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 Vectors are 3‐vectors 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 Matrices • 2D array of numbers A = CSE 167, Winter 2018 9 Matrix addition • Matrices must be the same size • Matrix subtraction is similar CSE 167, Winter 2018 10 Matrix‐scalar multiplication CSE 167, Winter 2018 11 Matrix‐matrix multiplication CSE 167, Winter 2018 12 Matrix‐vector multiplication • Same as matrix‐matrix multiplication – Example: 3x3 matrix multiplied with 3‐vector CSE 167, Winter 2018 13 Transpose • AT is the matrix A flipped over its diagonal – Example • Vectors can also be transposed to convert between column and row vectors – Example CSE 167, Winter 2018 14 The identity matrix CSE 167, Winter 2018 15 Matrix inverse • The inverse of a square matrix M is a matrix M‐1 such that • A square matrix has an inverse if and only if its determinant is nonzero • The inverse of a product of matrices is Example using three matrices CSE 167, Winter 2018 16 Representing points using vectors • 2D point • 3D point CSE 167, Winter 2018 17 Geometric transformations in 2D • Operations on vectors (or points) – Translation – Linear transformation • Scale • Shear • Rotation • Any combination of these – Affine transformation • Linear transformation followed by translation CSE 167, Winter 2018 18 2D translation • Translation of vector v to v’ under translation t CSE 167, Winter 2018 19 2D uniform scale • Scale x and y the same CSE 167, Winter 2018 20 2D nonuniform scale • Scale x and y independently CSE 167, Winter 2018 21 2D shear • Shear in x direction (horizontal) CSE 167, Winter 2018 22 2D rotation • Positive angles rotate counterclockwise where CSE 167, Winter 2018 23 2D rotation about a point CSE 167, Winter 2018 24 2D rotation about a point 1. Translate point 2. Rotate about 3. Translate origin to the origin the origin back to point CSE 167, Winter 2018 25 2D rotation about a point • This can be accomplished with one transformation matrix, if we use homogeneous coordinates • A 2D point using affine homogeneous coordinates is a 3‐vector with 1 as the last element CSE 167, Winter 2018 26 2D translation using homogeneous coordinates • 2D translation using a 3x3 matrix • Inverse of 2D translation is inverse of 3x3 matrix ‐ ‐ CSE 167, Winter 2018 27 2D rotation using homogeneous coordinates • 2D rotation using homogenous coordinates CSE 167, Winter 2018 28 2D rotation about a point using homogeneous coordinates ‐ ‐ Important: transformation matrices are applied right to left CSE 167, Winter 2018 29 2D rotation about a point using homogeneous coordinates ‐ ‐ M ‐ where M ‐ CSE 167, Winter 2018 30.