Adding an angle variance to a 3d vector

Question

I am converting a particle emitter from 2d to 3d.

In the 2d system the original coder used a random angle variance and added it to the angle. So it would be something like:

thisAngle = emitterAngle + angleVariance * RANDOM_MINUS_1_TO_1();

I want to do the same thing in 3d using an emission vector (vec3) and again an angleVariance. My specific variance for this particle will still be angleVariance * RANDOM_MINUS_1_TO_1(). But how do I apply that angle a 3d vector so that the new vector could be any value X degrees off of center. So if the original vector is straight up (0,1,0) the new vector could be off in the X or the Z and still normalized to 1.

Any hints? I have all the matrix and vector functions at my disposal.

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Thanks to nathan I was able to write this code. Not sure if I've over complicated it somewhere. Once I test it I'll checkmark his answer:

    // first pick a random Z from [cos(angleVariance), 1]
    GLfloat varianceZ = [self randomFloatBetween:myCosf(angleVariance) andLargerFloat:1.0];
    // and a random angle for the azimuth
    GLfloat rndA = [self randomFloatBetween:0.0 andLargerFloat:360.0];

    // then calculate the X & Y to make this a vector off of the +Z axis with a max width of angleVariance
    GLfloat varianceX = sqrt(1 - varianceZ*varianceZ) * myCosf(rndA); // my functions input degrees not radians
    GLfloat varianceY = sqrt(1 - varianceZ*varianceZ) * mySinf(rndA);

Up to here the above code works. It makes a variance angle off center but it is aligned to the +z axis. The next part is supposed to rotate/transform the angle to my emissionVector angle so it's a variance off of that. But so far this part isn't working.

    // next find an axis to rotate the point around to put it in line with the emmissionVector
    vec3 rotationAxis = normalizeVec3(crossProduct(vec3Make(0.0, 0.0, 1.0), emissionVector));
    GLfloat rotationAngle = acos(degreesToRadians(emissionVector.z));

    // make a quaternion out of the rotation axis and angle
    vec4 q = quaternionVector(rotationAxis, rotationAngle);
    // turn the quaternion into a matrix
    mat4 rotMat4;
    mat3 rotMat3;
    matrixQuaternion(q, rotMat4);
    getMat3FromMat4(rotMat4, rotMat3);
    // multiply the variance vector by the matrix to get a resulting vector with the variance added
    vector = matrixTransformVec3(rotMat3, vec3Make(varianceX, varianceY, varianceZ));

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The above code doesn't appear to be correct. I'm still working on the solution and will fix the above code when I get it working.

Answer 1

There is a formula for picking a random point on a sphere which can be adapted for your purposes. Generating a random point on a sphere can be done by picking a z-coordinate uniformly at random in [–1, 1], then selecting a random angle θ in [0, 2π] and calculating:

x = sqrt(1 - z^2) * cos θ
y = sqrt(1 - z^2) * sin θ

This can be extended to pick a random point on a spherical cap with any angular width, by simply choosing z uniformly at random within [cos(angleVariance), 1] instead of [–1, 1]. Then apply the preceding formulas for x and y as usual, and you'll end up with a random point distributed around the +Z axis but staying within angleVariance of it.

Finally, you can rotate the resulting point to align the +Z axis with your desired emission vector. To do this, you can rotate around the axis normalize(cross(float3(0, 0, 1), emissionVector)) by the angle acos(emissionVector.z).

Answer 2

In the 2D case your random values could be sampled from an arc/half a cricle and you have only a one degree of freedom. In the 3D case you can sample half a sphere, so in this case we need to use spherical coordinates.

Let's assume we have X,Y,Z in Cartesian coordinates, and r, theta, and alpha in spherical coordinates.

Recall that

• x = r * sin(theta)*cos(alpha)
• y = r * sin(theta)*sin(alpha)
• z = r * cos(theta)

Assuming r is 1 (unit sphere), now we have 2 variables we can choose randomly theta and alpha. According to mathworld it is incorrect to sample alpha and theta to be [0,pi) and [0,2pi) because the samples will bunched on the polars. so they suggest the following function, which guarantees every part of the sphere to be applicable,

• theta = 2 * pi* u
• alpha = cos_inv( 2*v - 1)

Where u,v are chosen randomly between (0,1). Now inorder to limit the selection to the half of the sphere, you can simply drop 2 in the theta = 2 * pi* u to become theta = pi* u.

I also came by this interesting article which might be of use.