Geometric Theory for 3D Graphics

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What is a 3D model?

A 3D model is a mathematical representation of a shape in 3D space. Cubes, spheres and cylinders are all examples of basic 3D models but they can also be very complex as you can see in these examples;

There are two main types of 3D models in use in the film and games industries these are NURBS surfaces and polygonal models. We’re mostly going to focus on polygonal models but I will give a quick explanation of NURBS surface modelling.

NURBS surfaces

A NURBS (Non-uniform rational B-spline) surface is a model created through the use of bezier curves (like the ones use to create vector graphics). To forms a NURBS surface the software interpolates between two or more curves and creates a smooth face to fill the space. Models constructed using NURBS are very mathematically accurate and are commonly used when modelling for engineering and automotive design.

Polygonal models

These types of models are more common in the film, animations and games industries and it’s these that we’ll focus on for the rest of this handout. Polygonal models are made up of vertices, edges and faces.

Vertices

These are points in 3D space. Each of these points is known a a vertex. Each vertex has its own coordinate information (x, y, z) and moving these vertices is a common method of shaping models. Here you can see vertices highlighted.

Edges

Edges are used to join vertices together. This creates a wireframe like the one here.

Faces

The thing that defines polygonal modelling is that the meshes are faceted. That means that they are made up of faces. Faces are used to fill the space between vertices and edges. In efficient modelling these faces are three or four sided (modelling in tris or quads). The faces that make up a 3D model is known as a mesh. The number of faces in a mesh is known as the poly-count and the polygon density is referred to as the resolution.

3D coordinate space and axes

The 3 dimensional space that games and applications such as Maya rely on is based on the Cartesian coordinate system. This is like a map for defining the position of points in 3d space. The system was developed by French brainbox Rene Descartes. Using this system space is defined using 3 axes, known as x, y and z. These represent the width, height and depth respectively. The coordinates for a point are written numerically in the order of x, y and z and look like this - (2,4,1).

The zero point of the axes is called the origin (0,0,0). This is where each of the three axes intersect. Heres an example of how this coordinate system looks and works.