Coordinate Systems

Wikipedia:
Defining a coordinate system based on another one

In geometry and kinematics, coordinate systems are used not only to describe the (linear) position of points, but also to describe the angular position of axes, planes, and rigid bodies. In the latter case, the orientation of a second (typically referred to as "local") coordinate system, fixed to the node, is defined based on the first (typically referred to as "global" or "world" coordinate system). For instance, the orientation of a rigid body can be represented by an orientation matrix, which includes, in its three columns, the Cartesian coordinates of three points. These points are used to define the orientation of the axes of the local system; they are the tips of three unit vectors aligned with those axes.

To read the coordinate system you have to know what side is "n" (the bottom side with numbers) then you go from "n" to whatever your number is.




Some CAD softwares use the terms coordinate system and plane interchangeably.  Any plane which has not only a normal direction, but also a rotational orientation (an "x" direction) is, technically speaking, a coordinate system.  For example, we can tell that the construction plane in Rhinoceros (c-plane) is actually a coordinate system in that it is inscribed with an x and y direction.





Wikipedia:
A coordinate transformation is a conversion from one system to another, to describe the same space.

With every bijection from the space to itself two coordinate transformations can be associated:

    * such that the new coordinates of the image of each point are the same as the old coordinates of the original point (the formulas for the mapping are the inverse of those for the coordinate transformation)
    * such that the old coordinates of the image of each point are the same as the new coordinates of the original point (the formulas for the mapping are the same as those for the coordinate transformation)

For example, in 1D, if the mapping is a translation of 3 to the right, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to -3, so that the coordinate of each point becomes 3 more.

Some Interesting Coordinate Systems

http://en.wikipedia.org/wiki/Cylindrical_coordinate_system
http://en.wikipedia.org/wiki/Spherical_coordinate_system

http://en.wikipedia.org/wiki/Parabolic_coordinates
      '' define how we interpret the x,y,z of our inPt in terms of the new coord system
      '' inPt.x = theta
      '' inPt.y = rho (expecting input as 0->1, need output in -1.0->1.0)
      '' inPt.z = phi (expecting input as 0->1, need output in radians -3.14->3.14)   
      Dim theta As Double = inPt.x
      Dim rho As Double = (inPt.y * 2) - 1
      Dim phi As Double = (inPt.z * (PI * 2)) + (PI / 2)

http://en.wikipedia.org/wiki/Elliptic_coordinates
      a = distance of foci from origin
      x = a * cosh(rho) * cos(phi)
      y = a * sinh(rho) * sin(phi)
      z = z

http://en.wikipedia.org/wiki/Bipolar_coordinates
      a = distance of foci from origin
      x = a * ( sinh(rho) / (cosh(rho)-cos(phi) )
      y = a * ( sinh(phi) / (cosh(rho)-cos(phi) )
      z = z