Finding Normal of Sphere in Ray Tracer Porgram

Question

I'm working on a ray tracer for spheres and I'm trying to implement an illuminate function to calculate the light intensity per ray. I'm currently stuck on calculating the diffuse reflection:

Given a ray R, a sphere S, a point P where R intersects S, and a light source L

I understand that to use Lambert's Law to calculate diffuse reflection, I need the light direction vector and the normal vector.

I know I can get the light direction vector by calculating L - P. I'm stuck now on calculating the normal.

I know I need to use the inverse of the S transform matrix but I don't understand conceptually what inverting the S transform matrix does so I was hoping to get some guidance on how to do this.

I think its light direction = normalized(L - P) and the normal = normalize(P - center of S); — WacławJasper
P - center of S is the surface normal at P. The light direction is L's position - P. What source are you learning from? — molbdnilo
could you please explains what this `S transform matrix" is for you ? — Guiroux
@Guiroux it would be the translate * scale matrix to position the sphere in 3d space — Chang Liu
@WacławJasper thank you, sorry I wrote the equations wrong. I looked at some code to do this and noticed that some people took the scale transform matrix and transposed it and then inverted it. And then they multiplied (invertedScale * invertedScale * normal) to get the normal vector. Do you know why that is? Especially the double multiplication. — Chang Liu

LJᛃ LJᛃ · Accepted Answer · 2016-03-15T17:58:16

As stated in the answer to your other question, using the inverse of the sphere matrix will move things transformed by it into sphere space(where the sphere is at origin[0,0,0]).

To calculate the normal you transform your light position into sphere space using S⁻¹, if you're not using anisotropic scalings you can just normalize your transformed light position to get the normal:

otherwise you'll need to transform using the inverted scaling matrix.

I noticed that some people took the normal calculated from P - center of S

It depends on what data and in what space that data is stored. This way assumes the primitives data is available in world space, so a sphere is defined by its center and its radius. From your previous question I assumed you only have a transform matrix describing your sphere.

and multiplied it by the inverse of the sphere's scale matrix squared like so: normalize(inv(S.scale) * inv(S.scale) * (P - center of S)).

Squaring the inverse makes the scaling agnostic to its sign, e.g. when the sphere is scaled uniformly by -1 the calculated normal will be the same as if no scaling was applied. When not squaring the scaling matrix a negative scaling would result in the normal pointing in the opposite direction and (under proper lighting conditions) resemble the inner surface of the sphere(-shell).

Finding Normal of Sphere in Ray Tracer Porgram

1 Answers