Calculating view matrix from rotation, elevation and location

Question

I have the rotation(A), elevation(B) and location(C) of a camera in a left handed 3d space. +x is right, +y is down and +z is inwards. Initially the camera is pointing in +z direction with up direction being -y. This camera is rotated by angle B along x axis, rotated by angle A along y axis and then translated by vector C.

How do I calculate the view matrix for such a camera?

Answer 1

You can compute your new right vector using the unit circle (assuming your angle A is measured in radians):

right.x = - cos(A)
right.y = 0
right.z = sin(A)

Here I've used a negative on the x component, because you said the camera is initially pointed in the +z direction. That would mean its "right" vector points in the world's "left" direction when angle A is zero.

We can use a similar trick for the forward and up vectors, using spherical coordinate formulas and a cross product:

forward.x = cos(B) * sin(A)
forward.y = sin(B)
forward.z = cos(B) * cos(A)

up = cross(forward, right)

Now a 3x3 matrix with (right, up, forward) as its columns will rotate an object by angles A and B.

Expand it to a 4x4 matrix and make the last column (C.x, C.y, C.z, 1) and now you have a complete local to world matrix, with both rotation and translation.

Invert this matrix to get a world to local matrix, to transform the worldspace positions of objects into the camera's view space.