Supplementary Material: The Rotation Matrix

Size: px

Start display at page:

Download "Supplementary Material: The Rotation Matrix"

Victor Hodges
5 years ago
Views:

1 Supplementary Material: The Rotation Matrix Computer Science 4766/6778 Department of Computer Science Memorial University of Newfoundland January 16, 2014 COMP 4766/6778 (MUN) The Rotation Matrix January 16, / 4

2 Assume we have a vertex at (x, y) which is to be rotated counterclockwise about the origin by an angle θ COMP 4766/6778 (MUN) The Rotation Matrix January 16, / 4

3 Assume we have a vertex at (x, y) which is to be rotated counterclockwise about the origin by an angle θ In polar coordinates, this vertex is at (r, φ); We can express this in Cartesian coordinates: COMP 4766/6778 (MUN) The Rotation Matrix January 16, / 4

4 Assume we have a vertex at (x, y) which is to be rotated counterclockwise about the origin by an angle θ In polar coordinates, this vertex is at (r, φ); We can express this in Cartesian coordinates: COMP 4766/6778 (MUN) The Rotation Matrix January 16, / 4

5 Assume we have a vertex at (x, y) which is to be rotated counterclockwise about the origin by an angle θ In polar coordinates, this vertex is at (r, φ); We can express this in Cartesian coordinates: x = r cos φ y = r sin φ COMP 4766/6778 (MUN) The Rotation Matrix January 16, / 4

6 Assume we have a vertex at (x, y) which is to be rotated counterclockwise about the origin by an angle θ In polar coordinates, this vertex is at (r, φ); We can express this in Cartesian coordinates: x = r cos φ y = r sin φ Now the rotation by θ can be understood as an addition of angles: COMP 4766/6778 (MUN) The Rotation Matrix January 16, / 4

7 Assume we have a vertex at (x, y) which is to be rotated counterclockwise about the origin by an angle θ In polar coordinates, this vertex is at (r, φ); We can express this in Cartesian coordinates: x = r cos φ y = r sin φ Now the rotation by θ can be understood as an addition of angles: COMP 4766/6778 (MUN) The Rotation Matrix January 16, / 4

8 Assume we have a vertex at (x, y) which is to be rotated counterclockwise about the origin by an angle θ In polar coordinates, this vertex is at (r, φ); We can express this in Cartesian coordinates: x = r cos φ y = r sin φ Now the rotation by θ can be understood as an addition of angles: y y r θ φ x r x COMP 4766/6778 (MUN) The Rotation Matrix January 16, / 4

9 The coordinates of the rotated vertex are as follows:

10 The coordinates of the rotated vertex are as follows:

11 The coordinates of the rotated vertex are as follows: x = r cos(θ + φ) y = r sin(θ + φ)

12 The coordinates of the rotated vertex are as follows: x = r cos(θ + φ) y = r sin(θ + φ) We can now make use of the following trigonometric identities:

13 The coordinates of the rotated vertex are as follows: x = r cos(θ + φ) y = r sin(θ + φ) We can now make use of the following trigonometric identities:

14 The coordinates of the rotated vertex are as follows: x = r cos(θ + φ) y = r sin(θ + φ) We can now make use of the following trigonometric identities: cos(a + b) = cos a cos b sin a sin b sin(a + b) = sin a cos b + cos a sin b

15 The coordinates of the rotated vertex are as follows: x = r cos(θ + φ) y = r sin(θ + φ) We can now make use of the following trigonometric identities: cos(a + b) = cos a cos b sin a sin b sin(a + b) = sin a cos b + cos a sin b After a few steps (COVERED ON BOARD) we obtain,

16 The coordinates of the rotated vertex are as follows: x = r cos(θ + φ) y = r sin(θ + φ) We can now make use of the following trigonometric identities: cos(a + b) = cos a cos b sin a sin b sin(a + b) = sin a cos b + cos a sin b After a few steps (COVERED ON BOARD) we obtain,

17 The coordinates of the rotated vertex are as follows: x = r cos(θ + φ) y = r sin(θ + φ) We can now make use of the following trigonometric identities: cos(a + b) = cos a cos b sin a sin b sin(a + b) = sin a cos b + cos a sin b After a few steps (COVERED ON BOARD) we obtain, x = x cos θ y sin θ y = x sin θ + y cos θ

18 In vector form, this can be written as: COMP 4766/6778 (MUN) The Rotation Matrix January 16, / 4

19 In vector form, this can be written as: COMP 4766/6778 (MUN) The Rotation Matrix January 16, / 4

20 In vector form, this can be written as: [ ] [ x cos θ sin θ y = sin θ cos θ ] [ x y ] COMP 4766/6778 (MUN) The Rotation Matrix January 16, / 4

21 In vector form, this can be written as: [ ] [ x cos θ sin θ y = sin θ cos θ ] [ x y ] or v = R ccw (θ)v COMP 4766/6778 (MUN) The Rotation Matrix January 16, / 4

22 In vector form, this can be written as: [ ] [ x cos θ sin θ y = sin θ cos θ ] [ x y ] or v = R ccw (θ)v COMP 4766/6778 (MUN) The Rotation Matrix January 16, / 4

23 In vector form, this can be written as: [ ] [ x cos θ sin θ y = sin θ cos θ ] [ x y ] or v = R ccw (θ)v where v is the original vertex, v is the rotated vertex, and R ccw is the counter-clockwise (hence ccw ) rotation matrix COMP 4766/6778 (MUN) The Rotation Matrix January 16, / 4

Unit 2: Trigonometry. This lesson is not covered in your workbook. It is a review of trigonometry topics from previous courses.

Unit 2: Trigonometry. This lesson is not covered in your workbook. It is a review of trigonometry topics from previous courses. Unit 2: Trigonometry This lesson is not covered in your workbook. It is a review of trigonometry topics from previous courses. Pythagorean Theorem Recall that, for any right angled triangle, the square