Mulawa - 5 
features, such as conic sections and general space curves. These can be represented by an 
arc length parameter s to describe a probability surface around those linear features. 
Additionally, this projection method can be used to determine a probability surface about a 
measured surface such as a three dimensional plane or sphere. In the case of an estimated 
surface, the orthogonal projection matrix E = lyp 1 is based on the normal tj at the point 
P being collapsed. 
The Best Fixed Point C® on the Line DL 
Since the projected covariance matrix Zpp is a quadratic function of the arc length 
parameters, it is reasonable to ask if there is a point C® on the line IL that in some way 
has the smallest covariance matrix. This point will be called the best fixed point C®. The 
criteria for determined the best fixed point C® will be based on determination of the point 
P on the line IL whose projected covariance matrix 2pp has the minimum trace tr(2pp). 
Thus, 
( 10 ) 
= tr(Zcc) + 2str(Zcß) + s 2 tr(2ßß) 
and the minimum 
(tr(Zpp)) is determined by setting the derivative — to zero: 
(ID 
= 2tr(Zcß) + 2str(2ßß) 
implies: 
( 12 ) 
or: 
s 
.© 
(13) 
Then, the best fixed point C® is associated with s® by: 
C® =C + s®p 
(14)