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**Polygon mesh**

A **polygon mesh** or unstructured grid is a collection of vertices, edges and faces that defines the shape of a polyhedral object in 3D computer graphics and solid modeling. The faces usually consist of triangles, quadrilaterals or other simple convex polygons, since this simplifies rendering, but may also be composed of more general concave polygons, or polygons with holes.

The study of polygon meshes is a large sub-field of computer graphics and geometric modeling. Different representations of polygon meshes are used for different applications and goals. The variety of operations performed on meshes may include Boolean logic, smoothing, simplification, and many others. Network representations, "streaming" and "progressive" meshes, are used to transmit polygon meshes over a network. Volumetric meshes are distinct from polygon meshes in that they explicitly represent both the surface and volume of a structure, while polygon meshes only explicitly represent the surface (the volume is implicit). As polygonal meshes are extensively used in computer graphics, algorithms also exist for ray tracing, collision detection, and rigid-body dynamics of polygon meshes.

## Elements of Mesh Modeling

Objects created with polygon meshes must store different types of elements. These include vertices, edges, faces, polygons and surfaces. In many applications, only vertices, edges and either faces or polygons are stored. A renderer may support only 3-sided faces, so polygons must be constructed of many of these, as shown in Figure 1. However, many renderers either support quads and higher-sided polygons, or are able to triangulate polygons to triangles on the fly, making it unnecessary to store a mesh in a triangulated form. Also, in certain applications like head modeling, it is desirable to be able to create both 3- and 4-sided polygons.

A **vertex** is a position along with other information such as color, normal vector and texture coordinates. An **edge** is a connection between two vertices. A **face** is a closed set of edges, in which a *triangle face* has three edges, and a *quad face* has four edges. A **polygon** is a set of faces. In systems that support multi-sided faces, polygons and faces are equivalent. However, most rendering hardware supports only 3- or 4-sided faces, so polygons are represented as multiple faces. Mathematically a polygonal mesh may be considered an unstructured grid, or undirected graph, with addition properties of geometry, shape and topology.

**Surfaces**, more often called **smoothing groups**, are useful, but not required to group smooth regions. Consider a cylinder with caps, such as a soda can. For smooth shading of the sides, all surface normals must point horizontally away from the center, while the normals of the caps must point in the +/-(0,0,1) directions. Rendered as a single, Phong-shaded surface, the crease vertices would have incorrect normals. Thus, some way of determining where to cease smoothing is needed to group smooth parts of a mesh, just as polygons group 3-sided faces. As an alternative to providing surfaces/smoothing groups, a mesh may contain other data for calculating the same data, such as a splitting angle (polygons with normals above this threshold are either automatically treated as separate smoothing groups or some technique such as splitting or chamfering is automatically applied to the edge between them). Additionally, very high resolution meshes are less subject to issues that would require smoothing groups, as their polygons are so small as to make the need irrelevant. Further, another alternative exists in the possibility of simply detaching the surfaces themselves from the rest of the mesh. Renderers do not attempt to smooth edges across noncontiguous polygons.

Mesh format may or may not define other useful data. **Groups** may be defined which define separate elements of the mesh and are useful for determining separate sub-objects for skeletal animation or separate actors for non-skeletal animation. Generally **materials** will be defined, allowing different portions of the mesh to use different shaders when rendered. Most mesh formats also suppose some form of UV coordinates**which are a separate 2d representation of the mesh "unfolded" to show what portion of a 2-dimensional texture map to apply to different polygons of the mesh.**

## Representations

Polygon meshes may be represented in a variety of ways, using different methods to store the vertex, edge and face data. These include:

- Face-vertex meshes: A simple list of vertices, and a set of polygons that point to the vertices it uses.
- Winged-edge meshes, in which each edge points to two vertices, two faces, and the four (clockwise and counterclockwise) edges that touch it. Winged-edge meshes allow constant time traversal of the surface, but with higher storage requirements.
- Half-edge meshes: Similar to winged-edge meshes except that only half the edge traversal information is used.
- Quad-edge meshes, which store edges, half-edges, and vertices without any reference to polygons. The polygons are implicit in the representation, and may be found by traversing the structure. Memory requirements are similar to half-edge meshes.
- Corner-tables, which store vertices in a predefined table, such that traversing the table implicitly defines polygons. This is in essence the "triangle fan" used in hardware graphics rendering. The representation is more compact, and more efficient to retrieve polygons, but operations to change polygons are slow. Furthermore, corner-tables do not represent meshes completely. Multiple corner-tables (triangle fans) are needed to represent most meshes.
- Vertex-vertex meshes: A "
`VV`" mesh represents only vertices, which point to other vertices. Both the edge and face information is implicit in the representation. However, the simplicity of the representation allows for many efficient operations to be performed on meshes.

Each of the representations above have particular advantages and drawbacks, further discussed in Smith (2006).

The choice of the data structure is governed by the application, the performance required, size of the data, and the operations to be performed. For example, it is easier to deal with triangles than general polygons, especially in computational geometry. For certain operations it is necessary to have a fast access to topological information such as edges or neighboring faces; this requires more complex structures such as the winged-edge representation. For hardware rendering, compact, simple structures are needed; thus the corner-table (triangle fan) is commonly incorporated into low-level rendering APIs such as DirectX and OpenGL.

**Equilateral polygon**

In geometry, an **equilateral polygon** is a polygon which has all sides of the same length.

For instance, an equilateral triangle is a triangle of equal edge lengths. All equilateral triangles are similar to each other, and have 60 degree internal angles.

An equilateral quadrilateral is a rhombus, which includes the square as a special case.

An equilateral polygon which is cyclic (its vertices are on a circle) is a regular polygon. All equilateral quadrilaterals are convex, but there exist equilateral polygons with five sides (pentagons) which are concave, and similarly for every larger number of sides.

Viviani's theorem generalizes to equilateral polygons.

Equilateral hexagons, also known as a **triambus**, appear in the three triambic icosahedra:

## symbol

**Algorithm examples**

*This article* 'Algorithm examples **supplements**Algorithm and Algorithm characterizations.

## An example: Algorithm specification of addition m+n

**Choice of machine model:**

There is no â€œbestâ€�, or â€œpreferredâ€� model. The Turing machine, while considered the standard, is notoriously awkward to use. And different problems seem to require different models to study them. Many researchers have observed these problems, for example:

- â€œThe principal purpose of this paper is to offer a theory which is closely related to Turing's but is more economical in the basic operationsâ€� (Wang (1954) p. 63)
- â€œCertain features of Turing machines have induced later workers to propose alternative devices as embodiments of what is to be meant by effective computability.... a Turing machine has a certain opacity, its workings are known rather than seen. Further a Turing machine is inflexible ... a Turing machine is slow in (hypothetical) operation and, usually complicated. This makes it rather hard to design it, and even harder to investigate such matters as time or storage optimization or a comparison between efficiency of two algorithms.â€� (Melzak (1961) p. 281)
- Shepherdson-Sturgis (1963) proposed their register-machine model because â€œthese proofs [using Turing machines] are complicated and tedious to follow for two reasons: (1) A Turing machine has only one head... (2) It has only one tape....â€� They were in search of â€œa form of idealized computer which is sufficiently flexible for one to be able to convert an intuitive computational procedure into a program for such a machineâ€� (p. 218).
- â€œI would prefer something along the lines of the random access computers of Angluin and Valiant [as opposed to the pointer machine of SchÃ¶nhage]â€� (Gurivich 1988 p. 6)
- â€œShowing that a function is Turing computable directly...is rather laborious ... we introduce an ostensibly more flexible kind of idealized machine, an abacus machine...â€� (Boolos-Burgess-Jeffrey 2002 p.45).

About all that one can insist upon is that the algorithm-writer specify in exacting detail (i) the machine model to be used and (ii) its instruction set.

**Atomization of the instruction set:**

The Turing machine model is primitive, but not as primitive as it can be. As noted in the above quotes this is a source of concern when studying complexity and equivalence of algorithms. Although the observations quoted below concern the Random access machine model â€“ a Turing-machine equivalent â€“ the problem remains for any Turing-equivalent model:

- â€œ...there hardly exists such a thing as an â€˜innocentâ€™ extension of the standard RAM model in the uniform time measure; either one only has additive arithmetic, or one might as well include all multiplicative and/or bitwise Boolean instructions on small operands....â€� (van Emde Boas (1992) p. 26)
- â€œSince, however, the computational power of a RAM model seems to depend rather sensitively on the scope of its instruction set, we nevertheless will have to go into detail...
- â€œOne important principle will be to admit only such instructions which can be said to be of an
*atomistic*nature. We will describe two versions of the so-called*successor*RAM, with the successor function as the only arithmetic operation....the RAM0 version deserves special attention for its extreme simplicity; its instruction set consists of only a few one letter codes, without any (explicit) addressing.â€� (SchÃ¶nhage (1980) p.494)

Example #1: The most general (and original) Turing machine â€“ single-tape with left-end, multi-symbols, 5-tuple instruction format â€“ can be atomized into the Turing machine of Boolos-Burgess-Jeffrey (2002) â€“ single-tape with no ends, two "symbols" { B, | } (where B symbolizes "blank square" and | symbolizes "marked square"), and a 4-tuple instruction format. This model in turn can be further atomized into a Post-Turing machineâ€“ single-tape with no ends, two symbols { B, | }, and a 0- and 1-parameter instruction set ( e.g. { Left, Right, Mark, Erase, Jump-if-marked to instruction xxx, Jump-if-blank to instruction xxx, Halt } ).

Example #2: The RASP can be reduced to a RAM by moving its instructions off the tape and (perhaps with translation) into its finite-state machine â€œtableâ€� of instructions, the RAM stripped of its indirect instruction and reduced to a 2- and 3-operand â€œabacusâ€� register machine; the abacus in turn can be reduced to the 1- and 2-operand Minsky (1967)/Shepherdson-Sturgis (1963) counter machine, which can be further atomized into the 0- and 1-operand instructions of SchÃ¶nhage (and even a 0-operand SchÃ¶nhage-like instruction set is possible).

**Cost of atomization**:

Atomization comes at a (usually severe) cost: while the resulting instructions may be â€œsimplerâ€�, atomization (usually) creates *more* instructions and the need for *more computational steps*. As shown in the following example the increase in computation steps may be significant (i.e. orders of magnitude â€“ the following example is â€œtameâ€�), and atomization may (but not always, as in the case of the Post-Turing model) reduce the usability and readability of â€œthe machine codeâ€�. For more see Turing tarpit.

Example: The single register machine instruction "INC 3" â€“ increment the contents of register #3, i.e. increase its count by 1 â€“ can be atomized into the 0-parameter instruction set of SchÃ¶nhage, but with the equivalent number of steps to accomplish the task increasing to 7; this number is directly related to the register number â€œnâ€� i.e. 4+n):

More examples can be found at the pages Register machine and Random access machine where the addition of "convenience instructions" CLR h and COPY h_{1},h_{1} are shown to reduce the number of steps dramatically. Indirect addressing is the other significant example.

## Precise specification of Turing-machine algorithm m+n

As described in Algorithm characterizations per the specifications of Boolos-Burgess-Jeffrey (2002) and Sipser (2006), and with a nod to the other characterizations we proceed to specify:

- (i)
**Number format**: unary strings of marked squares (a "marked square" signfied by the symbol 1) separated by single blanks (signified by the symbol B) e.g. â€œ2,3â€� = B11B111B - (ii)
**Machine type**: Turing machine: single-tape left-ended or no-ended, 2-symbol { B, 1 }, 4-tuple instruction format. - (iii)
**Head location**: See more at â€œImplementation Descriptionâ€� below. A symbolic representation of the head's location in the tape's symbol string will put the current state to the*right*of the scanned symbol. Blank squares may be included in this protocol. The state's number will appear with brackets around it, or sub-scripted. The head is shown as

From Encyclopedia

polygon closed plane figure bounded by straight line segments as sides. A polygon is convex if any two points inside the polygon can be connected by a line segment that does not intersect any side. If a side is intersected, the polygon is called concave. In a regular polygon the sides are of equal length and meet at equal angles; all other polygons are not regular, although either their sides or their angles may be equal, as in the cases of the rhombus and the rectangle. The simplest regular polygons are the equilateral triangle , the square , the regular pentagon (of 5 sides), and the regular hexagon (of 6 sides). Although the Greeks had developed methods of constructing these four polygons using only a straightedge and compass, they were unable to do the same for the regular heptagon (of 7 sides). In the 19th cent. C. F. Gauss showed that a regular heptagon was impossible to construct in this way. He proved that a regular polygon is constructible with a straightedge and compass only when the number of sides p is a prime number (see number theory ) of the form p â€‰=â€‰2 2n â€‰+â€‰1 or a product of such primes. The first five regular polygons with a prime number of sides that can be constructed using a straightedge and compass have 3, 5, 17, 257, and 65,537 sides.

From Yahoo Answers

Answers:5: Pentagram, pentomino, pentathalon, pentatuch. Look up each of those words - they all involve 5. 6: Actually, sex- is a far more common prefix than hex-, so you can think of it for now as a sexagon. you can then think of sextuplets - 6 babies. I can't think of any other good examples. 7: Heptathalon, SEPtember. But be careful! The Romans started their calendar in March. March was 1, April was 2, and so on...so SEPTember would be 7, OCTober would be 8, NO(N)vember would be9, and DECember would be 10. You may think of September as the 9th month, but the Romans were 2 months ahead! 8: October, as I said before. 10 - 2 = 8. 9: NOvember. 11 - 2 - 9. 10: DECember, 12 - 2 = 10. Decade, decathalon, DECimate. (Romans would kill 1 out of every 10 soldiers for some reason.) 11: Not a common shape, nor a common prefix. Can't help ya there. 12: No real good do-deca prefixes there. Just remember the 'do' means 2 more than the decagon.

Answers:The definition of a triangle is "a polygon with three sides and three angles.". that answers the question, my friend!

Answers:90: (x,y) becomes (-y, x) so A would go to (-2, -4) 180 (x,y) becomes (-x, -y) so A goes to (4, -2) 270 (x,y) becomes (y, -x) so A goes to (2, 4) and 360 it stays the same do the others like that

Answers:This comes from Latin: EQUI = equal LATERAL = side ANGULAR = angle equilateral = equal sides equiangular = equal angles If it is both, you could either call it "equilateral equiangular" or just something called "regular" which means both. Fun fact: for triangles, equilateral and equiangular are the same. If the three sides are the same, the three angles are the same, and vice-versa. This is only true for triangles. You want examples? Okay, how about a rhombus for equal sides but unequal angles? http://www.thesaurus.maths.org/mmkb/media/png/Rhombus.png And the other way, a rectangle? http://www.lil-fingers.com/coloring/images/rectangle.gif And the only one with both properties? A square: http://images.encarta.msn.com/xrefmedia/AEncMed%5CTargets%5CIllus%5CIFG%5C000f26be.gif

From Youtube

**CREATING REGULAR POLYGON DESIGNS :***The center of the circle is point O. *First you have to draw a diameter and label it, segment XY *Then construct its perpendicular bisector, label the points where it intersects the circle as A and Z. *Now, bisect the segment OY, label its midpoint as M. *Place the compass point on A and pencil on point M *Draw an arc that intersects the circle above Y, label the intersection as point B *Keep the compass equal to the distance AM. *Start al point B and mark 3 more congruent arcs on the circle. *Label the 3 point of intersection C, D and E in that order. *Finally draw the segments AB, BC, CD, DE and EA. ABCDE should be a regular pentagon. *The side of the regular pentagon should be equal to each other. *THE DIRECTION IN THE PROJECT PAPER HAS A MISTAKE BECAUSE THE PENTAGON COMES OUT WRONG BUT IT SHOULD LOOK A PENTAGON. *By using the regular pentagon construction as a basis, you can try to construct a regular decagon, too. *Also, regular polygon constructions can be the starting point for many attractive designs. *These are some of the examples that I have made.

**Triangles in Polygons :**Nicolet film showing that a rightangled triangle can be constructed by making use of the sides of three regular polygons (a pentagon, a hexagon and a decagon). Questions can be set, for example, asking whether these are the only polygons from which a right-angled triangle can be constructed. This might be followed by questions of approximation; eg, When two polygons are chosen, is it possible to find a third which produces a triangle approximating a right-angled triangle? It should be noted that the polygons are inscribed in the same given circle.