Next: Specific Solutions of the
Up: The Fixed Viewpoint Constraint
Previous: Derivation of the Fixed
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}](img32.gif) |
(9) |
The next step is to substitute
and set
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}](img35.gif) |
(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}](img36.gif) |
(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}](img37.gif) |
(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}](img38.gif) |
(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}](img39.gif) |
(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}](img40.gif) |
(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