Rotate About An Arbitrary Axis 3 Dimensions =LINK=
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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).
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 axispassing through the origin and a point (a,b,c) on the unit sphere inthree-dimensions is a linear transformation, and hence can berepresented by matrix multiplication. We will give a very slickmethod for determining this matrix, but to appreciate the compactnessof the formula it will be wise to start with a few remarks.
Rotations in three-dimensions are rather special lineartransformations, not least because they preserve the lengths ofvectors and also (when two vectors are rotated) the angles between thevectors. Such transformations are called "orthogonal" and they arerepresented by orthogonal matrices:
so that it coincides with one of the "standard" unit vectors, perhapsk (the positve z-axis). Now we know how to rotate around the z-axis;it's a matter of doing the usual 2x2 transformation on the x,ycoordinates alone:
To perform a 3D rotation, you simply need to offset the point of rotation to the origin and sequentially rotate around each axis, storing the results between each axis rotation for use with the next rotation operation. The algorithm looks like as follows:
When the object is rotated about an axis that is not parallel to any one of co-ordinate axis, i.e., x, y, z. Then additional transformations are required. First of all, alignment is needed, and then the object is being back to the original position. Following steps are required
How is this matrix generated? I've thought a lot about how to rotate around an arbitrary axis, but the only thing I could come up with was to break the 3D vector extending from the origin into its x, y and z components, and then perform three different rotations, which seems pretty inefficient.
If you only want to rotate around an arbitrary axis away from origin (that is, around a line), look for the item "6.2 The normalized matrix for rotation about an arbitrary line" in the 2nd URL ( ://inside.mines.edu:80/~gmurray/ArbitraryAxisRotation/ArbitraryAxisRotation.html).
rotates points in the xy plane counterclockwise through an angle θ with respect to the positive x axis about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates v = (x, y), it should be written as a column vector, and multiplied by the matrix R:
If x and y are the endpoint coordinates of a vector, where x is cosine and y is sine, then the above equations become the trigonometric summation angle formulae. Indeed, a rotation matrix can be seen as the trigonometric summation angle formulae in matrix form. One way to understand this is say we have a vector at an angle 30° from the x axis, and we wish to rotate that angle by a further 45°. We simply need to compute the vector endpoint coordinates at 75°.
For column vectors, each of these basic vector rotations appears counterclockwise when the axis about which they occur points toward the observer, the coordinate system is right-handed, and the angle θ is positive. Rz, for instance, would rotate toward the y-axis a vector aligned with the x-axis, as can easily be checked by operating with Rz on the vector (1,0,0):
For odd dimensions n = 2k + 1, a proper rotation R will have an odd number of eigenvalues, with at least one λ = 1 and the axis of rotation will be an odd dimensional subspace. Proof:
We can also generate a uniform distribution in any dimension using the subgroup algorithm of Diaconis & Shashahani (1987) harvtxt error: no target: CITEREFDiaconisShashahani1987 (help). This recursively exploits the nested dimensions group structure of SO(n), as follows. Generate a uniform angle and construct a 2 × 2 rotation matrix. To step from n to n + 1, generate a vector v uniformly distributed on the n-sphere Sn, embed the n × n matrix in the next larger size with last column (0, ..., 0, 1), and rotate the larger matrix so the last column becomes v.
Unit quaternions, known as versors, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation about an arbitrary axis. Rotation and orientation quaternions have applications in computer graphics,[1] computer vision, robotics,[2] navigation, molecular dynamics, flight dynamics,[3] orbital mechanics of satellites,[4] and crystallographic texture analysis.[5]
In order to visualize the space of rotations, it helps to consider a simpler case. Any rotation in three dimensions can be described by a rotation by some angle about some axis; for our purposes, we will use an axis vector to establish handedness for our angle. Consider the special case in which the axis of rotation lies in the xy plane. We can then specify the axis of one of these rotations by a point on a circle through which the vector crosses, and we can select the radius of the circle to denote the angle of rotation.
Similarly, a rotation whose axis of rotation lies in the xy plane can be described as a point on a sphere of fixed radius in three dimensions. Beginning at the north pole of a sphere in three-dimensional space, we specify the point at the north pole to be the identity rotation (a zero angle rotation). Just as in the case of the identity rotation, no axis of rotation is defined, and the angle of rotation (zero) is irrelevant. A rotation having a very small rotation angle can be specified by a slice through the sphere parallel to the xy plane and very near the north pole. The circle defined by this slice will be very small, corresponding to the small angle of the rotation. As the rotation angles become larger, the slice moves in the negative z direction, and the circles become larger until the equator of the sphere is reached, which will correspond to a rotation angle of 180 degrees. Continuing southward, the radii of the circles now become smaller (corresponding to the absolute value of the angle of the rotation considered as a negative number). Finally, as the south pole is reached, the circles shrink once more to the identity rotation, which is also specified as the point at the south pole.
Notice that a number of characteristics of such rotations and their representations can be seen by this visualization. The space of rotations is continuous, each rotation has a neighborhood of rotations which are nearly the same, and this neighborhood becomes flat as the neighborhood shrinks. Also, each rotation is actually represented by two antipodal points on the sphere, which are at opposite ends of a line through the center of the sphere. This reflects the fact that each rotation can be represented as a rotation about some axis, or, equivalently, as a negative rotation about an axis pointing in the opposite direction (a so-called double cover). The "latitude" of a circle representing a particular rotation angle will be half of the angle represented by that rotation, since as the point is moved from the north to south pole, the latitude ranges from zero to 180 degrees, while the angle of rotation ranges from 0 to 360 degrees. (the "longitude" of a point then represents a particular axis of rotation.) Note however that this set of rotations is not closed under composition. Two successive rotations with axes in the xy plane will not necessarily give a rotation whose axis lies in the xy plane, and thus cannot be represented as a point on the sphere. This will not be the case with a general rotation in 3-space, in which rotations do form a closed set under composition.
This visualization can be extended to a general rotation in 3-dimensional space. The identity rotation is a point, and a small angle of rotation about some axis can be represented as a point on a sphere with a small radius. As the angle of rotation grows, the sphere grows, until the angle of rotation reaches 180 degrees, at which point the sphere begins to shrink, becoming a point as the angle approaches 360 degrees (or zero degrees from the negative direction). This set of expanding and contracting spheres represents a hypersphere in four dimensional space (a 3-sphere). Just as in the simpler example above, each rotation represented as a point on the hypersphere is matched by its antipodal point on that hypersphere. The "latitude" on the hypersphere will be half of the corresponding angle of rotation, and the neighborhood of any point will become "flatter" (i.e. be represented by a 3-D Euclidean space of points) as the neighborhood shrinks. This behavior is matched by the set of unit quaternions: A general quaternion represents a point in a four dimensional space, but constraining it to have unit magnitude yields a three-dimensional space equivalent to the surface of a hypersphere. The magnitude of the unit quaternion will be unity, corresponding to a hypersphere of unit radius. The vector part of a unit quaternion represents the radius of the 2-sphere corresponding to the axis of rotation, and its magnitude is the cosine of half the angle of rotation. Each rotation is represented by two unit quaternions of opposite sign, and, as in the space of rotations in three dimensions, the quaternion product of two unit quaternions will yield a unit quaternion. Also, the space of unit quaternions is "flat" in any infinitesimal neighborhood of a given unit quaternion. 2b1af7f3a8
