Appendix B
This Appendix derives the parametric equations for a length-contracted circular contour in frame K’. in Fig. B.1
q is the angle (with the positive x’-axis) subtended by a point on the circular contour. f is the angle subtended by the corresponding (y=y’) point on the length-contracted contour. Whereas R, the radius of the circular contour, is constant, R’ varies with f. In the main document f is varied in whole degree increments and is therefore known. And of course R is a given. The objective is to determine x’ and y’ in terms of the known quantities f and R.Figure B.1
Circular and Ovoid Contours
By length contraction,
.
(B.1)
But
.
(B.2)
Dividing Eq. B.2 by B.1 produces
.
(B.3)
Thus
.
(B.4)
And from Eq. B.2,
.
(B.5)
Therefore
(B.6)
and
(B.7)
.