Step Up to 4x4

3D points become (x, y, z, 1) and transformations are 4x4 matrices in homogeneous coordinates.

Translation T(tx, ty, tz)

[ 1 0 0 tx ] [ 0 1 0 ty ] [ 0 0 1 tz ] [ 0 0 0 1 ]

Scaling S(sx, sy, sz) about origin

[ sx 0 0 0 ] [ 0 sy 0 0 ] [ 0 0 sz 0 ] [ 0 0 0 1 ]

Rotation about X-axis Rx(theta)

[ 1 0 0 0 ] [ 0 cos -sin 0 ] [ 0 sin cos 0 ] [ 0 0 0 1 ]

Rotation about Y-axis Ry(theta)

[ cos 0 sin 0 ] [ 0 1 0 0 ] [ -sin 0 cos 0 ] [ 0 0 0 1 ]

(Sign of sin matches right-handed coordinate system, looking down +Y.)

Rotation about Z-axis Rz(theta)

[ cos -sin 0 0 ] [ sin cos 0 0 ] [ 0 0 1 0 ] [ 0 0 0 1 ]

Rotation about Arbitrary Axis

Compose: align axis with a coordinate axis (typically Z), rotate, then undo the alignment. M = T(p) R_align^-1 Rz(theta) R_align T(-p), where p is a point on the axis.

Reflection (about coordinate planes)

• About XY plane: scale Z by -1.
• About YZ plane: scale X by -1.
• About XZ plane: scale Y by -1.

Shear (about XY plane)

[ 1 0 shx 0 ] [ 0 1 shy 0 ] [ 0 0 1 0 ] [ 0 0 0 1 ]

Sample Composition: Model Matrix for an Object

M_model = T_world R_orientation S_scale. Apply to a vertex: v' = M_model * v.

Right-Handed vs Left-Handed

OpenGL/GLM and most math books use right-handed coordinates; DirectX uses left-handed by default. The rotation matrices change sign on sin to account for the handedness.

Composition Rules Are Identical

Order, associativity, and inverse rules from 2D apply unchanged. Real engines store the camera as view = inverse(camera world transform) because moving the camera right is equivalent to moving the world left.