🚀 go-pugleaf

I have hit a brick wall trying to solve a problem that at first seemed like
simple 2-D geometry. I was wondering if some of you could help.

The problem is associated with moving a straight robot link within
its "sensing envelope" while in contact with an obstacle. I shall explain
it purely gemoetrically though.

I have a segment of finite length L with endpoints P (its tip) and B
(its base) whose coords are known.

Within this segment I define a point Q whose coordinates are known and
*fixed*.

I start "dragging" point P (the tip of the segment) along a straight line
of known direction (call it the pull line), with the constraint
that during such a continuous motion, the segment is always passing through
point Q.

FACT: the locus of B due to the motion of P is a curve of fourth
degree called the conchoid of Nichomedes (if anybody disagrees please let
me know).

QUESTION 1: How much should P move along the pull line if the LENGTH OF
	    THE LOCUS of B (i.e., the integral over the conchoid) is to be
	    exactly equal to some pre-defined constant, say SMAX ???

QUESTION 2: How much should P move along the pull line if the DISTANCE
	    BETWEEN B before and after the straight motion of P
	    (i.e., a chord within the conchoid) is to be exactly equal to
	    some pre-defined constant, say SMAX ???

For question 1, I have been able to come up with a non-integrable
expression and for question 2 I came up with a system of two equations
of 4th degree. In the least, "ugly" ! Any help is to be greatly appreciated.

Dan Reznik
Robotics Research
UW-Madison