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.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.8: Determinants
Problem 25E
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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.
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