I want to represent ovaloids with ray-marching. I have ovaloids defined with an implicit equation.

For instance, we can consider the equation of an ellipsoid given as: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} -1 = 0, $$ where $a, b, c$ are constant.

Given this equation I need to caculate the normal for the shading of the suface.

• How can I calculate this normal from the implicit equation?
• How is it done in real-world systems?
• Can you give an explanation of the methods used and ideally point me some implementations that exist?

Please, answer in full generality and not only considering the example given above.

I want to represent ovaloids with ray-marching. I have ovaloids defined with an implicit equation.

For instance, we can consider the equation of an ellipsoid given as: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} -1 = 0, $$ where $a, b, c$ are constant.

Given this equation I need to caculate the normal for the shading of the suface. In an standard course of Curves and Surfaces I would take $f(x,y,z) = \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} -1 $ and compute $N(p) = \frac{\nabla f(p)}{|\nabla f(p)|}$ as the normal of the surface.

Here I have to check that $0$ is a regular value of $f$ and that $f $ is differentiable. However, the second check is not automate.

• How is it done in real-world systems?
• Can you give an explanation of the methods used and ideally point me some implementations that exist?

Please, answer in full generality and not only considering the example given above.

I'm making a project about how to represent ovaloids with ray-marching. I have ovaloids defined with an implicit equation and for shading, I need to calculate the normal of the surface.

How can I calculate this normal from the implicit equation? How is it done in real-world systems? Can you give an explanation of the methods used and ideally point me some implementations that exists?

I want to represent ovaloids with ray-marching. I have ovaloids defined with an implicit equation.

For instance, we can consider the equation of an ellipsoid given as: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} -1 = 0, $$ where $a, b, c$ are constant.

Given this equation I need to caculate the normal for the shading of the suface.

• How can I calculate this normal from the implicit equation?
• How is it done in real-world systems?
• Can you give an explanation of the methods used and ideally point me some implementations that exist?

Please, answer in full generality and not only considering the example given above.