheds
Rain gauge data is used to measure total precipitation over a watersheds, and there are different ways to interpret that data. If the gau...   » Read the article

Encyclopedia - Fuchsian Model: Encyclopedia Ii - Fuchsian Model - Nielsen Isomorphism Theorem
The Nielsen isomorphism theorem basically states that the algebraic topology of a closed Riemann surface is the same as its geometry. Mor...   » Read the article

Encyclopedia - Schwarz-christoffel Mapping: Encyclopedia Ii - Schwarz-christoffel Mapping - Definition
Consider a polygon in the complex plane. The Riemann mapping theorem implies that there is a bijective holomorphic mapping f from the upp...   » Read the article

Encyclopedia - Ramification: Encyclopedia Ii - Ramification - In Complex Analysis
In complex analysis, the basic model can be taken as the z → zn mapping in the complex plane, near z = 0. This is the standard local ...   » Read the article

Encyclopedia - Fundamental Domain: Encyclopedia Ii - Fundamental Domain - Example
The existence and description of a fundamental domain is in general something requiring painstaking work to establish. The diagram to the...   » Read the article

Encyclopedia - Ramification: Encyclopedia Ii - Ramification - In Algebraic Topology
In a covering map the Euler-Poincaré characteristic should multiply by the number of sheets; ramification can therefore be detected by s...   » Read the article

Encyclopedia - Ramification: Encyclopedia Ii - Ramification - In Algebraic Number Theory
Ramification in algebraic number theory means prime numbers factorising into some repeated prime ideal factors. Let R be the ring of inte...   » Read the article

Encyclopedia - Polytope: Encyclopedia Ii - Polytope - Simplicial Decomposition
Now given any convex hull in r-dimensional space (but not in any (r-1)-plane, say) we can take linearly independent subsets of the vertic...   » Read the article

Encyclopedia - Rasterisation: Encyclopedia Ii - Rasterisation - Acceleration Techniques
To extract the maximum performance out of any rasterization engine, a minimum number of polygons should be sent to the renderer. A number...   » Read the article

Encyclopedia - Area Geometry: Encyclopedia Ii - Area Geometry - How To Define Area
Although area seems to be one of the basic notions in geometry, it is not at all easy to define even in the Euclidean plane. Most textboo...   » Read the article

Encyclopedia - Polytope: Encyclopedia Ii - Polytope - Convex Polytopes
One special kind of polytope is a convex polytope, which is the convex hull of a finite set of points. Convex polytopes can also be repre...   » Read the article

Encyclopedia - Clipping Computer Graphics: Encyclopedia Ii - Clipping Computer Graphics - 3d Clipping
Clipping is non-trivial, especially for 3D animations: if the objects are built up of, say, polygons, a routine is needed that determines...   » Read the article

Encyclopedia - List Of Regular Polytopes: Encyclopedia Ii - List Of Regular Polytopes - Two-dimensional Regular Polytopes
The two dimensional polytopes are called polygons. Regular polygons are equilateral and cyclic. Usually a regular polygon is considered c...   » Read the article

Encyclopedia - System Shock 2: Encyclopedia Ii - System Shock 2 - Fan Modifications
A free fan-made add-on pack, called "Rebirth," is available on the Internet, which increases some of the polygon counts and provides new ...   » Read the article

Encyclopedia - Wild Arms 4: Encyclopedia Ii - Wild Arms 4 - Combat
Battles this time around are done in a polygonal style fashion. Where your party members and enemies stand on several polygon platforms. ...   » Read the article

Encyclopedia - Exact Trigonometric Constants: Encyclopedia Ii - Exact Trigonometric Constants - How Can The Trig Values For Sin And Cos Be Calculated?
Exact trigonometric constants - The trivial ones. In degree format: 0, 90, 45, 30 and 60 can be calculated from their triangles, using...   » Read the article

Encyclopedia - Line Mathematics: Encyclopedia Ii - Line Mathematics - Line Segment
In mathematics, a line segment is a part of a line that is bounded by two end points. See also interval (mathematics). When the end point...   » Read the article

Encyclopedia - Surface Normal: Encyclopedia Ii - Surface Normal - Calculating A Surface Normal
For a polygon (such as a triangle), a surface normal can be calculated as the vector cross product of two edges of the polygon. For a pla...   » Read the article

Encyclopedia - Uniform Polyhedron: Encyclopedia Ii - Uniform Polyhedron - Mathematics
Euclid was the first to show that for a convex polyhedron the vertex angles of the polygons at each vertex must add up to less than 360°...   » Read the article

Encyclopedia - Powervr: Encyclopedia Ii - Powervr - The Technology
The PowerVR chipset uses a unique approach to rendering a 3D scene, known as Tile Based Deferred Rendering (often abbreviated as TBDR). A...   » Read the article

Encyclopedia - Routing Eda: Encyclopedia Ii - Routing Eda - Types Of Routers
The task of all routers is the same. They are given some pre-existing polygons consisting of pins (also called terminals) on cells, and (...   » Read the article

Encyclopedia - Clipping Computer Graphics: Encyclopedia Ii - Clipping Computer Graphics - Video Games
Good clipping strategy is important in the development of video games in order to maximize the game's frame rate and visual quality. Desp...   » Read the article

Dictionary - Mandala: Wiccan Pagan Dictionary On Mandala
MANDALA - intricate pattern of concentric circles, squares, polygons and other geometric or artistic symbols representing the Circles ...   » Read the article

Dictionary - Kara: Spiritual - Theosophy Dictionary On Kara
Kara(m) (Sanskrit) (from the verbal root kri to do; cf Latin cerus a former)   A maker, causer; hence a hand. Also the side of a ...   » Read the article