HA4 Task 3 - Geometric Theory

What are the fundamental principles behind the creation of virtual 3D space? Describe and explain 3D geometry.

Remember that you are trying to comprehensively explain the theory and applications of 3D with elucidated examples and consistently using subject terminology correctly.

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The Cartesian Coordinates System

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.




http://en.wikipedia.org/wiki/Cartesian_coordinate_system

Geometric Theory and Polygons

In Mesh modeling the basic object used is a vertex which is a point in three dimensional space. When two vertices are connected by a straight line it becomes an edge. When three vertices are connected to each other by three edges it creates a triangle which is the simplest polygon in Euclidean space. More complex polygons can be created out of multiple triangles, or as a single object with more than 3 vertices. Four sided polygons (generally referred to as quads) and triangles are the most common shapes used in polygonal modeling. A group of polygons, connected to each other by shared vertices, is generally referred to as an element. Each of the polygons making up an element is called a face.

In Euclidean geometry, any three non-collinear points determine a plane. For this reason, triangles always inhabit a single plane. This is not necessarily true of more complex polygons, however. The flat nature of triangles makes it simple to determine their surface normal, a three-dimensional vector perpendicular to the triangle's surface. Surface normals are useful for determining light transport in ray tracing.

A group of polygons which are connected by shared vertices is referred to as a mesh, often ferred to as a wireframe model. 

In order for a mesh to appear attractive when rendered, it is desirable that it be non-self-intersecting, meaning that no edge passes through a polygon. Another way of looking at this is that the mesh cannot pierce itself. It is also desirable that the mesh not contain any errors such as doubled vertices, edges, or faces. For some purposes it is important that the mesh be a manifold – that is, that it does not contain holes or singularities (locations where two distinct sections of the mesh are connected by a single vertex).

http://en.wikipedia.org/wiki/Polygonal_modeling

3D Primitives

Primitives are basic geometric forms that you can use as is or modify with transforms and Booleans. It's possible to create these objects by from 2D shapes by lathering them, however, most software packages build them in for speed and convenience.

The most common 3D primitives are the cube, pyramid, cone, sphere, and torus. Like 2D shapes, these primitives can have a resolution level assigned to them so that you can make them look smoother by boosting the number of sides and steps used to define them.

image from http://docs.autodesk.com/AMECH_PP/2013/ENU/index.html?url=filesACD/GUID-A6C31606-37F1-4D83-A325-C1399488AD42.htm,topicNumber=ACDd30e63652

http://www.peachpit.com/articles/article.aspx?p=30594&seqNum=5 

Surfaces 

Polygons can be defined as specific surfaces and that can have colour, texture or photographic maps added to them to create the desired look.