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## General Solution of the Constraint Equation

The first step in the solution of the fixed viewpoint constraint equation is to solve it as a quadratic to yield an expression for the surface slope:
 (9)
The next step is to substitute and set which yields:
 (10)
Then, we substitute 2 r x = y2 + r2 - b2, which when differentiated gives:
 (11)
and so we have:
 (12)
Rearranging this equation yields:
 (13)
Integrating both sides with respect to r results in:
 (14)
where C is the constant of integration. Hence,
 (15)
where k = 2 eC > 0 is a constant. By back substituting, rearranging, and simplifying we arrive at the two equations which comprise the general solution of the fixed viewpoint constraint equation:

In the first of these two equations, the constant parameter k is constrained by (rather than k>0) since 0 < k < 2 leads to complex solutions.

Next: Specific Solutions of the Up: The Fixed Viewpoint Constraint Previous: Derivation of the Fixed
Simon Baker
1/22/1998