Properties of Euclidean Space
Euclidean space has the following properties:
- There is no prefered origin in euclidean space. Any point would be as good as any other as a choice for the origin.
- There is no prefered direction in euclidean space.
- There is no specific way to define a point at infinity.
- The 'metric' for euclidean space. That is a function, for a given space, that defines the distance between points. For euclidean space, if p and q are two points then:
||p - q||² = (p-q)•(p-q) - Euclidean space is flat
- Euclidean space is linear
- Euclidean space is continous (differeniatable)
Euclidean n-space is the most elementry example of an n dimensional manifold.
Rotations in Euclidean Space
The Cartesian coordinate system allows us to specify directions, but what about direction of rotation? Which direction of rotation do we consider positive? This is an arbitrary decision in that it does not matter as long as we are consistent so, on this website, I have chosen to use the right hand rule. This is because that is the convention used by the VRML/x3d standards.
A rotation can be specified by a vector:
If the thumb of the right hand is pointed in the direction of vector, the positive direction of rotation is given by the curl of the fingers.
Rotations can be specified in many ways, we could use axis and angle in which case the positive angle direction is as described here. Another alternative is to use Euler Angles where we will use the right hand rule for the positive angle about each base positive coordinate direction.
