In computer graphics and robotics, we can define (2D) scaling, rotations, and translations in Cartesian coordinates using the following operations: x = 0 S L] cos Ꮎ - sinᎾ x X = + sinᎾ cos s Ꮎ t ין 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 Sycos 0 tx] [x- ty 90-0 y = 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. HELP WITH -> Write a rigid body transformation in homogeneous coordinates that rotates the body 45 degrees (CCW) and translates it two units in the x direction and three units in the y direction.
In computer graphics and robotics, we can define (2D) scaling, rotations, and translations in Cartesian coordinates using the following operations: x = 0 S L] cos Ꮎ - sinᎾ x X = + sinᎾ cos s Ꮎ t ין 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 Sycos 0 tx] [x- ty 90-0 y = 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. HELP WITH -> Write a rigid body transformation in homogeneous coordinates that rotates the body 45 degrees (CCW) and translates it two units in the x direction and three units in the y direction.
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![In computer graphics and robotics, we can define (2D) scaling, rotations, and
translations in Cartesian coordinates using the following operations:
x
=
0
S
L]
cos Ꮎ - sinᎾ x
X
=
+
sinᎾ
cos
s Ꮎ
t
ין
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
Sycos
0
tx] [x-
ty
90-0
y =
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.
HELP
WITH ->
Write a rigid body transformation in homogeneous coordinates that rotates the
body 45 degrees (CCW) and translates it two units in the x direction and three
units in the y direction.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F963e3034-6a1f-473c-a768-afd904a0c57d%2Fbc8878ab-8e55-4910-bf9c-ad389c61461a%2Fv43yprj_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
S
L]
cos Ꮎ - sinᎾ x
X
=
+
sinᎾ
cos
s Ꮎ
t
ין
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
Sycos
0
tx] [x-
ty
90-0
y =
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.
HELP
WITH ->
Write a rigid body transformation in homogeneous coordinates that rotates the
body 45 degrees (CCW) and translates it two units in the x direction and three
units in the y direction.
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