问题
I've been trying to get the correct normals for a sphere I'm messing with using a vertex shader. The algorithm can be boiled down simply to
vert.xyz += max(0, sin(time + 0.004*vert.x))*10*normal.xyz
This causes a wave to roll across the sphere. In order to make my normals correct, I need to transform them as well. I can take the tangent vector at a given x,y,z, get a perpendicular vector (0, -vert.z, vert.y), and then cross the tangent with the perp vector.
I've been having some issue with the math though, and it's become a personal vendetta at this point. I've solved for the derivative hundreds of times but I keep getting it incorrect. How can I get the tangent? Breaking down the above line, I can make a math function
f(x,y,z) = max(0, sin(time + 0.004*x))*10*Norm(x,y,z) + (x,y,z)
where Norm(..) is Normalize((x,y,z) - CenterOfSphere)
After applying f(x,y,z), unchanged normals
What is the correct f '(x,y,z)?
I've accounted for the weirdness caused by the max in f(...), so that's not the issue.
Edit: The most successful algorithm I have right now is as follows:
Tangent vector.x = 0.004*10*cos(0.004*vert.x + time)*norm.x + 10*sin(0.004*vert.x + time) + 1
Tangent vector.y = 10*sin(0.004*vert.x + time) + 1
Tangent vector.z = 10*sin(0.004*vert.x + time) + 1
2nd Tangent vector.x = 0
2nd Tangent vector.y = -norm.z
2nd Tangent vector.z = norm.y
Normalize both, and perform Cross(Tangent2, Tangent1). Normalize again, and done (it should be Cross(Tangent1, Tangent2), but this seems to have better results... more hints of an issue in my math!).
This yields this
回答1:
Get tangent/normal by derivate of function can sometimes fail if your surface points are nonlinearly distributed and or some math singularity is present or if you make a math mistake (which is the case in 99.99%). Anyway you can always use the geometric approach:
1. you can get the tangents easy by
U(x,y,z)=f(x+d,y,z)-f(x,y,z);V(x,y,z)=f(x,y+d,z)-f(x,y,z);- where
dis some small enough step - and
f(x,y,z)is you current surface point computation - not sure why you use 3 input variables I would use just 2
- but therefore if the shifted point is the same as unshifted
- use this instead
=f(x,y,z+d)-f(x,y,z); - at the end do not forget to normalize
U,Vsize to unit vector
2. next step
- if bullet 1 leads to correct normals
- then you can simply solve the U,V algebraically
- so rewrite
U(x,y,z)=f(x+d,y,z)-f(x,y,z);to full equation - by substituting
f(x,y,z)with the surface point equation - and simplify
[notes]
- sometimes well selected
dcan simplify normalization to multipliyng by a constant - you should add normals visualization for example like this:
- to actually see what is really happening (for debug purposses)
来源：https://stackoverflow.com/questions/28258406/calculating-normals-across-a-sphere-with-a-wave-like-vertex-shader