Rotate point by a direction vector

Question

Given a vector, A, which represents a point, and a unit vector, B, which represents the looking direction, how would I rotate A so that B is the axis that's looked down on? Let's say the looking-axis is originally (0,0,1) and I want it to become B by rotating A accordingly. So it needs to rotate by (0,0,1) - B.

Answer 1

You can directly construct a look-at matrix without using trigometry (but using vector math). First, you need an up-vector to specify which direction should be up from the camera's point of view. This would typically be (0, 1, 0) or (0, 0, 1), depending on your choice of coordinates.

Then, you can calculate the orthonormal basis for your camera:

float3 right = normalize(cross(dir, up));
float3 upTrue = normalize(cross(right, dir));

Beware! If dir is parallel or antiparallel to up, this will break.

Then, you can construct the (left-handed) camera-to-world matrix by shoving right, upTrue, and dir into the first three rows (if using row vectors) or columns (if using column vectors) of a 4x4 matrix. Put the camera position into the fourth row/column.

Then calculate the inverse of this 4x4 matrix to get the world-to-camera matrix. If you just need the rotation part, use a 3x3 matrix, and you can take the transpose instead of doing a full inverse.

This matrix can then be used as a view matrix in a 3D graphics engine, or you can transform a point yourself by multiplying with this matrix.

Answer 2

Given direction as d and point as p

d = normalize(d);
double yaw = Math.Atan2(d.X, d.Y);
double pitch = Math.Atan2(d.Z, Math.Sqrt((d.X * d.X) + (d.Y * d.Y)));
p.rotateXYZ(pitch, yaw, 0); //Roll == 0

Well, this should do :)