Source code for attitude.geom.test_offset_ellipse

"""
This test module is designed to show the exact relationship between the PCA error lengths
defined from covariance matrices and the hyperbolic and ellipsoidal error bounds
"""

import numpy as N
from .util import vector, dot
from ..error.axes import sampling_axes, noise_axes
from .conics import Conic, conic
import pytest

origin = vector(0,0)

def test_angular_shadow():
    """
    Make sure we can successfully recover an angular shadow
    from an ellipse offset from the origin, a simple two
    dimensional case of the more general 3d solution.

    BASIS: an ellipse with semiaxes [1,1] centered at
    [0,2] will have a half angular shadow of 30º
    """
    # Create an mean-centered conic with the requisite
    # semiaxial lengths
    _ = N.identity(3)
    _[-1,-1] = -1
    ell0 = conic(_)

    # Translate conic to new center
    ell = ell0.translate(vector(0,2))

    # Get the polar plane to the origin (plane intersecting
    # the conic at all points of tangency to the origin)
    plane = ell.polar_plane(vector(0,0))
    assert N.linalg.norm(plane.offset()) == 1.5

    # Get degenerate conic with d-1 dimensionality
    # (in this case, a line) and distance from projection
    # point of tangent
    con, transform, offset = ell.slice(plane)
    # this will be the half-length of the cut of the ellipse
    l = float(con.major_axes())
    dist = N.linalg.norm(offset)

    assert N.allclose(N.degrees(N.arctan2(l,dist)), 30)

    # Same with generalized function
    assert N.allclose(N.degrees(ell.angle_subtended()), 30)


@pytest.mark.xfail(reason="Mathematical basis seems to be incorrect")
def test_ellipse_offset():
    """
    An error ellipse should be offset a set amount off axis
    """
    a,b = 10,5
    axial_lengths = N.array([a,b])

    c = Conic.from_axes(1/axial_lengths**2)

    # angular error for hyperbolic representation of this conic
    th = N.arctan2(b,a)

    # This is always true for ellipse offset to the proper position
    gamma = N.pi/4

    # This is the distance we shift the ellipse
    l = 1/b*(a**4+b**4)
    assert l > 0
    assert l > b

    center = vector(0,l)
    c0 = c.translate(center)
    # Check that we have the right center
    assert N.allclose(c0.center(), center)

    origin = vector(0,0)
    # Plane of tangency
    plane = c0.polar_plane(origin)
    # find offset from origin,
    # it should be within the bottom half of the ellipse
    offs = plane.hessian_normal()[-1]

    # Check with offset computed algebraically
    assert N.allclose(l-b**2/l, offs)

    assert l-b < offs < l

    # Check geometrically
    #offs2 = a**2*N.cos(gamma)/b
    #assert N.allclose(offs2, offs)

    # Check that the projection of the ellipse from the
    # origin spans the same angle as the hyperbolic angular error


    xv = a*N.sqrt(1-b**2/l**2)

    angular_shadow = N.arctan2(xv,offs)
    assert th == angular_shadow