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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:
\begin{displaymath}
\frac{\mathrm{d} z}{\mathrm{d} r} \ = \
\frac{(z^{2} - r^{2} -cz) \pm \sqrt{r^{2}c^{2} + (z^{2} + r^{2}
-cz)^{2}}}{r(2z-c)}.\end{displaymath} (9)
The next step is to substitute $y = z - \frac{c}{2}$ and set $b=\frac{c}{2}$ which yields:
\begin{displaymath}
\frac{\mathrm{d} y}{\mathrm{d} r} \ = \
\frac{(y^{2} - r^{2}...
 ...{2}) \pm \sqrt{4r^{2}b^{2} + (y^{2} + r^{2}
-b^{2})^{2}}}{2ry}.\end{displaymath} (10)
Then, we substitute 2 r x = y2 + r2 - b2, which when differentiated gives:
\begin{displaymath}
2 y \frac{\mathrm{d} y}{\mathrm{d} r} \ = \
2 x + 2 r \frac{\mathrm{d} x}{\mathrm{d} r} - 2 r\end{displaymath} (11)
and so we have:
\begin{displaymath}
2 x + 2 r \frac{\mathrm{d} x}{\mathrm{d} r} - 2 r \ = \
\frac{ 2r x - 2 r^{2} \pm \sqrt{4 r^{2} b^{2} + 4 r^{2} x^{2}}}{r}.\end{displaymath} (12)
Rearranging this equation yields:
\begin{displaymath}
\frac{1}{\sqrt{b^{2} + x^{2}}} \frac{\mathrm{d} x}{\mathrm{d} r} \ = \
\pm \frac{1}{r}.\end{displaymath} (13)
Integrating both sides with respect to r results in:
\begin{displaymath}
\ln \left( x + \sqrt{ b^{2} + x^{2} } \right) \ = \ \pm \ln r + C\end{displaymath} (14)
where C is the constant of integration. Hence,
\begin{displaymath}
x + \sqrt{ b^{2} + x^{2} } \ =\ \frac{k}{2} r^{\pm 1} \end{displaymath} (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 $k \geq 2$ (rather than k>0) since 0 < k < 2 leads to complex solutions.


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