3D rotation steps and perspective projection

Steps to rotate about an arbitrary axis

Let the axis pass through P1(x1, y1, z1) and P2(x2, y2, z2). Unit vector u = (a, b, c) = (P2 − P1)/|P2 − P1|, d = √(b² + c²).

1. Translate P1 to the origin: T(−x1, −y1, −z1).
2. Rotate about the x-axis by α (cosα = c/d, sinα = b/d) so the axis lies in the xz plane.
3. Rotate about the y-axis by β (cosβ = d, sinβ = a) so the axis coincides with the z-axis.
4. Rotate about the z-axis by the required angle θ.
5. Apply the inverse rotations in reverse order: Ry(−β), then Rx(−α).
6. Translate back: T(x1, y1, z1).

M = T(P1) · Rx(−α) · Ry(−β) · Rz(θ) · Ry(β) · Rx(α) · T(−P1)

Perspective projection

Perspective projection projects 3D points onto a view plane along lines that meet at the centre of projection (COP). Distant objects appear smaller, which gives a realistic picture.

With the COP at (0, 0, zprp) and the view plane at z = zvp, dp = zprp − zvp:

xp = x·dp/(zprp − z), yp = y·dp/(zprp − z), zp = zvp

| 1 0  0        0           |
| 0 1  0        0           |
| 0 0 -zvp/dp   zvp·zprp/dp |
| 0 0 -1/dp     zprp/dp     |

The result is divided by the homogeneous factor h = (zprp − z)/dp.

Features:

• Parallel lines that are not parallel to the view plane converge at vanishing points.
• Types: one-point, two-point and three-point perspective.
• Size is not preserved, so it is not used for measurements.