3D Math Primer for Graphic and Game Development problem with Upright Space
$begingroup$
In the book in Chapter 3 he mentions Upright Space. I understood everything so far and I did every exercise from Chapter 2 with vectors correctly. But I cannot understand how did he get Upright Space unit vectors for x and y axis. Can someone help me here please?
So his example was this:
The origin of his object in world space is (4.5, 1.5) . .What I dont understand is how did he obtain these two vectors: x axis unit vector is (0.87, 0.50) and y: (−0.50, 0.87) . Can someone give me step by step solution please?
3.3.2 Specifying Coordinate Spaces
We are almost ready to talk about transformations. But there’s actually
one more basic question we should answer first: exactly how do we specify
a coordinate space relative to another coordinate space?7 Recall from Section
1.2.2 that a coordinate system is defined by its origin and axes. The
origin defines the position of the coordinate space, and the axes describe
its orientation. (Actually, the axes can describe other information, such as
scale and skew. For the moment, we assume that the axes are perpendicular
and the units used by the axes are the same as the units used by the
parent coordinate space.) So if we can find a way to describe the origin and
the axes, then we have fully documented the coordinate space.
Specifying the position of the coordinate space is straightforward. All
we have to do is describe the location of the origin. We do this just like we
do for any other point. Of course, we must express this point relative to the
parent coordinate space, not the local child space. The origin of the child
space, by definition, is always (0, 0, 0) when expressed in child coordinate
space. For example, consider the position of the 2D robot in Figure 3.2. To
establish a scale for the diagram, let’s say the robot is around 5 1/2 feet
tall. Then the world-space coordinates of her origin are close to (4.5, 1.5).
Specifying the orientation of a coordinate space in 3D is only slightly
more complicated. The axes are vectors (directions), and can be specified
like any other direction vector. Going back to our robot example, we could
describe her orientation by telling what directions the green vectors labeled
+x and +y were pointing—these are the axes of the robot’s object space.
(Actually, we would use vectors with unit length. The axes in the diagrams
were drawn as large as possible, but, as we see in just a moment, unit vectors
are usually used to describe the axes.) Just as with position, we do not
use the object space itself to describe the object-space axis directions, since
those coordinates are [1, 0] and [0, 1] by definition. Instead, the coordinates
are specified in upright space. In this example, unit vectors in the +x and
+y object-space directions have upright-space coordinates of [0.87, 0.50] and
[−0.50, 0.87], respectively.
What we have just described is one way to specify the orientation of
a coordinate space, but there are others. For example, in 2D, rather than
listing two 2D vectors, we could give a single angle. (The robot’s object axes
are rotated clockwise 30o relative to the upright axes.) In 3D, describing
orientation is considerably more complicated, and in fact we have devoted
all of Chapter 8 to the subject.
We specify a coordinate space by describing its origin and axes. The origin
is a point that defines the position of the space and can be described just
like any other point. The axes are vectors and describe the orientation of
the space (and possibly other information such as scale), and the usual tools
for describing vectors can be used. The coordinates we use to measure the
origin and axes must be relative to some other coordinate space.
EXERCISE AT THE END I KNOW FROM A TO E AFTER THAT NO IDEA:
Assume that the robot is at the position (1, 10, 3), and her right, up, and forward
vectors expressed in upright space are [0.866, 0,−0.500], [0, 1, 0], and
[0.500, 0, 0.866], respectively. (Note that these vectors form an orthonormal
basis.) The following points are expressed in object space. Calculate the
coordinates for these points in upright and world space.
(a) (−1, 2, 0)
(b) (1, 2, 0)
(c) (0, 0, 0)
(d) (1, 5, 0.5)
(e) (0, 5, 10)
The coordinates below are in world space. Transform these coordinates
from world space to upright space and object space.
(f) (1, 10, 3)
(g) (0, 0, 0)
(h) (2.732, 10, 2.000)
(i) (2, 11, 4)
(j) (1, 20, 3)
vectors coordinate-systems
asked Nov 30 '18 at 21:52
GoldSparkGoldSpark
235
$endgroup$
|
show 3 more comments
$begingroup$
In the book in Chapter 3 he mentions Upright Space. I understood everything so far and I did every exercise from Chapter 2 with vectors correctly. But I cannot understand how did he get Upright Space unit vectors for x and y axis. Can someone help me here please?
So his example was this:
The origin of his object in world space is (4.5, 1.5) . .What I dont understand is how did he obtain these two vectors: x axis unit vector is (0.87, 0.50) and y: (−0.50, 0.87) . Can someone give me step by step solution please?
3.3.2 Specifying Coordinate Spaces
We are almost ready to talk about transformations. But there’s actually
one more basic question we should answer first: exactly how do we specify
a coordinate space relative to another coordinate space?7 Recall from Section
1.2.2 that a coordinate system is defined by its origin and axes. The
origin defines the position of the coordinate space, and the axes describe
its orientation. (Actually, the axes can describe other information, such as
scale and skew. For the moment, we assume that the axes are perpendicular
and the units used by the axes are the same as the units used by the
parent coordinate space.) So if we can find a way to describe the origin and
the axes, then we have fully documented the coordinate space.
Specifying the position of the coordinate space is straightforward. All
we have to do is describe the location of the origin. We do this just like we
do for any other point. Of course, we must express this point relative to the
parent coordinate space, not the local child space. The origin of the child
space, by definition, is always (0, 0, 0) when expressed in child coordinate
space. For example, consider the position of the 2D robot in Figure 3.2. To
establish a scale for the diagram, let’s say the robot is around 5 1/2 feet
tall. Then the world-space coordinates of her origin are close to (4.5, 1.5).
Specifying the orientation of a coordinate space in 3D is only slightly
more complicated. The axes are vectors (directions), and can be specified
like any other direction vector. Going back to our robot example, we could
describe her orientation by telling what directions the green vectors labeled
+x and +y were pointing—these are the axes of the robot’s object space.
(Actually, we would use vectors with unit length. The axes in the diagrams
were drawn as large as possible, but, as we see in just a moment, unit vectors
are usually used to describe the axes.) Just as with position, we do not
use the object space itself to describe the object-space axis directions, since
those coordinates are [1, 0] and [0, 1] by definition. Instead, the coordinates
are specified in upright space. In this example, unit vectors in the +x and
+y object-space directions have upright-space coordinates of [0.87, 0.50] and
[−0.50, 0.87], respectively.
What we have just described is one way to specify the orientation of
a coordinate space, but there are others. For example, in 2D, rather than
listing two 2D vectors, we could give a single angle. (The robot’s object axes
are rotated clockwise 30o relative to the upright axes.) In 3D, describing
orientation is considerably more complicated, and in fact we have devoted
all of Chapter 8 to the subject.
We specify a coordinate space by describing its origin and axes. The origin
is a point that defines the position of the space and can be described just
like any other point. The axes are vectors and describe the orientation of
the space (and possibly other information such as scale), and the usual tools
for describing vectors can be used. The coordinates we use to measure the
origin and axes must be relative to some other coordinate space.
EXERCISE AT THE END I KNOW FROM A TO E AFTER THAT NO IDEA:
Assume that the robot is at the position (1, 10, 3), and her right, up, and forward
vectors expressed in upright space are [0.866, 0,−0.500], [0, 1, 0], and
[0.500, 0, 0.866], respectively. (Note that these vectors form an orthonormal
basis.) The following points are expressed in object space. Calculate the
coordinates for these points in upright and world space.
(a) (−1, 2, 0)
(b) (1, 2, 0)
(c) (0, 0, 0)
(d) (1, 5, 0.5)
(e) (0, 5, 10)
The coordinates below are in world space. Transform these coordinates
from world space to upright space and object space.
(f) (1, 10, 3)
(g) (0, 0, 0)
(h) (2.732, 10, 2.000)
(i) (2, 11, 4)
(j) (1, 20, 3)
vectors coordinate-systems
asked Nov 30 '18 at 21:52
GoldSparkGoldSpark
235
$endgroup$
$begingroup$
A solution to what? Please read How To Ask A Good Question and update yours accordingly.
$endgroup$
– amd
Dec 1 '18 at 0:12
$begingroup$
Upright Space is simply the world space with a translated origin. The coordinate axis directions are the same in both.
$endgroup$
– amd
Dec 1 '18 at 0:13
$begingroup$
Ah sorry yeah I updated I just want to know how did he get those two vectors.. I tried to convert 4.5 and 1.5 to unit but it did not gice me values like those...
$endgroup$
– GoldSpark
Dec 1 '18 at 0:53
$begingroup$
Of course not. The world-space coordinates of the object in and of themselves don’t tell you anything about the axis directions. In both world space and upright space the unit axis vectors are simply $(1,0)$ and $(0,1)$, so those other vectors are relative to some other coordinate system that you haven’t described here. Once again, without your providing more information in your question, there’s little anyone can do to help you.
$endgroup$
– amd
Dec 1 '18 at 1:15
$begingroup$
Ah okay... I dont know how to give this additional information because that was all. I can quote what he wrote there in that section of the book?
$endgroup$
– GoldSpark
Dec 1 '18 at 1:30
|
show 3 more comments
$begingroup$
In the book in Chapter 3 he mentions Upright Space. I understood everything so far and I did every exercise from Chapter 2 with vectors correctly. But I cannot understand how did he get Upright Space unit vectors for x and y axis. Can someone help me here please?
So his example was this:
The origin of his object in world space is (4.5, 1.5) . .What I dont understand is how did he obtain these two vectors: x axis unit vector is (0.87, 0.50) and y: (−0.50, 0.87) . Can someone give me step by step solution please?
3.3.2 Specifying Coordinate Spaces
We are almost ready to talk about transformations. But there’s actually
one more basic question we should answer first: exactly how do we specify
a coordinate space relative to another coordinate space?7 Recall from Section
1.2.2 that a coordinate system is defined by its origin and axes. The
origin defines the position of the coordinate space, and the axes describe
its orientation. (Actually, the axes can describe other information, such as
scale and skew. For the moment, we assume that the axes are perpendicular
and the units used by the axes are the same as the units used by the
parent coordinate space.) So if we can find a way to describe the origin and
the axes, then we have fully documented the coordinate space.
Specifying the position of the coordinate space is straightforward. All
we have to do is describe the location of the origin. We do this just like we
do for any other point. Of course, we must express this point relative to the
parent coordinate space, not the local child space. The origin of the child
space, by definition, is always (0, 0, 0) when expressed in child coordinate
space. For example, consider the position of the 2D robot in Figure 3.2. To
establish a scale for the diagram, let’s say the robot is around 5 1/2 feet
tall. Then the world-space coordinates of her origin are close to (4.5, 1.5).
Specifying the orientation of a coordinate space in 3D is only slightly
more complicated. The axes are vectors (directions), and can be specified
like any other direction vector. Going back to our robot example, we could
describe her orientation by telling what directions the green vectors labeled
+x and +y were pointing—these are the axes of the robot’s object space.
(Actually, we would use vectors with unit length. The axes in the diagrams
were drawn as large as possible, but, as we see in just a moment, unit vectors
are usually used to describe the axes.) Just as with position, we do not
use the object space itself to describe the object-space axis directions, since
those coordinates are [1, 0] and [0, 1] by definition. Instead, the coordinates
are specified in upright space. In this example, unit vectors in the +x and
+y object-space directions have upright-space coordinates of [0.87, 0.50] and
[−0.50, 0.87], respectively.
What we have just described is one way to specify the orientation of
a coordinate space, but there are others. For example, in 2D, rather than
listing two 2D vectors, we could give a single angle. (The robot’s object axes
are rotated clockwise 30o relative to the upright axes.) In 3D, describing
orientation is considerably more complicated, and in fact we have devoted
all of Chapter 8 to the subject.
We specify a coordinate space by describing its origin and axes. The origin
is a point that defines the position of the space and can be described just
like any other point. The axes are vectors and describe the orientation of
the space (and possibly other information such as scale), and the usual tools
for describing vectors can be used. The coordinates we use to measure the
origin and axes must be relative to some other coordinate space.
EXERCISE AT THE END I KNOW FROM A TO E AFTER THAT NO IDEA:
Assume that the robot is at the position (1, 10, 3), and her right, up, and forward
vectors expressed in upright space are [0.866, 0,−0.500], [0, 1, 0], and
[0.500, 0, 0.866], respectively. (Note that these vectors form an orthonormal
basis.) The following points are expressed in object space. Calculate the
coordinates for these points in upright and world space.
(a) (−1, 2, 0)
(b) (1, 2, 0)
(c) (0, 0, 0)
(d) (1, 5, 0.5)
(e) (0, 5, 10)
The coordinates below are in world space. Transform these coordinates
from world space to upright space and object space.
(f) (1, 10, 3)
(g) (0, 0, 0)
(h) (2.732, 10, 2.000)
(i) (2, 11, 4)
(j) (1, 20, 3)
vectors coordinate-systems
asked Nov 30 '18 at 21:52
GoldSparkGoldSpark
235
$endgroup$
In the book in Chapter 3 he mentions Upright Space. I understood everything so far and I did every exercise from Chapter 2 with vectors correctly. But I cannot understand how did he get Upright Space unit vectors for x and y axis. Can someone help me here please?
So his example was this:
The origin of his object in world space is (4.5, 1.5) . .What I dont understand is how did he obtain these two vectors: x axis unit vector is (0.87, 0.50) and y: (−0.50, 0.87) . Can someone give me step by step solution please?
3.3.2 Specifying Coordinate Spaces
We are almost ready to talk about transformations. But there’s actually
one more basic question we should answer first: exactly how do we specify
a coordinate space relative to another coordinate space?7 Recall from Section
1.2.2 that a coordinate system is defined by its origin and axes. The
origin defines the position of the coordinate space, and the axes describe
its orientation. (Actually, the axes can describe other information, such as
scale and skew. For the moment, we assume that the axes are perpendicular
and the units used by the axes are the same as the units used by the
parent coordinate space.) So if we can find a way to describe the origin and
the axes, then we have fully documented the coordinate space.
Specifying the position of the coordinate space is straightforward. All
we have to do is describe the location of the origin. We do this just like we
do for any other point. Of course, we must express this point relative to the
parent coordinate space, not the local child space. The origin of the child
space, by definition, is always (0, 0, 0) when expressed in child coordinate
space. For example, consider the position of the 2D robot in Figure 3.2. To
establish a scale for the diagram, let’s say the robot is around 5 1/2 feet
tall. Then the world-space coordinates of her origin are close to (4.5, 1.5).
Specifying the orientation of a coordinate space in 3D is only slightly
more complicated. The axes are vectors (directions), and can be specified
like any other direction vector. Going back to our robot example, we could
describe her orientation by telling what directions the green vectors labeled
+x and +y were pointing—these are the axes of the robot’s object space.
(Actually, we would use vectors with unit length. The axes in the diagrams
were drawn as large as possible, but, as we see in just a moment, unit vectors
are usually used to describe the axes.) Just as with position, we do not
use the object space itself to describe the object-space axis directions, since
those coordinates are [1, 0] and [0, 1] by definition. Instead, the coordinates
are specified in upright space. In this example, unit vectors in the +x and
+y object-space directions have upright-space coordinates of [0.87, 0.50] and
[−0.50, 0.87], respectively.
What we have just described is one way to specify the orientation of
a coordinate space, but there are others. For example, in 2D, rather than
listing two 2D vectors, we could give a single angle. (The robot’s object axes
are rotated clockwise 30o relative to the upright axes.) In 3D, describing
orientation is considerably more complicated, and in fact we have devoted
all of Chapter 8 to the subject.
We specify a coordinate space by describing its origin and axes. The origin
is a point that defines the position of the space and can be described just
like any other point. The axes are vectors and describe the orientation of
the space (and possibly other information such as scale), and the usual tools
for describing vectors can be used. The coordinates we use to measure the
origin and axes must be relative to some other coordinate space.
EXERCISE AT THE END I KNOW FROM A TO E AFTER THAT NO IDEA:
Assume that the robot is at the position (1, 10, 3), and her right, up, and forward
vectors expressed in upright space are [0.866, 0,−0.500], [0, 1, 0], and
[0.500, 0, 0.866], respectively. (Note that these vectors form an orthonormal
basis.) The following points are expressed in object space. Calculate the
coordinates for these points in upright and world space.
(a) (−1, 2, 0)
(b) (1, 2, 0)
(c) (0, 0, 0)
(d) (1, 5, 0.5)
(e) (0, 5, 10)
The coordinates below are in world space. Transform these coordinates
from world space to upright space and object space.
(f) (1, 10, 3)
(g) (0, 0, 0)
(h) (2.732, 10, 2.000)
(i) (2, 11, 4)
(j) (1, 20, 3)
vectors coordinate-systems
vectors coordinate-systems
asked Nov 30 '18 at 21:52
GoldSparkGoldSpark
235
asked Nov 30 '18 at 21:52
GoldSparkGoldSpark
235
edited Dec 2 '18 at 18:08
GoldSpark
asked Nov 30 '18 at 21:52
GoldSparkGoldSpark
235
asked Nov 30 '18 at 21:52
GoldSparkGoldSpark
235
asked Nov 30 '18 at 21:52
GoldSparkGoldSpark
235
235
$begingroup$
A solution to what? Please read How To Ask A Good Question and update yours accordingly.
$endgroup$
– amd
Dec 1 '18 at 0:12
$begingroup$
Upright Space is simply the world space with a translated origin. The coordinate axis directions are the same in both.
$endgroup$
– amd
Dec 1 '18 at 0:13
$begingroup$
Ah sorry yeah I updated I just want to know how did he get those two vectors.. I tried to convert 4.5 and 1.5 to unit but it did not gice me values like those...
$endgroup$
– GoldSpark
Dec 1 '18 at 0:53
$begingroup$
Of course not. The world-space coordinates of the object in and of themselves don’t tell you anything about the axis directions. In both world space and upright space the unit axis vectors are simply $(1,0)$ and $(0,1)$, so those other vectors are relative to some other coordinate system that you haven’t described here. Once again, without your providing more information in your question, there’s little anyone can do to help you.
$endgroup$
– amd
Dec 1 '18 at 1:15
$begingroup$
Ah okay... I dont know how to give this additional information because that was all. I can quote what he wrote there in that section of the book?
$endgroup$
– GoldSpark
Dec 1 '18 at 1:30
|
show 3 more comments
$begingroup$
A solution to what? Please read How To Ask A Good Question and update yours accordingly.
$endgroup$
– amd
Dec 1 '18 at 0:12
$begingroup$
Upright Space is simply the world space with a translated origin. The coordinate axis directions are the same in both.
$endgroup$
– amd
Dec 1 '18 at 0:13
$begingroup$
Ah sorry yeah I updated I just want to know how did he get those two vectors.. I tried to convert 4.5 and 1.5 to unit but it did not gice me values like those...
$endgroup$
– GoldSpark
Dec 1 '18 at 0:53
$begingroup$
Of course not. The world-space coordinates of the object in and of themselves don’t tell you anything about the axis directions. In both world space and upright space the unit axis vectors are simply $(1,0)$ and $(0,1)$, so those other vectors are relative to some other coordinate system that you haven’t described here. Once again, without your providing more information in your question, there’s little anyone can do to help you.
$endgroup$
– amd
Dec 1 '18 at 1:15
$begingroup$
Ah okay... I dont know how to give this additional information because that was all. I can quote what he wrote there in that section of the book?
$endgroup$
– GoldSpark
Dec 1 '18 at 1:30
$begingroup$
A solution to what? Please read How To Ask A Good Question and update yours accordingly.
$endgroup$
– amd
Dec 1 '18 at 0:12
$begingroup$
A solution to what? Please read How To Ask A Good Question and update yours accordingly.
$endgroup$
– amd
Dec 1 '18 at 0:12
$begingroup$
Upright Space is simply the world space with a translated origin. The coordinate axis directions are the same in both.
$endgroup$
– amd
Dec 1 '18 at 0:13
$begingroup$
Upright Space is simply the world space with a translated origin. The coordinate axis directions are the same in both.
$endgroup$
– amd
Dec 1 '18 at 0:13
$begingroup$
Ah sorry yeah I updated I just want to know how did he get those two vectors.. I tried to convert 4.5 and 1.5 to unit but it did not gice me values like those...
$endgroup$
– GoldSpark
Dec 1 '18 at 0:53
$begingroup$
Ah sorry yeah I updated I just want to know how did he get those two vectors.. I tried to convert 4.5 and 1.5 to unit but it did not gice me values like those...
$endgroup$
– GoldSpark
Dec 1 '18 at 0:53
$begingroup$
Of course not. The world-space coordinates of the object in and of themselves don’t tell you anything about the axis directions. In both world space and upright space the unit axis vectors are simply $(1,0)$ and $(0,1)$, so those other vectors are relative to some other coordinate system that you haven’t described here. Once again, without your providing more information in your question, there’s little anyone can do to help you.
$endgroup$
– amd
Dec 1 '18 at 1:15
$begingroup$
Of course not. The world-space coordinates of the object in and of themselves don’t tell you anything about the axis directions. In both world space and upright space the unit axis vectors are simply $(1,0)$ and $(0,1)$, so those other vectors are relative to some other coordinate system that you haven’t described here. Once again, without your providing more information in your question, there’s little anyone can do to help you.
$endgroup$
– amd
Dec 1 '18 at 1:15
$begingroup$
Ah okay... I dont know how to give this additional information because that was all. I can quote what he wrote there in that section of the book?
$endgroup$
– GoldSpark
Dec 1 '18 at 1:30
$begingroup$
Ah okay... I dont know how to give this additional information because that was all. I can quote what he wrote there in that section of the book?
$endgroup$
– GoldSpark
Dec 1 '18 at 1:30
|
show 3 more comments
1 Answer
1
active
oldest
votes
$begingroup$
The key sentence is a parenthetical remark toward the end of that excerpt:
The robot’s object axes are rotated clockwise 30° relative to the
upright axes.
So, relative to both the world and upright spaces, which share the same axis directions, the object space unit $x$-vector is $$(cos(30°),sin(30°)) = left(frac{sqrt3}2,frac12right) approx (0.87,0.50)$$ and the unit $y$-vector is $$(-sin(30°),cos(30°)) = left(-frac12,frac{sqrt3}2right) approx (-0.50,0.87).$$ (I’ve inferred that in this text positive angles represent clockwise rotations.) I expect that you’ll be able to find a detailed explanation if you need one of why these are the vectors that result from a 30-degree rotation of the coordinate axes back in Chapter 2.
answered Dec 2 '18 at 0:04
amdamd
29.5k21050
$endgroup$
$begingroup$
Thank a lot man!! I thought I didnt have to use trigonometry there but seems like I have. Thank you!
$endgroup$
– GoldSpark
Dec 2 '18 at 15:22
$begingroup$
Also what he meant : "This is one way to specify orientation that we have just described" after that he wrote your solution which is another way?
$endgroup$
– GoldSpark
Dec 2 '18 at 15:40
$begingroup$
Because exercise at the end does not give angle so I have no idea what to do..I updated the description .. I know from a) to e) but from f) to j) I have no idea what to do
$endgroup$
– GoldSpark
Dec 2 '18 at 18:06
$begingroup$
And answer to g) is Upright: (−1.000,−10.000,−3.000); Object: (0.634,−10.000,−3.098) .... and g) is only 0,0,0 .. WHAT THE HELL
$endgroup$
– GoldSpark
Dec 2 '18 at 18:22
1
$begingroup$
@GoldSpark The key to solving the problems in the exercise is to remember that you can construct the appropriate rotation matrix directly from the right/up/forward vectors of the object.
$endgroup$
– amd
Dec 2 '18 at 20:02
|
show 1 more comment
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The key sentence is a parenthetical remark toward the end of that excerpt:
The robot’s object axes are rotated clockwise 30° relative to the
upright axes.
So, relative to both the world and upright spaces, which share the same axis directions, the object space unit $x$-vector is $$(cos(30°),sin(30°)) = left(frac{sqrt3}2,frac12right) approx (0.87,0.50)$$ and the unit $y$-vector is $$(-sin(30°),cos(30°)) = left(-frac12,frac{sqrt3}2right) approx (-0.50,0.87).$$ (I’ve inferred that in this text positive angles represent clockwise rotations.) I expect that you’ll be able to find a detailed explanation if you need one of why these are the vectors that result from a 30-degree rotation of the coordinate axes back in Chapter 2.
answered Dec 2 '18 at 0:04
amdamd
29.5k21050
$endgroup$
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Thank a lot man!! I thought I didnt have to use trigonometry there but seems like I have. Thank you!
$endgroup$
– GoldSpark
Dec 2 '18 at 15:22
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Also what he meant : "This is one way to specify orientation that we have just described" after that he wrote your solution which is another way?
$endgroup$
– GoldSpark
Dec 2 '18 at 15:40
$begingroup$
Because exercise at the end does not give angle so I have no idea what to do..I updated the description .. I know from a) to e) but from f) to j) I have no idea what to do
$endgroup$
– GoldSpark
Dec 2 '18 at 18:06
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And answer to g) is Upright: (−1.000,−10.000,−3.000); Object: (0.634,−10.000,−3.098) .... and g) is only 0,0,0 .. WHAT THE HELL
$endgroup$
– GoldSpark
Dec 2 '18 at 18:22
1
$begingroup$
@GoldSpark The key to solving the problems in the exercise is to remember that you can construct the appropriate rotation matrix directly from the right/up/forward vectors of the object.
$endgroup$
– amd
Dec 2 '18 at 20:02
|
show 1 more comment
$begingroup$
The key sentence is a parenthetical remark toward the end of that excerpt:
The robot’s object axes are rotated clockwise 30° relative to the
upright axes.
So, relative to both the world and upright spaces, which share the same axis directions, the object space unit $x$-vector is $$(cos(30°),sin(30°)) = left(frac{sqrt3}2,frac12right) approx (0.87,0.50)$$ and the unit $y$-vector is $$(-sin(30°),cos(30°)) = left(-frac12,frac{sqrt3}2right) approx (-0.50,0.87).$$ (I’ve inferred that in this text positive angles represent clockwise rotations.) I expect that you’ll be able to find a detailed explanation if you need one of why these are the vectors that result from a 30-degree rotation of the coordinate axes back in Chapter 2.
answered Dec 2 '18 at 0:04
amdamd
29.5k21050
$endgroup$
$begingroup$
Thank a lot man!! I thought I didnt have to use trigonometry there but seems like I have. Thank you!
$endgroup$
– GoldSpark
Dec 2 '18 at 15:22
$begingroup$
Also what he meant : "This is one way to specify orientation that we have just described" after that he wrote your solution which is another way?
$endgroup$
– GoldSpark
Dec 2 '18 at 15:40
$begingroup$
Because exercise at the end does not give angle so I have no idea what to do..I updated the description .. I know from a) to e) but from f) to j) I have no idea what to do
$endgroup$
– GoldSpark
Dec 2 '18 at 18:06
$begingroup$
And answer to g) is Upright: (−1.000,−10.000,−3.000); Object: (0.634,−10.000,−3.098) .... and g) is only 0,0,0 .. WHAT THE HELL
$endgroup$
– GoldSpark
Dec 2 '18 at 18:22
1
$begingroup$
@GoldSpark The key to solving the problems in the exercise is to remember that you can construct the appropriate rotation matrix directly from the right/up/forward vectors of the object.
$endgroup$
– amd
Dec 2 '18 at 20:02
|
show 1 more comment
$begingroup$
The key sentence is a parenthetical remark toward the end of that excerpt:
The robot’s object axes are rotated clockwise 30° relative to the
upright axes.
So, relative to both the world and upright spaces, which share the same axis directions, the object space unit $x$-vector is $$(cos(30°),sin(30°)) = left(frac{sqrt3}2,frac12right) approx (0.87,0.50)$$ and the unit $y$-vector is $$(-sin(30°),cos(30°)) = left(-frac12,frac{sqrt3}2right) approx (-0.50,0.87).$$ (I’ve inferred that in this text positive angles represent clockwise rotations.) I expect that you’ll be able to find a detailed explanation if you need one of why these are the vectors that result from a 30-degree rotation of the coordinate axes back in Chapter 2.
answered Dec 2 '18 at 0:04
amdamd
29.5k21050
$endgroup$
The key sentence is a parenthetical remark toward the end of that excerpt:
The robot’s object axes are rotated clockwise 30° relative to the
upright axes.
So, relative to both the world and upright spaces, which share the same axis directions, the object space unit $x$-vector is $$(cos(30°),sin(30°)) = left(frac{sqrt3}2,frac12right) approx (0.87,0.50)$$ and the unit $y$-vector is $$(-sin(30°),cos(30°)) = left(-frac12,frac{sqrt3}2right) approx (-0.50,0.87).$$ (I’ve inferred that in this text positive angles represent clockwise rotations.) I expect that you’ll be able to find a detailed explanation if you need one of why these are the vectors that result from a 30-degree rotation of the coordinate axes back in Chapter 2.
answered Dec 2 '18 at 0:04
amdamd
29.5k21050
answered Dec 2 '18 at 0:04
amdamd
29.5k21050
answered Dec 2 '18 at 0:04
amdamd
29.5k21050
answered Dec 2 '18 at 0:04
amdamd
29.5k21050
29.5k21050
$begingroup$
Thank a lot man!! I thought I didnt have to use trigonometry there but seems like I have. Thank you!
$endgroup$
– GoldSpark
Dec 2 '18 at 15:22
$begingroup$
Also what he meant : "This is one way to specify orientation that we have just described" after that he wrote your solution which is another way?
$endgroup$
– GoldSpark
Dec 2 '18 at 15:40
$begingroup$
Because exercise at the end does not give angle so I have no idea what to do..I updated the description .. I know from a) to e) but from f) to j) I have no idea what to do
$endgroup$
– GoldSpark
Dec 2 '18 at 18:06
$begingroup$
And answer to g) is Upright: (−1.000,−10.000,−3.000); Object: (0.634,−10.000,−3.098) .... and g) is only 0,0,0 .. WHAT THE HELL
$endgroup$
– GoldSpark
Dec 2 '18 at 18:22
1
$begingroup$
@GoldSpark The key to solving the problems in the exercise is to remember that you can construct the appropriate rotation matrix directly from the right/up/forward vectors of the object.
$endgroup$
– amd
Dec 2 '18 at 20:02
|
show 1 more comment
$begingroup$
Thank a lot man!! I thought I didnt have to use trigonometry there but seems like I have. Thank you!
$endgroup$
– GoldSpark
Dec 2 '18 at 15:22
$begingroup$
Also what he meant : "This is one way to specify orientation that we have just described" after that he wrote your solution which is another way?
$endgroup$
– GoldSpark
Dec 2 '18 at 15:40
$begingroup$
Because exercise at the end does not give angle so I have no idea what to do..I updated the description .. I know from a) to e) but from f) to j) I have no idea what to do
$endgroup$
– GoldSpark
Dec 2 '18 at 18:06
$begingroup$
And answer to g) is Upright: (−1.000,−10.000,−3.000); Object: (0.634,−10.000,−3.098) .... and g) is only 0,0,0 .. WHAT THE HELL
$endgroup$
– GoldSpark
Dec 2 '18 at 18:22
1
$begingroup$
@GoldSpark The key to solving the problems in the exercise is to remember that you can construct the appropriate rotation matrix directly from the right/up/forward vectors of the object.
$endgroup$
– amd
Dec 2 '18 at 20:02
$begingroup$
Thank a lot man!! I thought I didnt have to use trigonometry there but seems like I have. Thank you!
$endgroup$
– GoldSpark
Dec 2 '18 at 15:22
$begingroup$
Thank a lot man!! I thought I didnt have to use trigonometry there but seems like I have. Thank you!
$endgroup$
– GoldSpark
Dec 2 '18 at 15:22
$begingroup$
Also what he meant : "This is one way to specify orientation that we have just described" after that he wrote your solution which is another way?
$endgroup$
– GoldSpark
Dec 2 '18 at 15:40
$begingroup$
Also what he meant : "This is one way to specify orientation that we have just described" after that he wrote your solution which is another way?
$endgroup$
– GoldSpark
Dec 2 '18 at 15:40
$begingroup$
Because exercise at the end does not give angle so I have no idea what to do..I updated the description .. I know from a) to e) but from f) to j) I have no idea what to do
$endgroup$
– GoldSpark
Dec 2 '18 at 18:06
$begingroup$
Because exercise at the end does not give angle so I have no idea what to do..I updated the description .. I know from a) to e) but from f) to j) I have no idea what to do
$endgroup$
– GoldSpark
Dec 2 '18 at 18:06
$begingroup$
And answer to g) is Upright: (−1.000,−10.000,−3.000); Object: (0.634,−10.000,−3.098) .... and g) is only 0,0,0 .. WHAT THE HELL
$endgroup$
– GoldSpark
Dec 2 '18 at 18:22
$begingroup$
And answer to g) is Upright: (−1.000,−10.000,−3.000); Object: (0.634,−10.000,−3.098) .... and g) is only 0,0,0 .. WHAT THE HELL
$endgroup$
– GoldSpark
Dec 2 '18 at 18:22
1
1
$begingroup$
@GoldSpark The key to solving the problems in the exercise is to remember that you can construct the appropriate rotation matrix directly from the right/up/forward vectors of the object.
$endgroup$
– amd
Dec 2 '18 at 20:02
$begingroup$
@GoldSpark The key to solving the problems in the exercise is to remember that you can construct the appropriate rotation matrix directly from the right/up/forward vectors of the object.
$endgroup$
– amd
Dec 2 '18 at 20:02
|
show 1 more comment
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A solution to what? Please read How To Ask A Good Question and update yours accordingly.
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– amd
Dec 1 '18 at 0:12
$begingroup$
Upright Space is simply the world space with a translated origin. The coordinate axis directions are the same in both.
$endgroup$
– amd
Dec 1 '18 at 0:13
$begingroup$
Ah sorry yeah I updated I just want to know how did he get those two vectors.. I tried to convert 4.5 and 1.5 to unit but it did not gice me values like those...
$endgroup$
– GoldSpark
Dec 1 '18 at 0:53
$begingroup$
Of course not. The world-space coordinates of the object in and of themselves don’t tell you anything about the axis directions. In both world space and upright space the unit axis vectors are simply $(1,0)$ and $(0,1)$, so those other vectors are relative to some other coordinate system that you haven’t described here. Once again, without your providing more information in your question, there’s little anyone can do to help you.
$endgroup$
– amd
Dec 1 '18 at 1:15
$begingroup$
Ah okay... I dont know how to give this additional information because that was all. I can quote what he wrote there in that section of the book?
$endgroup$
– GoldSpark
Dec 1 '18 at 1:30