def rytz_axis_construction(d1: Vec3, d2: Vec3) -> Tuple[Vec3, Vec3, float]:
""" The Rytz’s axis construction is a basic method of descriptive Geometry
to find the axes, the semi-major axis and semi-minor axis, starting from two
conjugated half-diameters.
Source: `Wikipedia <https://en.m.wikipedia.org/wiki/Rytz%27s_construction>`_
Given conjugated diameter `d1` is the vector from center C to point P and
the given conjugated diameter `d2` is the vector from center C to point Q.
Center of ellipse is always ``(0, 0, 0)``. This algorithm works for
2D/3D vectors.
Args:
d1: conjugated semi-major axis as :class:`Vec3`
d2: conjugated semi-minor axis as :class:`Vec3`
Returns:
Tuple of (major axis, minor axis, ratio)
"""
Q = Vec3(d1) # vector CQ
# calculate vector CP', location P'
if math.isclose(d1.z, 0, abs_tol=1e-9) and math.isclose(
d2.z, 0, abs_tol=1e-9):
# Vec3.orthogonal() works only for vectors in the xy-plane!
P1 = Vec3(d2).orthogonal(ccw=False)
else:
extrusion = d1.cross(d2)
P1 = extrusion.cross(d2).normalize(d2.magnitude)
D = P1.lerp(Q) # vector CD, location D, midpoint of P'Q
radius = D.magnitude
radius_vector = (Q - P1).normalize(radius) # direction vector P'Q
A = D - radius_vector # vector CA, location A
B = D + radius_vector # vector CB, location B
if A.isclose(NULLVEC) or B.isclose(NULLVEC):
raise ArithmeticError('Conjugated axis required, invalid source data.')
major_axis_length = (A - Q).magnitude
minor_axis_length = (B - Q).magnitude
if math.isclose(major_axis_length, 0.) or math.isclose(
minor_axis_length, 0.):
raise ArithmeticError('Conjugated axis required, invalid source data.')
ratio = minor_axis_length / major_axis_length
major_axis = B.normalize(major_axis_length)
minor_axis = A.normalize(minor_axis_length)
return major_axis, minor_axis, ratio
def minor_axis(major_axis: Vec3, extrusion: Vec3, ratio: float) -> Vec3:
return extrusion.cross(major_axis).normalize(major_axis.magnitude * ratio)
