HA4 Task 3 - Geometric Theory

HA4 Task 3

Geometry

The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian.

In 2D vector artwork the computers draws the image by plotting points on the x and the y axes, and this points are joined with paths. The subsequent shapes can be filled in with colour and thickness if required.

3D programs operate on a grid of 3D co-ordinates. 3D co-ordinates are pretty much the same as 2D co-ordinates except there’s a third axis known as the Z or ‘depth’ axis

Geometric Theory and Polygons
The basic object used in mesh modelling is a vertex, a point in three dimensional space. Two vertices connected by a straight line become an edge. Three vertices, connected to each other by three edges, define 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 modelling. A group of polygons, connected to each other by shared vertices, is generally referred to as an element.

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 normal are useful for determining light transport in ray tracing.

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).

Primitives
In 3D applications, pre-made objects can be used to make models out of various shapes, the most basic of this shapes are the Standard Primitive Objects, or the Common Primitives, these shapes vary from the basic cube or box to spheres, cylinders, pyramids (both triangular and square based) and cones. They are used as the starting point for modelling. They can be edited once created.

Surfaces
Polygons can be defined as specific surfaces and then have colour, texture or photographic maps added to them to create the desired look. The example below shows how a map is displayed as if the object has been unwrapped.