1 Composite/ Hierarchical Transforms

• M1 transforms the teapot from its object space to the table’s object space
• M2 transforms the table from its object space to world space
• M2M1 transforms the teapot from its object space to world space

Concatenation of Transformations

Sequence of affine maps A1, A2, A3, ... Concatenation by matrix multiplication

• Very important for performance!

  

• Matrix multiplication not commutative, ordering is important!

Examples

Commutativity of Rotations—2D

• In 2D, order of rotations doesn’t matter:

Commutativity of Rotations—3D

• IN-CLASS ACTIVITY:
• Rotate 90°around Y, then 90°around Z, then 90°around X
• Rotate 90° around Z, then 90° around Y, then 90° around X
• Was there any difference?

CONCLUSION: bad things can happen if we’re not careful about the order in which we apply rotations!

Transforming normal vectors 【todo】