Geometric Theory

1. Geometric Theory By Eugene Adams

2. Vertices • Edges • Faces • Polygons • Meshes • 2D co-ordinates • 3D co-ordinates

3. Vertices A vertex is the singular form of vertices. The term is used in geometry to describe the corner point of a shape: In laymans terms, the point at which a line either terminates or is intersected is a vertex. In computer graphics a vertex is associated not only with the three spatial coordinates which dictate its location, but also with any other graphical information necessary to render the object correctly.

4. On the Tetrahedron to the left, I have circled all of the vertices on the shape. Being a point on a shape, a vertex has no standardised form and changes according to the shape it is a part of.

5. Edges In geometrical terms, an edge is a 1 dimensional line that connects two 0 dimensional vertices. When a minimum of 3 vertices are combined with a minimum of 3 edges, a 2D polygon (a Triangle in this case) or ‘face’ is created. Adding in a minimum of 1 more vertex and 3 more edges will transform the 2D polygon into a 3D polyhedron (in this case a Tetrahedron). This means that to create a 3D shape, a minimum of 4 vertices and 6 edges are required.

6. 2 vertices connected by a single edge. 3 vertices and 3 edges creating a face or triangular polygon. 4 vertices and 6 edges creating a tetrahedron.

7. Faces The term polyhedron is a greek term meaning “many faces”. Any 2D shape can be a face, combining several to create a polyhedron (e.g. 6 squares can combine to create a cube), thus making the term very appropriate. A face is created using 3 or more vertices and 3 or more edges in a closed pattern.

8. A single face (or ‘hedra’), in this case a square. A combination of 6 sqares to create a cube. Being made of 6 faces, the cube is thus a hexahedron.

9. Polygon The word “polygon” is derived from the Greek words “poly” meaning “many”, and “gonia”, meaning “knee” or “angle”. In moodern geometry, however, a polygon is usually defined according to its amount of sides rather than its angles. A polygon is always 2D, and many polygons can be used as faces to create 3D Polyhedrons. A polygon is composed of a closed circuit of vertices and edges, with a minimum of 3 vertices and 3 edges required.

10. Meshes A polygon mesh is a combination of vertices, edges and faces that form a Polyhedron. The faces used are usually triangles to simplify rendering, though (in theory) any shape can be used.

11. A dolphin made using a triangle mesh.