Distance Functions

Distance Functions

In this page, I will detail my procedure to create distance functions taking the example of a sphere:

1. Get yourself the function for the outside of the shape $(x+o_x)^2 + (y+o_y)^2 + (z+o_z)^2 = r^2$ ($o$ is the inverse of the offset of the sphere from the origin here)

2. Add a stretch factor to later solve for $(dx+o_x)^2 + (dy+o_y)^2 + (dz+o_z)^2 = r^2$

3. Solve for $d$

   • $d^2 x^2 +2dxo_x+o_x^2 +d^2 y^2 +2dyo_y+o_y^2 +d^2 z^2 +2dzo_z+o_z^2 =r^2$

   • $d^2 (x^2 +y^2 +z^2) + 2d(xo_x+yo_y+zo_z) + o_x^2 +o_y^2 +o_z^2 =r^2$

   • $d^2 (x^2 +y^2 +z^2) + 2d(xo_x+yo_y+zo_z) + (o_x^2 +o_y^2 +o_z^2 -r^2)=0$

   • $A=x²+y²+z²$, $B=2(xo_x+yo_y+zo_z)$, $C=o_x^2 + o_y^2 + o_z^2 - r^2$; $d^2A + 2dB + C=0$

   • $d=\frac{-B\pm\sqrt{B^2-4AC}}{2A}$

4. Now, just return $min(d)$ (because of the $\pm$ in the formula) so that you get the front of the sphere

Notes:

• You can use any kind of variable as a uniform for your function

• The vector is always assumed as the origin to simplify the calculations, however you must do that with your object too (offsets for the current vector are given)