CMSC 335 COMPUTER GRAPHICS LECTURE 4 LINEAR ALGEBRA

CMSC 335 COMPUTER GRAPHICS LECTURE 4 • LINEAR ALGEBRA REVIEW

BASIC GRAPHICS PIPELINE (VIEWING PIPELINE) y y y z x x z Projection Coordinates (Homogeneous) Device Coordinates x Model Coordinates z World Coordinates Camera Coordinates

MECHANISMS CONTROLLING VIEWING PIPELINE • Coordinate or reference frames • Cross products help define them • Transformations between the stages • Matrix multiplications • At different phases of this class, we will reason from different coordinate frames, and it is important for equations (e. g. , lighting) to know which frame you are in • Example: a common bug in lighting is using a light direction in workspace and a normal from model space and a view direction from camera space, yikes!

VECTOR CROSS PRODUCT • Right hand rule – fingers in the direction of the first vector, palm faces the second vector, and your thumb is the direction of the cross product • Uses: • Determining a reference frame or a vector that is perpendicular to the plane defined by two other vectors • Determining the side (right/left) that a vector lies in relation to another (sign of the result)

VECTOR CROSS PRODUCT •

VECTOR CROSS PRODUCT • Why are these important? • Bugs • Algebraic manipulation to a more efficient algorithm

MANAGING COORDINATE FRAMES •

ORTHONORMAL BASIS •

ORTHONORMAL BASES • Easy to "square-up" to remove error • How?

MATRICES • •

MATRIX ADDITION •

MATRIX SCALAR PRODUCT •

MATRIX MULTIPLICATION •

REPRESENTING VECTORS AS MATRICES •

MATRIX MULTIPLICATION •

IDENTITY MATRICES •

MATRIX INVERSION •

TRANSPOSE •