Quaternion that transforms a point like a 2D angle
$begingroup$
Am looking for a way to transpose a 2D solution of a problem to a 3D solution of the same problem.
The algorithm I've implemented in 2D works as follows:
Given the points $A (A_x, A_y)$ and $B (B_x, B_y)$.
- Compute the absolute angle of point $A$ via $text{atan2}(A_x, A_y) rightarrow A_alpha$
- Compute the absolute angle of point $B$ via $text{atan2}(B_x, B_y)rightarrow B_{alpha}$
- Compute the resulting point $C$ by point-distance of point $B$ and applying the angle $A_{alpha}+B_alpha$.
The effect is that $C$ is point $B$ further rotated around the origin $(0,0)$ by as much as point $A$ is rotated around origin $(0,0)$.
This image visualizes that:
2D Angles
What I need now is the same in the third dimension.
The points are threedimensional coordinates and obviously instead of angles I'll need quaternions.
Am already knowing how to apply quaternions to 3D points but I cannot get an idea how to compute the transformation quaternion that is the equivalent to $A_alpha$.
Could someone push me in the right direction perhaps?
Huge thanks in advance!
transformation rotations quaternions
edited Dec 11 '18 at 15:38
Widawensen
4,69321446
asked Dec 10 '18 at 16:16
DragonGamerDragonGamer
1163
$endgroup$
add a comment |
$begingroup$
Am looking for a way to transpose a 2D solution of a problem to a 3D solution of the same problem.
The algorithm I've implemented in 2D works as follows:
Given the points $A (A_x, A_y)$ and $B (B_x, B_y)$.
- Compute the absolute angle of point $A$ via $text{atan2}(A_x, A_y) rightarrow A_alpha$
- Compute the absolute angle of point $B$ via $text{atan2}(B_x, B_y)rightarrow B_{alpha}$
- Compute the resulting point $C$ by point-distance of point $B$ and applying the angle $A_{alpha}+B_alpha$.
The effect is that $C$ is point $B$ further rotated around the origin $(0,0)$ by as much as point $A$ is rotated around origin $(0,0)$.
This image visualizes that:
2D Angles
What I need now is the same in the third dimension.
The points are threedimensional coordinates and obviously instead of angles I'll need quaternions.
Am already knowing how to apply quaternions to 3D points but I cannot get an idea how to compute the transformation quaternion that is the equivalent to $A_alpha$.
Could someone push me in the right direction perhaps?
Huge thanks in advance!
transformation rotations quaternions
edited Dec 11 '18 at 15:38
Widawensen
4,69321446
asked Dec 10 '18 at 16:16
DragonGamerDragonGamer
1163
$endgroup$
$begingroup$
How is it “obvious” that you need a quaternion? There are many ways to express rotations. Anyway, you need to nail down some more details of how you want this rotation to work: Rotations take place in a plane. Which plane do you want to use in 3D? A natural choice is the plane defined by $A$ and $B$, but then from where are you measuring the angles?
$endgroup$
– amd
Dec 10 '18 at 23:48
$begingroup$
@amd Thank you for the comment! Well, the engine I'm working with (Robot Operating System ROS) is using almost exclusively quaternions for transformations. Sometimes euler angles. Hmm, in 2D my reference is a horizontal line. isn't there a trivial equivalent for that in 3D? In the 3D version I need that in the case that A.z and B.z are both 0, C.x und C.y should be exactly the same as if the whole setup were still 2D.
$endgroup$
– DragonGamer
Dec 11 '18 at 3:52
add a comment |
$begingroup$
Am looking for a way to transpose a 2D solution of a problem to a 3D solution of the same problem.
The algorithm I've implemented in 2D works as follows:
Given the points $A (A_x, A_y)$ and $B (B_x, B_y)$.
- Compute the absolute angle of point $A$ via $text{atan2}(A_x, A_y) rightarrow A_alpha$
- Compute the absolute angle of point $B$ via $text{atan2}(B_x, B_y)rightarrow B_{alpha}$
- Compute the resulting point $C$ by point-distance of point $B$ and applying the angle $A_{alpha}+B_alpha$.
The effect is that $C$ is point $B$ further rotated around the origin $(0,0)$ by as much as point $A$ is rotated around origin $(0,0)$.
This image visualizes that:
2D Angles
What I need now is the same in the third dimension.
The points are threedimensional coordinates and obviously instead of angles I'll need quaternions.
Am already knowing how to apply quaternions to 3D points but I cannot get an idea how to compute the transformation quaternion that is the equivalent to $A_alpha$.
Could someone push me in the right direction perhaps?
Huge thanks in advance!
transformation rotations quaternions
edited Dec 11 '18 at 15:38
Widawensen
4,69321446
asked Dec 10 '18 at 16:16
DragonGamerDragonGamer
1163
$endgroup$
Am looking for a way to transpose a 2D solution of a problem to a 3D solution of the same problem.
The algorithm I've implemented in 2D works as follows:
Given the points $A (A_x, A_y)$ and $B (B_x, B_y)$.
- Compute the absolute angle of point $A$ via $text{atan2}(A_x, A_y) rightarrow A_alpha$
- Compute the absolute angle of point $B$ via $text{atan2}(B_x, B_y)rightarrow B_{alpha}$
- Compute the resulting point $C$ by point-distance of point $B$ and applying the angle $A_{alpha}+B_alpha$.
The effect is that $C$ is point $B$ further rotated around the origin $(0,0)$ by as much as point $A$ is rotated around origin $(0,0)$.
This image visualizes that:
2D Angles
What I need now is the same in the third dimension.
The points are threedimensional coordinates and obviously instead of angles I'll need quaternions.
Am already knowing how to apply quaternions to 3D points but I cannot get an idea how to compute the transformation quaternion that is the equivalent to $A_alpha$.
Could someone push me in the right direction perhaps?
Huge thanks in advance!
transformation rotations quaternions
transformation rotations quaternions
edited Dec 11 '18 at 15:38
Widawensen
4,69321446
asked Dec 10 '18 at 16:16
DragonGamerDragonGamer
1163
edited Dec 11 '18 at 15:38
Widawensen
4,69321446
asked Dec 10 '18 at 16:16
DragonGamerDragonGamer
1163
edited Dec 11 '18 at 15:38
Widawensen
4,69321446
edited Dec 11 '18 at 15:38
Widawensen
4,69321446
edited Dec 11 '18 at 15:38
Widawensen
4,69321446
4,69321446
asked Dec 10 '18 at 16:16
DragonGamerDragonGamer
1163
asked Dec 10 '18 at 16:16
DragonGamerDragonGamer
1163
asked Dec 10 '18 at 16:16
DragonGamerDragonGamer
1163
1163
$begingroup$
How is it “obvious” that you need a quaternion? There are many ways to express rotations. Anyway, you need to nail down some more details of how you want this rotation to work: Rotations take place in a plane. Which plane do you want to use in 3D? A natural choice is the plane defined by $A$ and $B$, but then from where are you measuring the angles?
$endgroup$
– amd
Dec 10 '18 at 23:48
$begingroup$
@amd Thank you for the comment! Well, the engine I'm working with (Robot Operating System ROS) is using almost exclusively quaternions for transformations. Sometimes euler angles. Hmm, in 2D my reference is a horizontal line. isn't there a trivial equivalent for that in 3D? In the 3D version I need that in the case that A.z and B.z are both 0, C.x und C.y should be exactly the same as if the whole setup were still 2D.
$endgroup$
– DragonGamer
Dec 11 '18 at 3:52
add a comment |
$begingroup$
How is it “obvious” that you need a quaternion? There are many ways to express rotations. Anyway, you need to nail down some more details of how you want this rotation to work: Rotations take place in a plane. Which plane do you want to use in 3D? A natural choice is the plane defined by $A$ and $B$, but then from where are you measuring the angles?
$endgroup$
– amd
Dec 10 '18 at 23:48
$begingroup$
@amd Thank you for the comment! Well, the engine I'm working with (Robot Operating System ROS) is using almost exclusively quaternions for transformations. Sometimes euler angles. Hmm, in 2D my reference is a horizontal line. isn't there a trivial equivalent for that in 3D? In the 3D version I need that in the case that A.z and B.z are both 0, C.x und C.y should be exactly the same as if the whole setup were still 2D.
$endgroup$
– DragonGamer
Dec 11 '18 at 3:52
$begingroup$
How is it “obvious” that you need a quaternion? There are many ways to express rotations. Anyway, you need to nail down some more details of how you want this rotation to work: Rotations take place in a plane. Which plane do you want to use in 3D? A natural choice is the plane defined by $A$ and $B$, but then from where are you measuring the angles?
$endgroup$
– amd
Dec 10 '18 at 23:48
$begingroup$
How is it “obvious” that you need a quaternion? There are many ways to express rotations. Anyway, you need to nail down some more details of how you want this rotation to work: Rotations take place in a plane. Which plane do you want to use in 3D? A natural choice is the plane defined by $A$ and $B$, but then from where are you measuring the angles?
$endgroup$
– amd
Dec 10 '18 at 23:48
$begingroup$
@amd Thank you for the comment! Well, the engine I'm working with (Robot Operating System ROS) is using almost exclusively quaternions for transformations. Sometimes euler angles. Hmm, in 2D my reference is a horizontal line. isn't there a trivial equivalent for that in 3D? In the 3D version I need that in the case that A.z and B.z are both 0, C.x und C.y should be exactly the same as if the whole setup were still 2D.
$endgroup$
– DragonGamer
Dec 11 '18 at 3:52
$begingroup$
@amd Thank you for the comment! Well, the engine I'm working with (Robot Operating System ROS) is using almost exclusively quaternions for transformations. Sometimes euler angles. Hmm, in 2D my reference is a horizontal line. isn't there a trivial equivalent for that in 3D? In the 3D version I need that in the case that A.z and B.z are both 0, C.x und C.y should be exactly the same as if the whole setup were still 2D.
$endgroup$
– DragonGamer
Dec 11 '18 at 3:52
add a comment |
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$begingroup$
How is it “obvious” that you need a quaternion? There are many ways to express rotations. Anyway, you need to nail down some more details of how you want this rotation to work: Rotations take place in a plane. Which plane do you want to use in 3D? A natural choice is the plane defined by $A$ and $B$, but then from where are you measuring the angles?
$endgroup$
– amd
Dec 10 '18 at 23:48
$begingroup$
@amd Thank you for the comment! Well, the engine I'm working with (Robot Operating System ROS) is using almost exclusively quaternions for transformations. Sometimes euler angles. Hmm, in 2D my reference is a horizontal line. isn't there a trivial equivalent for that in 3D? In the 3D version I need that in the case that A.z and B.z are both 0, C.x und C.y should be exactly the same as if the whole setup were still 2D.
$endgroup$
– DragonGamer
Dec 11 '18 at 3:52