HoaMatrixRotation | SuperCollider 3.13.0 Help

Description

A class to generate a matrix representing a rotation of real or complex spherical harmonics, around three axes, in a specified order of rotations.

Use of HoaMatrixXformer's various rotation methods, which use this class internally, is more convenient for most rotation operations and is advised. See HoaMatrixXformer: *newRotate, *newRTT, *newYPR, *newRotateAxis.

Class Methods

.new

Generate an HoaMatrixRotation

Arguments:

r1	The first rotation angle, in radians.
r2	The second rotation angle, in radians.
r3	The third rotation angle, in radians.
axes	A Symbol defining the order of rotations, by axis. E.g. `'zyz'`.
order	Ambisonic order. Defaults to `AtkHoa.defaultOrder`.

Instance Methods

.matrix

Return the Matrix object representing the rotation.

.r3x3

Return the Matrix representing the original 3x3 rotation of degree l = 1, which is then used to recursively build the full rotation matrix of higher order. The user will not typically need this rotation kernel, but is provided for introspection.

.order

Return the ambisonic order of the rotation matrix.

.buildComplexSHRotMtx

Return a rotation Matrix of complex spherical harmonics.

Examples

Test axial rotations via HoaMatrixXformer: -newRotateAxis(which uses HoaMatrixRotation internally)./* Generate planewave encoding coefficients for directions */ ( var pwDirs = [ [0, 0], // front [90, 0], // left [-90, 0], // right [0, -90] // nadir ].degrad; ~pwCoeffs = pwDirs.collect({ |dir| HoaMatrixEncoder.newDirection(*dir).matrix.getCol(0) }) ) /* Rotate */ ( // Create a 90-degree rotation around Z axis var rotationMatrix = HoaMatrixXformer.newRotateAxis(\z, 90.degrad); var forwardPw = ~pwCoeffs[0]; // planewave coefficients for forward direction var leftPw = ~pwCoeffs[1]; // planewave coefficients for left direction // Use the matrix to rotate the coefficients of a frontal planewave var rotatedPw = MatrixArray.with( rotationMatrix.matrix.asArray ).mixCoeffs(forwardPw); // Compare rotated planewave with a planewave directly encoded // in the rotation direction rotatedPw.round(0.001) == leftPw.round(0.001) ) /* Tilt */ ( // Create a 90-degree rotation around X axis var rotationMatrix = HoaMatrixXformer.newRotateAxis(\x, 90.degrad); var leftPw = ~pwCoeffs[1]; var downPw = ~pwCoeffs[3]; var rotatedPw = MatrixArray.with( rotationMatrix.matrix.asArray ).mixCoeffs(downPw); rotatedPw.round(0.001) == leftPw.round(0.001) ) /* Tumble */ ( // Create a 90-degree rotation around Y axis var rotationMatrix = HoaMatrixXformer.newRotateAxis(\y, 90.degrad); var forwardPw = ~pwCoeffs[0]; var downPw = ~pwCoeffs[3]; var rotatedPw = MatrixArray.with( rotationMatrix.matrix.asArray ).mixCoeffs(downPw); rotatedPw.round(0.001) == forwardPw.round(0.001) )

Attribution

The underlying utilities used here are largely a port of spherical harmonic rotations from Archontis Politis's Spherical-Harmonic-Transform Library for Matlab/Octave. ¹ ²

The technique was originally authored by Ivanic and Ruedenberg. ³ ⁴

Politis's code for real SH rotations is itself a port of Bing Jian's implementations of the above technique. ⁵

[1] - https://github.com/polarch/Spherical-Harmonic-Transform.

[2] - Specifically: euler2rotationMatrix.m, getSHrotMtx.m, complex2realSHMtx.m

[3] - Ivanic, J., Ruedenberg, K. (1996). Rotation Matrices for Real Spherical Harmonics. Direct Determination by Recursion. The Journal of Physical Chemistry, 100(15), 6342-6347.

[4] - With corrections: Ivanic, J., Ruedenberg, K. (1998). Rotation Matrices for Real Spherical Harmonics. Direct Determination by Recursion Page: Additions and Corrections. Journal of Physical Chemistry A, 102(45), 9099-9100.

[5] - http://www.mathworks.com/matlabcentral/fileexchange/15377-real-valued-spherical-harmonics

helpfile source: HelpSource/Classes/HoaMatrixRotation.schelp
link::atk-sc3/Classes/HoaMatrixRotation::

HoaMatrixRotationExtension