Make Use of $\angle G$

The gradient direction $\angle G$ of an edge pixel can be used to reduce the computation in the parameter space. First recall that the tangent direction or the derivative $dy/dx$ of an implicit function $f(x,y)=0$ can be found as:

$$\tan\; \phi = \frac{dy}{dx}=-\frac{f'_x(x,y)}{f'_y(x,y)}$$

where $f'_x(x,y)$ and $f'_y(x,y)$ are the partial derivatives of the function $f(x,y)$ with respect to $x$ and $y$, respectively. Also note that at an arbitrary point on the curve specified by the equation $f(x,y)=0$, the gradient direction is always perpendicular to the tangent direction:

$$\phi = \angle G \pm \frac{\pi}{2}$$

i.e.,

$$\frac{dy}{dx}=-\frac{f'_x(x,y)}{f'_y(x,y)} = \tan \;\phi = \tan\;(\angle G \pm \frac{\pi}{2}) = -\cot\; \angle G$$

or

$$\frac{f'_x(x,y)}{f'_y(x,y)} =\cot\; \angle G$$

as shown in the figure:

Now the gradient direction $\angle G$ can be used to establish a second equation, in addition to the original equation describing the shape, and thereby reducing the dimensions of the hyper-surface in the parameter space that needs to be incremented by 1.

$$\left\{ \begin{array}{l} f(x,y,\alpha_1,\cdots,\alpha_n)=0 \... ...\alpha_1,\cdots,\alpha_n) =cot\; \angle G \end{array} \right.$$