Download Geometric Theory

By Eugene Adams
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Vertices
Edges
Faces
Polygons
Meshes
2D co-ordinates
3D co-ordinates
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.
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.
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.
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.
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.
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.
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.
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.
A dolphin made using a triangle mesh.