3D - The Basics: Geometric Theory

                                                  Geometric Theory

Geometry

The branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogues.

In computer graphics many applications need to alter or manipulate a picture, for example, by changing its size, position or orientation. This can be done by applying a geometric transformation to the coordinate point’s defining the picture. These notes cover the basic theory of two-dimensional (2D) geometric transformations.

Cartesian coordinate system

A coordinate system in which the coordinates of a point are its distances from a set of perpendicular lines that intersect at an origin, such as two lines in a plane or three in space.

Third Axis

(in a three-dimensional Cartesian coordinate system) the axis along which values of z are measured and at which both xand y equal zero.



Geometric Theory & Polygons

A vertex is a corner point of a polygon, polyhedron, or other higher dimensional polytope, formed by the intersection of edges, faces or facets of the object.

In a polygon, a vertex is called "convex" if the internal angle of the polygon, that is, the angle formed by the two edges at the vertex, with the polygon inside the angle, is less than π radians; otherwise, it is called "concave" or "reflex". More generally, a vertex of a polyhedron or polytope is convex if the intersection of the polyhedron or polytope with a sufficiently small sphere centered at the vertex is convex, and concave otherwise.

Mesh

Also called mesh topology or a mesh network, mesh is a network topology in which devices are connected with many redundant interconnections between network nodes. In a true mesh topology every node has a connection to every other node in the network.

There are two types of mesh topologies: full mesh and partial mesh.

Full mesh topology occurs when every node has a circuit connecting it to every other node in a network. Full mesh is very expensive to implement but yields the greatest amount of redundancy, so in the event that one of those nodes fails, network traffic can be directed to any of the other nodes. Full mesh is usually reserved for backbone networks.

Primitives

The term geometric primitive in computer graphics and CAD systems is used in various senses, with the common meaning of the simplest (i.e. 'atomic' or irreducible) geometric objects that the system can handle (draw, store). Sometimes the subroutines that draw the corresponding objects are called "geometric primitives" as well. The most "primitive" primitives are point and straight line segment, which were all that early vector graphics systems had.




Surfaces

In technical applications of 3D computer graphics (CAx) such as computer-aided design and computer-aided manufacturing, surfaces are one way of representing objects. The other ways are wireframe (lines and curves) and solids. Point clouds are also sometimes used as temporary ways to represent an object, with the goal of using the points to create one or more of the three permanent representations.

















Bibliography

Geometry - http://oxforddictionaries.com/definition/english/geometry

Cartesian coordinate system - http://www.thefreedictionary.com/Cartesian+coordinate+system

Z Axis - http://wiki.guildwars.com/images/thumb/f/fb/Z_axis.jpg/300px-Z_axis.jpg

Vertex - http://en.wikipedia.org/wiki/Vertex_(geometry)

Mesh - http://www.webopedia.com/TERM/M/mesh.html

Wireframe Model - http://b.vimeocdn.com/ts/275/404/275404818_640.jpg

Wireframe model 2 - http://ufoseries.com/computerGraphics/geoffInterceptorMesh.jpg

Geometric Primitives - http://en.wikipedia.org/wiki/Geometric_primitive

Primitive objects -http://www.cem.brighton.ac.uk/staff/alb14/CI219/Lectures%20&%20Tutorials/images/standard.jpg

Surfaces - http://en.wikipedia.org/wiki/Computer_representation_of_surfaces

Suface 1 - http://www.cl.cam.ac.uk/research/rainbow/projects/subdnurbs/

Surface 2 -  http://worldarts.info/wp-content/uploads/2012/10/3d-model.jpg