Linear algebra and geometric transformations in 2D
Linear algebra and geometric transformations in 2D
Computer Graphics CSE 167 Lecture 2
CSE 167: Computer Graphics
? Linear algebra
? Vectors ? Matrices
? Points as vectors ? Geometric transformations in 2D
? Homogeneous coordinates
CSE 167, Winter 2018
2
Vectors
? Represent magnitude and direction in multiple dimensions
? Examples
? Translation of a point ? Surface normal vectors (vectors orthogonal to
surface)
CSE 167, Winter 2018
3
Based on slides courtesy of Jurgen Schulze
Vectors and arithmetic
Examples using 3-vectors
Vectors are column vectors
Vectors must be the same length
CSE 167, Winter 2018
4
Magnitude of a vector
? The magnitude of a vector is its norm
Example using 3-vector
? A vector if magnitude 1 is called a unit vector ? A vector can be unitized by dividing by its
norm
CSE 167, Winter 2018
5
Dot product of two vectors
Angle between two vectors
CSE 167, Winter 2018
6
Cross product of two 3-vectors
? The cross product of two 3-vectors a and b results in another 3-vector that is orthogonal (using right hand rule) to the two vectors
CSE 167, Winter 2018
7
Cross product of two 3-vectors
CSE 167, Winter 2018
8
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