Geometry is concerned with the properties of space and the shapes and relationship of things in it. An important topic for this site. Its interesting how much of maths is related to geometry. If an algebra can be any set of objects represented by abstract symbols and a set of rules, the only criteria is that the algebra is consistent (has no contradictions) no requirement to represent reality, so why does so much of it have a geometrical interpretation?
Traditional Geometry
The traditional geometry that many of us were taught at school involves two dimensional constructions with points, lines between points, angles between lines and (of course) trigonometry. This type of geometry is very useful and we use a lot of it on this site (here), examples are, proving Pythagoras theorem and the double angle formula.
For other uses we will need to employ other geometrical ideas because the above approach does have downsides. It gets a lot more complicated if we want to work in three or more dimensions, or if we want to work in different coordinate systems, or if we want to transform complex shapes in different ways.
Vector Space
Vectors can be very useful. A vector has direction and magnitude and can therefore be used to specify the position of points relative to the origin of the coordinate system. Using vectors we can do different operations such as transforming a shape much more automatically by transforming all the vertices of the shape. we can also specify physics laws using vectors. Such operations on vectors can be specified using vector algebra, alternatively linear transforms on vectors can be specified using matrices or quaternions.
This can all be very powerful for computer graphics but there are still complications, especially if we are mixing rotations and translations, or using physics rules or swapping between coordinate systems, then there could be advantages if we can have the option of other geometry approaches available to us.
Dimensions
We are interested in two dimensional space (the main subject of traditional geometry) where we are limited to a particular plane. Here points require 2 scalar values to specify them and rotation requires one scalar value.
We are also interested in three dimensional space of course (the world that we live in and are modeling). This requires three scalar values to represent a point and three scalar values to represent a rotation.
We are also interested in higher numbers of dimensions, why? Well there are some physics theories that postulate more than 3 dimensions, which is interesting, but you may not feel this justifies a lot of your time to study it. More practically we can embed our 3D world in a higher number of dimensions to simplify the calculations, for instance, rotations in 3D are non-linear and messy to combine and work with. Embedding them in a subset of a higher dimensional space allows us to calculate the result without these problems and then translate the result back to 3D to use. For example quaternions (Note: I'm not suggesting that quaternions are vectors but they do contain 4 scalar values) we will also looks at homogeneous and conformal spaces to calculate the movement of solid objects.
Also looking at things in 'n' dimensions seems to show us an underlying structure of things, properties of 3D space that seem arbitrary when studied alone seem to fall into a pattern. Clifford Algebra is a good way of working in 'n' dimensions and group theory helps us categorise the properties. Of course, this is a more advanced option, something to study after the basics.
Elements of 'n' Dimensional Space
We will therefore investigate methods which start with very simple subsets of 'n' dimensional space (on this page);
- a point at the origin
- an infinite length line which goes through the origin
- an infinite plane which goes through the origin.
- in higher dimensions, we can extend this to hyperplanes which are like volumes in a higher number of dimensions
We can refer to these primitive linear subspaces as flats and extend them in the following ways;
- associate the flats with a scalar quantity - directed length (line associated with a scalar quantity) directed area (plane associated with a scalar quantity -area).
- offset flats - modify the primitives so they are offset from the origin.
We can then combine these shapes with intersections and unions.
The advantage of this approach is that it has the same properties for any number of dimensions. This means we can embed a three dimensional space in a higher dimensional model which allows us to deal with rotational and transnational transforms in a linear way.
Other Spaces
We are used to the idea of Euclidian Space in two or three dimensions, this is a flat space which is closest to our intuitive understanding of the world we live in. However we will find it useful to investigate other types of space, especially to simplify various transformations and interactions at the cost of embedding our model in a higher dimensional space.
| space | coordinates | metric |
|---|---|---|
| Euclidian Space | Cartesian Coordinates | ||p - q||² = (p-q)•(p-q) |
| Affine Space | ||
| Projective Space | Homogeneous Coordinates | |
| Conformal Space | ||
| Minkowski Space | ||
| Barycentric Coordinates |
Where Next?
Probably the best place to start is coordinate systems.
Other Geometry Pages
- coordinate systems
- Curves - Bezier Curves,B-Spline,NURBS (Non-Uniform Rational B-Spline)
- Shapes - Dodecahedron -Icosahedon
- transformations
- rotations (orthogonal transformations)
- rotations and translations (affine transformations)
- trigonometry
related topics
- Vector algebra
- Rotations
Concepts
