User talk:Jengelh/Pastebin

step 1
$$\alpha^2 r_{x}^2 + 2 \alpha r_x x_0 + x_0^2 +$$ $$\alpha^2 r_{y}^2 + 2 \alpha r_y y_0 + y_0^2 +$$ $$\alpha^2 r_{z}^2 + 2 \alpha r_z z_0 + z_0^2 = 1^2$$

step 2
Quadratic equation:

$$f \alpha^2 + g \alpha + h = 0$$

with

$$f = r_x^2 + r_y^2 + r_z^2$$ $$g = 2 (r_x x_0 + r_y y_0 + r_z z_0)$$ $$h = x_0^2 + y_0^2 + z_0^2 - 1^2$$

step 3
Divide by f to obtain a PQ-able formula.

$$\alpha^2 + \frac{g}{f} \alpha + \frac{h}{f} = 0$$

step 4
Apply PQ.

$$\alpha^{+} = -\frac{\frac{g}{f}}{2} + \sqrt{\frac{\frac{g}{f}^2}{4} - \frac{h}{f}}$$ oder irgendwie so.