Euclidean Space Definitions
We can define Euclidean Space (coordinate system, definition of distance, etc.) in various ways.
Classical Geometry (points, lines, angles, etc.) Approach
We can distill Euclids postulates down to 5 postulates which define Euclidean Space:
- A straight line may be drawn from any one point to any other point (any 2 points determine a unique line).
- A finite straight line may be produced to any length in a straight line.
- A circle may be described with any centre at any distance from that centre.
- All right angles are equal.
- If a straight line meets two other straight lines, so as to make the two interior angles on one side of it together less than two right angles, the other straight lines will meet if produced on that side on which the angles are less than two right angles.
This last postulate, known as the parallel postulate, can be stated in different ways. If we change this, such as allowing more than one line through a point to be parallel to a given line, then we get interesting non-euclidean space defined on this page.
Euclidean Vector Space
We can define a geometry in a more analytical way based on vectors, such as the following properties:
- v1•v2= v2•v1
- (v1+v2)•v3= v1•v3 + v2•v3
- s*(v1•v2)= (s*v1)•v2 = v1•(s*v2)
- v1•v2 > 0 if and only if v1≠0
Euclidean Metric
A quite fundamental property of spaces is the scalar value that represents the distance between two points.
v1•v2 = |v1|•|v2|*cosθ
Properties of Euclidean Space
Euclidean space has the following properties:
- There is no preferred origin in euclidean space. Any point would be as good as any other as a choice for the origin.
- There is no preferred 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 continuous (differeniatable)
Euclidean n-space is the most elementary 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.
