CS 4300 Computer Graphics Prof Harriet Fell Fall
- Slides: 36
CS 4300 Computer Graphics Prof. Harriet Fell Fall 2011 Lecture 11 – September 29, 2011 11 June 2021 ©College of Computer and Information Science, Northeastern University 1
Today’s Topics • Linear Algebra Review § Matrices § Transformations • New Linear Algebra § Homogeneous Coordinates 11 June 2021 ©College of Computer and Information Science, Northeastern University 2
Matrices • We use 2 x 2, 3 x 3, and 4 x 4 matrices in computer graphics. • We’ll start with a review of 2 D matrices and transformations. 11 June 2021 ©College of Computer and Information Science, Northeastern University 3
Basic 2 D Linear Transforms 11 June 2021 ©College of Computer and Information Science, Northeastern University 4
Scale by. 5 (1, 0) (0. 5, 0) (0, 1) (0, 0. 5) 11 June 2021 ©College of Computer and Information Science, Northeastern University 5
Scaling by. 5 y y x 11 June 2021 ©College of Computer and Information Science, Northeastern University x 6
General Scaling y y x 11 June 2021 ©College of Computer and Information Science, Northeastern University x 7
General Scaling y y 1 sy x 1 11 June 2021 x sx ©College of Computer and Information Science, Northeastern University 8
Rotation φ sin(φ) cos(φ) -sin(φ) φ cos(φ) 11 June 2021 ©College of Computer and Information Science, Northeastern University 9
Rotation y y x 11 June 2021 φ ©College of Computer and Information Science, Northeastern University x 10
Reflection in y-axis 11 June 2021 ©College of Computer and Information Science, Northeastern University 11
Reflection in y-axis y y x 11 June 2021 ©College of Computer and Information Science, Northeastern University x 12
Reflection in x-axis 11 June 2021 ©College of Computer and Information Science, Northeastern University 13
Reflection in x-axis y y x 11 June 2021 ©College of Computer and Information Science, Northeastern University x 14
Shear-x s 11 June 2021 ©College of Computer and Information Science, Northeastern University 15
Shear x y y x 11 June 2021 ©College of Computer and Information Science, Northeastern University x 16
Shear-y s 11 June 2021 ©College of Computer and Information Science, Northeastern University 17
Shear y y y x 11 June 2021 ©College of Computer and Information Science, Northeastern University x 18
Linear Transformations • Scale, Reflection, Rotation, and Shear are all linear transformations • They satisfy: T(au + bv) = a. T(u) + b. T(v) § u and v are vectors § a and b are scalars • If T is a linear transformation § T((0, 0)) = (0, 0) 11 June 2021 ©College of Computer and Information Science, Northeastern University 19
Composing Linear Transformations • If T 1 and T 2 are transformations § T 2 T 1(v) =def T 2( T 1(v)) • If T 1 and T 2 are linear and are represented by matrices M 1 and M 2 § T 2 T 1 is represented by M 2 M 1 § T 2 T 1(v) = T 2( T 1(v)) = (M 2 M 1)(v) 11 June 2021 ©College of Computer and Information Science, Northeastern University 20
Reflection About an Arbitrary Line (through the origin) y y x 11 June 2021 ©College of Computer and Information Science, Northeastern University x 21
Reflection as a Composition y x 11 June 2021 ©College of Computer and Information Science, Northeastern University 22
Decomposing Linear Transformations • Any 2 D Linear Transformation can be decomposed into the product of a rotation, a scale, and a rotation if the scale can have negative numbers. • M = R 1 SR 2 11 June 2021 ©College of Computer and Information Science, Northeastern University 23
Rotation about an Arbitrary Point y y φ φ x x This is not a linear transformation. The origin moves. 11 June 2021 ©College of Computer and Information Science, Northeastern University 24
Translation (x, y) (x+a, y+b) y y (a, b) x x This is not a linear transformation. The origin moves. 11 June 2021 ©College of Computer and Information Science, Northeastern University 25
Homogeneous Coordinates y y Embed the xy-plane in R 3 at z = 1. (x, y) (x, y, 1) x x z 11 June 2021 ©College of Computer and Information Science, Northeastern University 26
2 D Linear Transformations as 3 D Matrices Any 2 D linear transformation can be represented by a 2 x 2 matrix or a 3 x 3 matrix 11 June 2021 ©College of Computer and Information Science, Northeastern University 27
2 D Linear Translations as 3 D Matrices Any 2 D translation can be represented by a 3 x 3 matrix. This is a 3 D shear that acts as a translation on the plane z = 1. 11 June 2021 ©College of Computer and Information Science, Northeastern University 28
Translation as a Shear y y x x z 11 June 2021 ©College of Computer and Information Science, Northeastern University 29
2 D Affine Transformations • An affine transformation is any transformation that preserves co-linearity (i. e. , all points lying on a line initially still lie on a line after transformation) and ratios of distances (e. g. , the midpoint of a line segment remains the midpoint after transformation). • With homogeneous coordinates, we can represent all 2 D affine transformations as 3 D linear transformations. • We can then use matrix multiplication to transform objects. 11 June 2021 ©College of Computer and Information Science, Northeastern University 30
Rotation about an Arbitrary Point y y φ φ x 11 June 2021 ©College of Computer and Information Science, Northeastern University x 31
Rotation about an Arbitrary Point y T(cx, cy) R(φ)T(-cx, -cy) φφ φφ 11 June 2021 x ©College of Computer and Information Science, Northeastern University 32
Windowing Transforms (A, B) (a, b) translate (A-a, B-b) scale (C, D) (C-c, D-d) translate (c, d) 11 June 2021 ©College of Computer and Information Science, Northeastern University 33
3 D Transformations Remember: A 3 D linear transformation can be represented by a 3 x 3 matrix. 11 June 2021 ©College of Computer and Information Science, Northeastern University 34
3 D Affine Transformations 11 June 2021 ©College of Computer and Information Science, Northeastern University 35
3 D Rotations 11 June 2021 ©College of Computer and Information Science, Northeastern University 36
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