我想用射線行進來表示橢圓形。我有一個用隱式方程定義的橢圓。
例如，我們可以考慮橢圓體的方程，其給出方式為： $$ \ frac {x ^ 2} {a ^ 2} + \ frac {y ^ 2} {b ^ 2} + \ frac {z ^ 2} {c ^ 2} -1 = 0，$$ 其中 $ a，b，c $ 是恆定的。
鑑於此等式，我需要計算表面陰影的法線。在曲線和曲面的標準過程中，我將採用 $ f（x，y，z）= \ frac {x ^ 2} {a ^ 2} + \ frac {y ^ 2} {b ^ 2} + \ frac {z ^ 2} {c ^ 2} -1 $ 併計算 $ N（p）= \ frac {\ nabla f（p）} {| \ nabla f（p）|} $ 作為表面的法線。
在這裡，我必須檢查 $ 0 $ 是 $ f $ 的常規值，並且 $ f $ 是可區分的。但是，第二項檢查不是自動化的。
請全面回答，而不僅要考慮上面給出的示例。
The usual way of doing this with raymarching is to define your surface as a signed-distance field and use finite differences to get the gradient of the distance function at the point you’re sampling. In other words, if you have a function map(p) that returns the signed distance value at a point p, the normal at p is given by:
float epsilon = 0.001; // arbitrary — should be smaller than any surface detail in your distance function, but not so small as to get lost in float precision
float centerDistance = map(p);
float xDistance = map(p + float3(epsilon, 0, 0));
float yDistance = map(p + float3(0, epsilon, 0));
float zDistance = map(p + float3(0, 0, epsilon));
float3 normal = (float3(xDistance, yDistance, zDistance) - centerDistance) / epsilon;
Note that you can improve the accuracy by taking an extra 3 samples on the other “side” of the center position and subtracting those from the first three, then dividing by 2× the epsilon value, but if your distance function is complicated then that can be a lot of additional work for not much visible benefit.
The sample function you provide isn’t a signed-distance function as-is, and I don’t know of a straightforward way to convert it into one, but there’s an article by Íñigo Quílez (kind of a legend in this field) here with various functions you can make use of, including one for an ellipsoid. There’s also a lot of reference material available on Shadertoy—search for “raymarching” and you’ll find plenty of examples.