Order of transforms

The same translate, rotate, and scale can be multiplied in more than one order. Matrix products are read from right to left, so the rightmost matrix touches the point first.

Read the product from right to left

The rightmost matrix touches the point first. The dark dot is the origin. The origin does not move with the arrow.

M = T · R · S

Scale first. Then rotate. Then translate. The arrow turns about its tail. Then it steps along X.

Translate X 1.05 Rotate Z 35° Scale Y 1.55 Show all three

T · R · S
R · T · S
S · R · T

Each line names one order. The rightmost column of its matrix is where the tail went. That point is the image of the local origin.

Matrix for T · R · S
—	—	—	—
—	—	—	—
—	—	—	—
—	—	—	—

Where this is used

A robotics engineer places a robot hand by combining the moves of each joint. The order of those moves changes where the hand goes. This is called forward kinematics.
A character rigger stacks bones. Each bone moves in the frame of the bone above it.
A 3D tool stores a scene as a tree. A move on a parent carries every child.
A mechanical designer places parts in an assembly. A turn and then a slide is not the same as a slide and then a turn.

Common mistakes

Reading the product from left to right applies the moves in the wrong order.
A turn and then a slide is not the same as a slide and then a turn.
The dark dot is the origin. It does not move with the arrow.
The three slider numbers stay the same when only the order changes.

Morph targets · Next: point vs direction