In computer graphics and robotics, we can define (2D) scaling, rotations, and translations in Cartesian coordinates using the following operations: x 0 X x x' cos-sinð x x' = = sinᎾ cos Ꮎ + [M]-6 ©D] M-0-0 S That is, scaling and rotations are done using matrix multiplication and translation by adding a vector. We can combine these three operations (in that order, scaling, roation, and then translation) into a single matrix multiplication using homogeneous coordinates. In homogeneous coordinates, we add a third coordinate to a point (vector). Instead of being represented by a pair of numbers (x, y), each point (vector) is represented as a triple (x, y, w) where w 0. For convenience, w is usually taken to be 1. Thus, [Sx cos - sysin 0 Sx sin e 0 Sycose 0 tx] [x₁ 96-8 ty = transforms the vector (x, y)' into a new vector that has been scaled in the x direction by S and in the y direction by sy, rotated counter-clockwise by 0, and translated by (tx, ty)'. Note that in robotics, the these operations are done on rigid bodies so the scaling is always unity.
In computer graphics and robotics, we can define (2D) scaling, rotations, and translations in Cartesian coordinates using the following operations: x 0 X x x' cos-sinð x x' = = sinᎾ cos Ꮎ + [M]-6 ©D] M-0-0 S That is, scaling and rotations are done using matrix multiplication and translation by adding a vector. We can combine these three operations (in that order, scaling, roation, and then translation) into a single matrix multiplication using homogeneous coordinates. In homogeneous coordinates, we add a third coordinate to a point (vector). Instead of being represented by a pair of numbers (x, y), each point (vector) is represented as a triple (x, y, w) where w 0. For convenience, w is usually taken to be 1. Thus, [Sx cos - sysin 0 Sx sin e 0 Sycose 0 tx] [x₁ 96-8 ty = transforms the vector (x, y)' into a new vector that has been scaled in the x direction by S and in the y direction by sy, rotated counter-clockwise by 0, and translated by (tx, ty)'. Note that in robotics, the these operations are done on rigid bodies so the scaling is always unity.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
Help by write cartesian transformation in homogeneous coordinates that scales the object by two in each direction, then rotates the body 30 degrees clockwise, and translates it four units in each direction. Read attached picture below for more context.
![In computer graphics and robotics, we can define (2D) scaling, rotations, and
translations in Cartesian coordinates using the following operations:
x
0
X
x
x'
cos-sinð
x
x'
=
=
sinᎾ
cos Ꮎ
+
[M]-6 ©D] M-0-0
S
That is, scaling and rotations are done using matrix multiplication and translation by adding a
vector. We can combine these three operations (in that order, scaling, roation, and then
translation) into a single matrix multiplication using homogeneous coordinates. In
homogeneous coordinates, we add a third coordinate to a point (vector). Instead of being
represented by a pair of numbers (x, y), each point (vector) is represented as a triple (x, y, w)
where w 0. For convenience, w is usually taken to be 1. Thus,
[Sx cos - sysin 0
Sx sin e
0
Sycose
0
tx] [x₁
96-8
ty
=
transforms the vector (x, y)' into a new vector that has been scaled in the x direction by
S and in the y direction by sy, rotated counter-clockwise by 0, and translated by (tx, ty)'.
Note that in robotics, the these operations are done on rigid bodies so the scaling is always
unity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F963e3034-6a1f-473c-a768-afd904a0c57d%2F7635dec9-6ae9-42f9-90bd-7829189bb833%2F5nfkv3_processed.png&w=3840&q=75)
Transcribed Image Text:In computer graphics and robotics, we can define (2D) scaling, rotations, and
translations in Cartesian coordinates using the following operations:
x
0
X
x
x'
cos-sinð
x
x'
=
=
sinᎾ
cos Ꮎ
+
[M]-6 ©D] M-0-0
S
That is, scaling and rotations are done using matrix multiplication and translation by adding a
vector. We can combine these three operations (in that order, scaling, roation, and then
translation) into a single matrix multiplication using homogeneous coordinates. In
homogeneous coordinates, we add a third coordinate to a point (vector). Instead of being
represented by a pair of numbers (x, y), each point (vector) is represented as a triple (x, y, w)
where w 0. For convenience, w is usually taken to be 1. Thus,
[Sx cos - sysin 0
Sx sin e
0
Sycose
0
tx] [x₁
96-8
ty
=
transforms the vector (x, y)' into a new vector that has been scaled in the x direction by
S and in the y direction by sy, rotated counter-clockwise by 0, and translated by (tx, ty)'.
Note that in robotics, the these operations are done on rigid bodies so the scaling is always
unity.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning