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:

$$\frac{\mathrm{d} z}{\mathrm{d} r} \ = \ \frac{(z^{2} - r^{2} -cz) \pm \sqrt{r^{2}c^{2} + (z^{2} + r^{2} -cz)^{2}}}{r(2z-c)}.$$ (9)

The next step is to substitute $y = z - \frac{c}{2}$ and set $b=\frac{c}{2}$ which yields:

$$\frac{\mathrm{d} y}{\mathrm{d} r} \ = \ \frac{(y^{2} - r^{2}... ...{2}) \pm \sqrt{4r^{2}b^{2} + (y^{2} + r^{2} -b^{2})^{2}}}{2ry}.$$ (10)

Then, we substitute 2 r x = y² + r² - b², which when differentiated gives:

$$2 y \frac{\mathrm{d} y}{\mathrm{d} r} \ = \ 2 x + 2 r \frac{\mathrm{d} x}{\mathrm{d} r} - 2 r$$ (11)

and so we have:

$$2 x + 2 r \frac{\mathrm{d} x}{\mathrm{d} r} - 2 r \ = \ \frac{ 2r x - 2 r^{2} \pm \sqrt{4 r^{2} b^{2} + 4 r^{2} x^{2}}}{r}.$$ (12)

Rearranging this equation yields:

$$\frac{1}{\sqrt{b^{2} + x^{2}}} \frac{\mathrm{d} x}{\mathrm{d} r} \ = \ \pm \frac{1}{r}.$$ (13)

Integrating both sides with respect to r results in:

$$\ln \left( x + \sqrt{ b^{2} + x^{2} } \right) \ = \ \pm \ln r + C$$ (14)

where C is the constant of integration. Hence,

$$x + \sqrt{ b^{2} + x^{2} } \ =\ \frac{k}{2} r^{\pm 1}$$ (15)

where k = 2 e^C > 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.