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Updated 05 Jul Generates the roto-translation matrix for the rotation around an arbitrary line in 3D. The line need not pass through the origin. Optionally, also, applies this transformation to a list of 3D coordinates. Counter-clockwise is defined using the right hand rule in reference to the direction of u. Matt J Retrieved April 12, Sorry, the previous question should have been asked with my account, could you please answer it here. How can I use your code for a 3D point cloud data, XYZthat should be rotated around the intersection of two planes in space?

Hi Matt J, I have a 3D data x,y,z, and color. They initially have a cube-shape. When I rotate the data around x-axis and point x0, I expect to see a cube-shape again, but the shape is different! For Y-Z plane, the shape is parallelogram instead of rectangle! Can you tell me what is going wrong? I want to preserve the degree between planes.

However, you must understand that deg specifies the amount of rotation. If you rotate everything by 0 degrees, then there is no rotation and nothing moves. The argument u does not represent a target for the new z-axis. It is the axis you want to rotate everything around. Matt J I don't understand the 'deg' well. Please point out where I have done wrong. Thank you very much for your code and time and energy to reply.

That depends on which axis you want to rotate about. You will need tformarray to rotate a 3D image, but you can compute the requisite tform using AxelRot. AxelRot does not rotate 2D or 3D images. It rotates coordinates. If using syntax 3, the Data must be 3xN whose columns form a list of 3D point coordinates.Documentation Help Center. When acting on a matrix, each column of the matrix represents a different vector.

Then let the matrix operate on a vector.

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Under a rotation around the x -axis, the x-component of a vector is invariant. Rotation angle specified as a real-valued scalar. The rotation angle is positive if the rotation is in the counter-clockwise direction when viewed by an observer looking along the x-axis towards the origin. Angle units are in degrees. In transforming vectors in three-dimensional space, rotation matrices are often encountered.

Rotation matrices are used in two senses: they can be used to rotate a vector into a new position or they can be used to rotate a coordinate basis or coordinate system into a new one.

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In this case, the vector is left alone but its components in the new basis will be different from those in the original basis. In Euclidean space, there are three basic rotations: one each around the x, y and z axes. Each rotation is specified by an angle of rotation. The rotation angle is defined to be positive for a rotation that is counterclockwise when viewed by an observer looking along the rotation axis towards the origin.

For example, the inverse of the x-axis rotation matrix is obtained by changing the sign of the angle:. This example illustrates a basic property: the inverse rotation matrix is the transpose of the original. Under rotations, vector lengths are preserved as well as the angles between vectors. We can think of rotations in another way. Consider the original set of basis vectors, ijkand rotate them all using the rotation matrix A. Using the transpose, you can write the new basis vectors as a linear combinations of the old basis vectors:.

Using algebraic manipulation, you can derive the transformation of components for a fixed vector when the basis or coordinate system rotates. This transformation uses the transpose of the rotation matrix. The next figure illustrates how a vector is transformed as the coordinate system rotates around the x-axis.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It only takes a minute to sign up. I am trying to understand the following statement. This fact explains the rather mysterious looking factor of two in the definition of the rotation matrices. I know the rotation matrices in terms of the Pauli matrices, i. But the difficulties for me start from here. How can I construct a concrete proof? Sign up to join this community.

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Rotations in Bloch Sphere about an arbitrary axis Ask Question. Asked 4 years ago. Active 2 years, 8 months ago. Viewed 3k times. Anuroop Kuppam. Anuroop Kuppam Anuroop Kuppam 51 1 1 silver badge 6 6 bronze badges.

That would also help us understand what factor of two you are talking about it is right now rather mysterious indeed However I am still not sure if it is clear enough. There surely has to be a better way. Are you familiar with the rotation group and Pauli-matrix algebra?

The dark mode beta is finally here. Change your preferences any time. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. I am programming Starcraft 2 custom maps and got some proglems with math in 3D. Currently I am trying to create and rotate a point around an arbitrary axis, given by x,y and z the xyz vector is normalized. I've been trying around a lot and read through a lot of stuff on the internet, but I just cant get how it works correctly.

My current script you probably dont know the language, but it's nothing special is the result of breaking everything for hours doesn't work correctly :. I just cant get my mind around the math. If you can explain it in simple terms that would be the best solution, a code snipped would be good as well but not quite as helpful, because I plan to do more 3D stuff in the future.

Look under the section Rotation matrix from axis and angle. For your convenience, here's the matrix you need. It's a bit hairy. A useful method for doing such rotations is to do them with quaternions. In practice, I've found them to be easier to use and have the added bonus of avoiding Gimbal lock. Here is a nice walk through that explains how and why they are used for rotation about an arbitrary axis it's the response to the user's question.

It's a bit higher level and would be good for someone who is new to the idea, so I recommend starting there. As you have no doubt already concluded, rotation around the axis passing through the origin and a point a,b,c on the unit sphere in three-dimensions is a linear transformation, and hence can be represented by matrix multiplication. We will give a very slick method for determining this matrix, but to appreciate the compactness of the formula it will be wise to start with a few remarks.

Rotations in three-dimensions are rather special linear transformations, not least because they preserve the lengths of vectors and also when two vectors are rotated the angles between the vectors.

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Such transformations are called "orthogonal" and they are represented by orthogonal matrices:. In other words the transpose of an orthogonal matrix is its inverse. Consider the data which is needed to define the transformation.

The only other datum is the angle of rotation, which for lack of a more natural character I will denote by r for rotation? Now the rotations are actually a bit special even among orthogonal transformations, and in fact they are also called special orthogonal transformations or matrices in virtue of their property of being "orientation preserving".

Compare them with reflections, which are also length and angle preserving, and you will find that the geometric characteristic of preserving orientation or "handedness" if you prefer has a numerical counterpart in the determinant of the matrix. A rotation's matrix has determinant 1, while a reflection's matrix has determinant It turns out that the product or composition of two rotations is again a rotation, which agrees with the fact that the determinant of a product is the product of the determinants or 1 in the case of a rotation.

Now we can describe a step by step approach that one might follow to construct the desired matrix before we shortcut the whole process and jump to the Answer!

Consider first a step in which we rotate the unit vector:. Now we know how to rotate around the z-axis; it's a matter of doing the usual 2x2 transformation on the x,y coordinates alone:. Finally we need to "undo" that initial rotation that took u to k, which is easy because the inverse of that transformation is we recall represented by the matrix transpose.

In other words, if matrix R represents a rotation taking u to k, then R' takes k to u, and we can write out the composition of transformations like this:.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. The dark mode beta is finally here. Change your preferences any time. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information.

I am programming Starcraft 2 custom maps and got some proglems with math in 3D. Currently I am trying to create and rotate a point around an arbitrary axis, given by x,y and z the xyz vector is normalized.

I've been trying around a lot and read through a lot of stuff on the internet, but I just cant get how it works correctly.

My current script you probably dont know the language, but it's nothing special is the result of breaking everything for hours doesn't work correctly :. I just cant get my mind around the math. If you can explain it in simple terms that would be the best solution, a code snipped would be good as well but not quite as helpful, because I plan to do more 3D stuff in the future.

Look under the section Rotation matrix from axis and angle. For your convenience, here's the matrix you need. It's a bit hairy. A useful method for doing such rotations is to do them with quaternions.

In practice, I've found them to be easier to use and have the added bonus of avoiding Gimbal lock. Here is a nice walk through that explains how and why they are used for rotation about an arbitrary axis it's the response to the user's question.

It's a bit higher level and would be good for someone who is new to the idea, so I recommend starting there. As you have no doubt already concluded, rotation around the axis passing through the origin and a point a,b,c on the unit sphere in three-dimensions is a linear transformation, and hence can be represented by matrix multiplication. We will give a very slick method for determining this matrix, but to appreciate the compactness of the formula it will be wise to start with a few remarks.

Rotations in three-dimensions are rather special linear transformations, not least because they preserve the lengths of vectors and also when two vectors are rotated the angles between the vectors. Such transformations are called "orthogonal" and they are represented by orthogonal matrices:.

How can I rotate a set of points in a plane by a certain angle about an arbitrary point?

In other words the transpose of an orthogonal matrix is its inverse. Consider the data which is needed to define the transformation. The only other datum is the angle of rotation, which for lack of a more natural character I will denote by r for rotation? Now the rotations are actually a bit special even among orthogonal transformations, and in fact they are also called special orthogonal transformations or matrices in virtue of their property of being "orientation preserving".

Compare them with reflections, which are also length and angle preserving, and you will find that the geometric characteristic of preserving orientation or "handedness" if you prefer has a numerical counterpart in the determinant of the matrix. A rotation's matrix has determinant 1, while a reflection's matrix has determinant It turns out that the product or composition of two rotations is again a rotation, which agrees with the fact that the determinant of a product is the product of the determinants or 1 in the case of a rotation.

Now we can describe a step by step approach that one might follow to construct the desired matrix before we shortcut the whole process and jump to the Answer! Consider first a step in which we rotate the unit vector:. Now we know how to rotate around the z-axis; it's a matter of doing the usual 2x2 transformation on the x,y coordinates alone:. Finally we need to "undo" that initial rotation that took u to k, which is easy because the inverse of that transformation is we recall represented by the matrix transpose.

In other words, if matrix R represents a rotation taking u to k, then R' takes k to u, and we can write out the composition of transformations like this:. It is easily verified that this product of matrices, when multiplied times u, gives u back again:.The problem of rotation about an arbitrary axis in three dimensions arises in many fields including computer graphics and molecular simulation.

In this article we give an algorithm and matrices for doing the movement. Many of the results were initially obtained with Mathematica. We will write our three-dimensional points in four homogeneous coordinates; i. This enables us to do coordinate transformations using 4x4 matrices. Note that these are really only necessary for translations, if we omitted translations from our movements we could do the motions with 3x3 rotation matrices obtained by deleting the last rows and last columns of the 4x4 matrices.

The general rotation matrix depends on the order of rotations.

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The first matrix rotates about xthen ythen z ; the second rotates about zthen ythen x. Note that we use the components to form expressions for the cosines and sines to avoid using inverse trigonometric functions.

We will define an arbitrary line by a point the line goes through and a direction vector. We can now write a transformation for the rotation of a point about this line. Glenn Murray's Home Page. Search this site. Navigation Glenn Murray, Ph. Rotation Matrices and Formulas.

Glenn Murray, Ph. Glenn Murray. Rotation Matrices and Formulas Sitemap.Documentation Help Center. The rotate function rotates a graphics object in three-dimensional space. Specify h as a surface, patch, line, text, or image object.

The default origin of the axis of rotation is the center of the plot box. This point is not necessarily the origin of the axes.

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Positive alpha is defined as the righthand-rule angle about the direction vector as it extends from the origin of rotation. If h is an array of handles, all objects must be children of the same axes. Create a surface plot of the peaks function and return the surface object. Rotate the surface plot 25 degrees around its x -axis and y -axis. The rotation transformation modifies the object's data. This technique is different from that used by view and rotate3dwhich modify only the viewpoint.

The axis of rotation is defined by an origin of rotation and a point P. Specify P as the spherical coordinates [theta phi] or as the Cartesian coordinates [x p ,y p ,z p ]. In the two-element form for directiontheta is the angle in the x-y plane counterclockwise from the positive x -axis. The three-element form for direction specifies the axis direction using Cartesian coordinates.

Linear Algebra 21i: How to Represent a Rotation with respect to an Arbitrary Axis

The direction vector is the vector from the origin of rotation to P. A modified version of this example exists on your system. Do you want to open this version instead? Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select:. Select the China site in Chinese or English for best site performance. Other MathWorks country sites are not optimized for visits from your location. Toggle Main Navigation.

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Search Support Support MathWorks. Search MathWorks. Off-Canvas Navigation Menu Toggle. Syntax rotate h,direction,alpha rotate Description The rotate function rotates a graphics object in three-dimensional space. Examples collapse all Rotate Plot Around x -Axis. Open Live Script. Rotate Plot Around y -Axis.

Rotate Plot Around x -Axis and y -Axis.