***** A Interesting Problem ****

[Robert V. Fleisig]

I am looking for a bit of mathematics advice regarding a problem for
which I have no solution.

The problem is as follows. Suppose you have a rigid body with a
Cartesian reference frame associated with the body. This body is
symmetric about one of the axes of the reference frame. To represent
the orientation of the body in three-space one therefore requires only
two degrees of freedom (the third is redundant). To represent the
orientation one could use spherical coordinates or Euler angles or a
unit vector. The range of possible values for the orientation lie on
the surface of a unit sphere centred at the origin.

However I wish to interpolate a parametric polynomial curve through
the orientations such that the orientation varies smoothly with some
parameter. Easy enough. If for example I use spherical coordinates I
run into a problem. Because the coorindate lines on the sphere are
not everywhere parallel I get different curves depending on where my
interpolatory points are located. This is a serious problem for my
particular application. What I need is a coordinate system which will
give me the same curve for points which are located relative to one
another no matter where they are on the sphere.

An suggestions, ideas, questions or recommendations would be
appreciated (especially references to where I might find the answer).
Please reply to my e-mail address: rob...@mcmaster.ca

Thanks in advance,

Robert V. Fleisig