transform

cart2sphere(x, y, z)	Return angles for Cartesian 3D coordinates x, y, and z.
euler_matrix(ai, aj, ak[, axes])	Return homogeneous rotation matrix from Euler angles and axis sequence.
sphere2cart(r, theta, phi)	Spherical to Cartesian coordinates.

cart2sphere

fury.transform.cart2sphere(x, y, z)

Return angles for Cartesian 3D coordinates x, y, and z.

See doc for sphere2cart for angle conventions and derivation of the formulae.

$0lethetamathrm{(theta)}lepi$ and $-pilephimathrm{(phi)}lepi$

Parameters

• x (array_like) – x coordinate in Cartesian space

• y (array_like) – y coordinate in Cartesian space

• z (array_like) – z coordinate

Returns

• r (array) – radius

• theta (array) – inclination (polar) angle

• phi (array) – azimuth angle

euler_matrix

fury.transform.euler_matrix(ai, aj, ak, axes='sxyz')

Return homogeneous rotation matrix from Euler angles and axis sequence.

Code modified from the work of Christoph Gohlke link provided here http://www.lfd.uci.edu/~gohlke/code/transformations.py.html

Parameters

• aj, ak (ai,) –

• axes (One of 24 axis sequences as string or encoded tuple) –

Returns

• matrix (ndarray (4, 4))

• Code modified from the work of Christoph Gohlke link provided here

• http (//www.lfd.uci.edu/~gohlke/code/transformations.py.html)

Examples

>>> import numpy
>>> R = euler_matrix(1, 2, 3, 'syxz')
>>> numpy.allclose(numpy.sum(R[0]), -1.34786452)
True
>>> R = euler_matrix(1, 2, 3, (0, 1, 0, 1))
>>> numpy.allclose(numpy.sum(R[0]), -0.383436184)
True
>>> ai, aj, ak = (4.0*math.pi) * (numpy.random.random(3) - 0.5)
>>> for axes in _AXES2TUPLE.keys():
...    _ = euler_matrix(ai, aj, ak, axes)
>>> for axes in _TUPLE2AXES.keys():
...    _ = euler_matrix(ai, aj, ak, axes)

sphere2cart

fury.transform.sphere2cart(r, theta, phi)

Spherical to Cartesian coordinates.

This is the standard physics convention where theta is the inclination (polar) angle, and phi is the azimuth angle.

Imagine a sphere with center (0,0,0). Orient it with the z axis running south-north, the y axis running west-east and the x axis from posterior to anterior. theta (the inclination angle) is the angle to rotate from the z-axis (the zenith) around the y-axis, towards the x axis. Thus the rotation is counter-clockwise from the point of view of positive y. phi (azimuth) gives the angle of rotation around the z-axis towards the y axis. The rotation is counter-clockwise from the point of view of positive z.

Equivalently, given a point P on the sphere, with coordinates x, y, z, theta is the angle between P and the z-axis, and phi is the angle between the projection of P onto the XY plane, and the X axis.

Geographical nomenclature designates theta as ‘co-latitude’, and phi as ‘longitude’

Parameters

• r (array_like) – radius

• theta (array_like) – inclination or polar angle

• phi (array_like) – azimuth angle

Returns

• x (array) – x coordinate(s) in Cartesion space

• y (array) – y coordinate(s) in Cartesian space

• z (array) – z coordinate

Notes

See these pages:

• http://en.wikipedia.org/wiki/Spherical_coordinate_system

• http://mathworld.wolfram.com/SphericalCoordinates.html

for excellent discussion of the many different conventions possible. Here we use the physics conventions, used in the wikipedia page.

Derivations of the formulae are simple. Consider a vector x, y, z of length r (norm of x, y, z). The inclination angle (theta) can be found from: cos(theta) == z / r -> z == r * cos(theta). This gives the hypotenuse of the projection onto the XY plane, which we will call Q. Q == r*sin(theta). Now x / Q == cos(phi) -> x == r * sin(theta) * cos(phi) and so on.

We have deliberately named this function sphere2cart rather than sph2cart to distinguish it from the Matlab function of that name, because the Matlab function uses an unusual convention for the angles that we did not want to replicate. The Matlab function is trivial to implement with the formulae given in the Matlab help.