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)