def test_convex(self):
mb_0 = Metaball(1.0, 2.0, 0.0)
mb_1 = Metaball(1.0, 2.0, 2.8)
interval = mb_1.lower, mb_0.upper
coeffs = mb_0.get_coeffs(False) + mb_1.get_coeffs(True)
self._test_result(coeffs, interval, *np_roots(coeffs, interval))
a, b = np_roots(coeffs, interval)[:2]
mid = (a + b) / 2
# Only left root.
interval = mb_1.lower, mid
self._test_result(coeffs, interval, a, np.nan)
# Only right root.
interval = mid, mb_0.upper
self._test_result(coeffs, interval, b, np.nan)
# Roots are interval ends.
# The root must be picked up either in interval i or i + 1
interval = a, b
result = self.get_roots(coeffs, interval)
np_result = np_roots(coeffs, interval)
interval = b, mb_0.upper
result = np.sort(np.concatenate((result,
self.get_roots(coeffs, interval))))
np_result = np.sort(np.concatenate((np_result,
np_roots(coeffs, interval))))
result = filter_close(result)
np_result = filter_close(np_result)
np.testing.assert_almost_equal(result, np_result,
decimal=self.precision_places)
def test_derivative_stationary(self):
"""Test the case when there are two roots and f' has stationary
points in the interval end points.
"""
mb = Metaball(1.0, 2.0, 0.0)
coeffs = mb.get_coeffs(True)
stat_left = self.get_stationary(coeffs, (mb.lower, 0))
stat_right = self.get_stationary(coeffs, (0, mb.upper))
self._test_result(coeffs, (mb.lower, stat_right), -1, 1)
self._test_result(coeffs, (stat_left, mb.upper), -1, 1)
self._test_result(coeffs, (stat_left, stat_right), -1, 1)
def test_derivative_stationary(self):
"""Test the case when there are two roots and f' has stationary
points in the interval end points.
"""
mb = Metaball(1.0, 2.0, 0.0)
coeffs = mb.get_coeffs(True)
stat_left = self.get_stationary(coeffs, (mb.lower, 0))
stat_right = self.get_stationary(coeffs, (0, mb.upper))
self._test_result(coeffs, (mb.lower, stat_right), -1, 1)
self._test_result(coeffs, (stat_left, mb.upper), -1, 1)
self._test_result(coeffs, (stat_left, stat_right), -1, 1)
def test_small_on_big_metaball(self):
mb_0 = Metaball(0.0, 0.5, 0.51)
mb_1 = Metaball(1.5, 100.0, -1.5)
# First interval, only big metaball, two roots.
coeffs = mb_1.get_coeffs(True)
interval = mb_1.lower, mb_0.lower
self._test_result(coeffs, interval, *np_roots(coeffs, interval))
# Last interval, nothing in it.
interval = mb_0.upper, mb_1.upper
self._test_result(coeffs, interval, np.nan, np.nan)
# Small and big metaballs, two roots.
coeffs = mb_0.get_coeffs(False) + mb_1.get_coeffs(True)
interval = mb_0.lower, mb_0.upper
self._test_result(coeffs, interval, *np_roots(coeffs, interval))
def test_stationary_far_from_center(self):
mb = Metaball(0.1, 2.0, 0.0)
# Far left and right beyond the object.
# Both roots.
coeffs = mb.get_coeffs(True)
interval = mb.lower, 1.5 * mb.r
self._test_result(coeffs, interval, -mb.r, mb.r)
# One root.
interval = mb.lower, 0
self._test_result(coeffs, interval, -mb.r, np.nan)
# Far right
interval = -1.5 * mb.r, mb.upper
self._test_result(coeffs, interval, -mb.r, mb.r)
# One root.
interval = 0, mb.upper
self._test_result(coeffs, interval, mb.r, np.nan)
def test_stationary_far_from_center(self):
mb = Metaball(0.1, 2.0, 0.0)
# Far left and right beyond the object.
# Both roots.
coeffs = mb.get_coeffs(True)
interval = mb.lower, 1.5 * mb.r
self._test_result(coeffs, interval, -mb.r, mb.r)
# One root.
interval = mb.lower, 0
self._test_result(coeffs, interval, -mb.r, np.nan)
# Far right
interval = -1.5 * mb.r, mb.upper
self._test_result(coeffs, interval, -mb.r, mb.r)
# One root.
interval = 0, mb.upper
self._test_result(coeffs, interval, mb.r, np.nan)
def test_convex(self):
mb_0 = Metaball(1.0, 2.0, 0.0)
mb_1 = Metaball(1.0, 2.0, 2.8)
interval = mb_1.lower, mb_0.upper
coeffs = mb_0.get_coeffs(False) + mb_1.get_coeffs(True)
self._test_result(coeffs, interval, *np_roots(coeffs, interval))
a, b = np_roots(coeffs, interval)[:2]
mid = (a + b) / 2
# Only left root.
interval = mb_1.lower, mid
self._test_result(coeffs, interval, a, np.nan)
# Only right root.
interval = mid, mb_0.upper
self._test_result(coeffs, interval, b, np.nan)
# Roots are interval ends.
# The root must be picked up either in interval i or i + 1
interval = a, b
result = self.get_roots(coeffs, interval)
np_result = np_roots(coeffs, interval)
interval = b, mb_0.upper
result = np.sort(
np.concatenate((result, self.get_roots(coeffs, interval))))
np_result = np.sort(
np.concatenate((np_result, np_roots(coeffs, interval))))
result = filter_close(result)
np_result = filter_close(np_result)
np.testing.assert_almost_equal(result,
np_result,
decimal=self.precision_places)
def test(offset):
mb = Metaball(1.0, 2.0, 0.0)
# Root into the stationary point.
coeffs = mb.get_coeffs(True)
coeffs = coeffs - \
np.array([0, 0, 0, 0, f(coeffs, mb.center) + offset])
# Tip of the quartic is the stationary point.
interval = mb.lower, mb.upper
self._test_result(coeffs, interval, *np_roots(coeffs, interval))
# Moreover, split the interval in the root
interval = mb.lower, 0.0
result = self.get_roots(coeffs, interval)
ground_truth = np_roots(coeffs, interval)
interval = 0.0, mb.upper
result = np.sort(
np.concatenate((result, self.get_roots(coeffs, interval))))
ground_truth = np.sort(
np.concatenate((ground_truth, np_roots(coeffs, interval))))
np.testing.assert_almost_equal(result,
ground_truth,
decimal=self.precision_places)
def test(offset):
mb = Metaball(1.0, 2.0, 0.0)
# Root into the stationary point.
coeffs = mb.get_coeffs(True)
coeffs = coeffs - \
np.array([0, 0, 0, 0, f(coeffs, mb.center) + offset])
# Tip of the quartic is the stationary point.
interval = mb.lower, mb.upper
self._test_result(coeffs, interval, *np_roots(coeffs, interval))
# Moreover, split the interval in the root
interval = mb.lower, 0.0
result = self.get_roots(coeffs, interval)
ground_truth = np_roots(coeffs, interval)
interval = 0.0, mb.upper
result = np.sort(np.concatenate((result,
self.get_roots(coeffs,
interval))))
ground_truth = np.sort(np.concatenate((ground_truth,
np_roots(coeffs,
interval))))
np.testing.assert_almost_equal(result, ground_truth,
decimal=self.precision_places)
def test_small_on_big_metaball(self):
mb_0 = Metaball(0.0, 0.5, 0.51)
mb_1 = Metaball(1.5, 100.0, -1.5)
# First interval, only big metaball, two roots.
coeffs = mb_1.get_coeffs(True)
interval = mb_1.lower, mb_0.lower
self._test_result(coeffs, interval, *np_roots(coeffs, interval))
# Last interval, nothing in it.
interval = mb_0.upper, mb_1.upper
self._test_result(coeffs, interval, np.nan, np.nan)
# Small and big metaballs, two roots.
coeffs = mb_0.get_coeffs(False) + mb_1.get_coeffs(True)
interval = mb_0.lower, mb_0.upper
self._test_result(coeffs, interval, *np_roots(coeffs, interval))
def test_one_metaball(self):
# Regular two roots.
for r in np.linspace(self.pixel_size / 4, 1.0 - self.pixel_size, 10):
mb = Metaball(r, 1.0, 0.0)
coeffs = mb.get_coeffs(True)
interval = (mb.lower, mb.upper)
self._test_result(coeffs, interval, -r, r)
# One root left
mb = Metaball(1.0, 2.0, 0.0)
coeffs = mb.get_coeffs(True)
interval = mb.lower, mb.center - mb.r / 2
self._test_result(coeffs, interval, -1.0, np.nan)
# One root right
interval = mb.center + mb.r / 2, mb.upper
self._test_result(coeffs, interval, 1, np.nan)
# Make interval end points stationary.
coeffs = mb.get_coeffs(True)
interval = (mb.lower, 0)
self._test_result(coeffs, interval, -1, np.nan)
interval = (0, mb.upper)
self._test_result(coeffs, interval, 1, np.nan)
# Put interval end points to roots.
# In right endpoint.
interval = mb.lower, -1.0
self._test_result(coeffs, interval, -1.0)
# In left endpoint.
interval = -1.0, 0.0
self._test_result(coeffs, interval)
# By epsilon too far -> no root expected
interval = mb.lower, -1 - self.pixel_size
self._test_result(coeffs, interval)
interval = 1.0, mb.upper
# By epsilon too far -> no root expected
interval = 1.0 + self.pixel_size, mb.upper
self._test_result(coeffs, interval, np.nan, np.nan)
# Left end point is a root.
coeffs = coeffs - np.array([0, 0, 0, 0, f(coeffs, mb.center)])
interval = 0.0, mb.upper
self._test_result(coeffs, interval, 0.0)
# Right end point is a root.
interval = mb.lower, 0.0
self._test_result(coeffs, interval, 0.0)
# No roots
coeffs = mb.get_coeffs(True)
coeffs = coeffs - np.array([0, 0, 0, 0, f(coeffs, mb.center) + 1])
interval = mb.lower, mb.upper
self._test_result(coeffs, interval, np.nan, np.nan)
# No roots on the left from the stationary point
coeffs = mb.get_coeffs(True)
interval = mb.lower, (mb.lower - mb.r) / 2
self._test_result(coeffs, interval, np.nan, np.nan)
# No roots on the right from the stationary point
coeffs = mb.get_coeffs(True)
interval = mb.upper - (mb.upper - mb.r) / 2, mb.upper
self._test_result(coeffs, interval, np.nan, np.nan)
# No roots because the metaball does not reach y = 1.
mb = Metaball(0.0, 0.5, 0.0)
coeffs = mb.get_coeffs(True)
interval = mb.lower, mb.upper
self._test_result(coeffs, interval, np.nan, np.nan)
def test_thickness(self):
def test(coeffs,
roots,
expected_thickness,
expected_previous,
previous=np.nan,
last_derivative_sgn=-2,
expected_last_derivative_sgn=None):
thickness_res, previous_res, last_derivative_sgn_res = \
self.get_thickness_addition(coeffs,
roots, previous,
last_derivative_sgn)
np.testing.assert_almost_equal(expected_thickness,
thickness_res,
decimal=self.precision_places)
np.testing.assert_almost_equal(expected_previous,
previous_res,
decimal=self.precision_places)
if expected_last_derivative_sgn is not None:
np.testing.assert_almost_equal(expected_last_derivative_sgn,
last_derivative_sgn_res)
mb_0 = Metaball(1.0, 2.0, 0.0)
coeffs = mb_0.get_coeffs(True)
# Simple cases, no previous, no last accounted value.
roots = 4 * [np.nan]
test(coeffs, roots, 0.0, np.nan, np.nan)
roots_base = [-2.5, -1, 1, 2.5, np.nan]
for i in range(1, 5):
roots[:i] = roots_base[:i]
if i % 2 == 0:
# Roots are coupled.
test(coeffs, roots + [np.nan], i / 2 * 1.5, np.nan)
else:
# Previous is created.
test(coeffs,
roots + [np.nan], (i - 1) / 2 * 1.5,
roots[i - 1],
expected_last_derivative_sgn=sgn(
f(derivative(coeffs), roots[i])))
# Previous is not nan.
roots = [-1, 1, 1.5, np.nan, np.nan]
previous = -3
test(coeffs,
roots,
2,
roots[1],
previous=previous,
last_derivative_sgn=f(derivative(coeffs), previous))
roots = [-1, 1, 2.5, 4, np.nan]
previous = -3
test(coeffs, roots, 3.5, np.nan, previous=previous)
# The leftmost root was coupled before, just skip the current
# first root.
roots = [1, 2, np.nan, np.nan, np.nan]
previous = np.nan
last_accounted = 1 - self.pixel_size / 2
last_derivative_sgn = sgn(f(derivative(coeffs), last_accounted))
test(coeffs, roots, 0, roots[1], previous, last_derivative_sgn)
# The leftmost root existed before, but was not coupled.
# Take the root into account.
roots = [1, 2, np.nan, np.nan, np.nan]
previous = 1 - self.pixel_size / 2
last_derivative_sgn = sgn(f(derivative(coeffs), previous))
test(coeffs, roots, 1, np.nan, previous, last_derivative_sgn)
# Extremum in the middle of the new roots, but the surrounding
# roots lie on the opposite sides, thus take them into account.
roots = [-2.1, -self.pixel_size / 4, self.pixel_size / 4, 2.1, np.nan]
test(coeffs, roots, roots[1] - roots[0] + roots[3] - roots[2], np.nan)
# Make the middle roots to be on the same slope of the extremum,
# thus do not take them both into account.
# First root is not coupled with anything, thus must be coupled
# with one of them.
roots = [-1, self.pixel_size / 4, self.pixel_size / 2, 2.5, np.nan]
test(coeffs, roots, roots[1] - roots[0], roots[-2])
# Create some previous value in order for the first root to be
# coupled with some previous, then one of the two roots is saved
# as previous.
roots = [-1, self.pixel_size / 4, self.pixel_size / 2, np.nan, np.nan]
previous = -2.5
test(coeffs, roots, roots[0] - previous, roots[1], previous=previous)
# Multiple roots.
roots = [1, 1, 1, 1, np.nan]
test(coeffs,
roots,
0,
1,
expected_last_derivative_sgn=sgn(f(derivative(coeffs), 1)))
# Multiple roots with previous.
previous = 1
last_derivative_sgn = sgn(f(derivative(coeffs), previous))
test(coeffs,
roots,
0,
previous,
expected_last_derivative_sgn=last_derivative_sgn)
# Left multiple roots.
roots = [-1, -1, 1.5, 1.7, np.nan]
test(coeffs, roots, roots[2] - roots[0], np.nan)
# Right multiple roots.
roots = [-1, 1, 2.5, 2.5, np.nan]
test(coeffs, roots, roots[1] - roots[0], roots[-2])
# Stationary points as roots which are not extrema.
roots = [-2, 2, np.nan, np.nan, np.nan]
test(coeffs, roots, 4, np.nan, np.nan)
# More than polynomial degree roots.
roots = [-2.5, -1, 1, 2.5, 2.5]
test(coeffs, roots, 3, np.nan)
def test_thickness(self):
def test(coeffs, roots, expected_thickness, expected_previous,
previous=np.nan, last_derivative_sgn=-2,
expected_last_derivative_sgn=None):
thickness_res, previous_res, last_derivative_sgn_res = \
self.get_thickness_addition(coeffs,
roots, previous,
last_derivative_sgn)
np.testing.assert_almost_equal(expected_thickness, thickness_res,
decimal=self.precision_places)
np.testing.assert_almost_equal(expected_previous, previous_res,
decimal=self.precision_places)
if expected_last_derivative_sgn is not None:
np.testing.assert_almost_equal(expected_last_derivative_sgn,
last_derivative_sgn_res)
mb_0 = Metaball(1.0, 2.0, 0.0)
coeffs = mb_0.get_coeffs(True)
# Simple cases, no previous, no last accounted value.
roots = 4 * [np.nan]
test(coeffs, roots, 0.0, np.nan, np.nan)
roots_base = [-2.5, -1, 1, 2.5, np.nan]
for i in range(1, 5):
roots[:i] = roots_base[:i]
if i % 2 == 0:
# Roots are coupled.
test(coeffs, roots + [np.nan], i / 2 * 1.5, np.nan)
else:
# Previous is created.
test(coeffs, roots + [np.nan], (i - 1) / 2 * 1.5, roots[i - 1],
expected_last_derivative_sgn=sgn(f(derivative(coeffs),
roots[i])))
# Previous is not nan.
roots = [-1, 1, 1.5, np.nan, np.nan]
previous = -3
test(coeffs, roots, 2, roots[1], previous=previous,
last_derivative_sgn=f(derivative(coeffs), previous))
roots = [-1, 1, 2.5, 4, np.nan]
previous = -3
test(coeffs, roots, 3.5, np.nan, previous=previous)
# The leftmost root was coupled before, just skip the current
# first root.
roots = [1, 2, np.nan, np.nan, np.nan]
previous = np.nan
last_accounted = 1 - self.pixel_size / 2
last_derivative_sgn = sgn(f(derivative(coeffs), last_accounted))
test(coeffs, roots, 0, roots[1], previous, last_derivative_sgn)
# The leftmost root existed before, but was not coupled.
# Take the root into account.
roots = [1, 2, np.nan, np.nan, np.nan]
previous = 1 - self.pixel_size / 2
last_derivative_sgn = sgn(f(derivative(coeffs), previous))
test(coeffs, roots, 1, np.nan, previous, last_derivative_sgn)
# Extremum in the middle of the new roots, but the surrounding
# roots lie on the opposite sides, thus take them into account.
roots = [-2.1, -self.pixel_size / 4, self.pixel_size / 4, 2.1, np.nan]
test(coeffs, roots, roots[1] - roots[0] + roots[3] - roots[2], np.nan)
# Make the middle roots to be on the same slope of the extremum,
# thus do not take them both into account.
# First root is not coupled with anything, thus must be coupled
# with one of them.
roots = [-1, self.pixel_size / 4, self.pixel_size / 2, 2.5, np.nan]
test(coeffs, roots, roots[1] - roots[0], roots[-2])
# Create some previous value in order for the first root to be
# coupled with some previous, then one of the two roots is saved
# as previous.
roots = [-1, self.pixel_size / 4, self.pixel_size / 2, np.nan, np.nan]
previous = -2.5
test(coeffs, roots, roots[0] - previous, roots[1], previous=previous)
# Multiple roots.
roots = [1, 1, 1, 1, np.nan]
test(coeffs, roots, 0, 1,
expected_last_derivative_sgn=sgn(f(derivative(coeffs), 1)))
# Multiple roots with previous.
previous = 1
last_derivative_sgn = sgn(f(derivative(coeffs), previous))
test(coeffs, roots, 0, previous,
expected_last_derivative_sgn=last_derivative_sgn)
# Left multiple roots.
roots = [-1, -1, 1.5, 1.7, np.nan]
test(coeffs, roots, roots[2] - roots[0], np.nan)
# Right multiple roots.
roots = [-1, 1, 2.5, 2.5, np.nan]
test(coeffs, roots, roots[1] - roots[0], roots[-2])
# Stationary points as roots which are not extrema.
roots = [-2, 2, np.nan, np.nan, np.nan]
test(coeffs, roots, 4, np.nan, np.nan)
# More than polynomial degree roots.
roots = [-2.5, -1, 1, 2.5, 2.5]
test(coeffs, roots, 3, np.nan)
def test_one_metaball(self):
# Regular two roots.
for r in np.linspace(self.pixel_size / 4, 1.0 - self.pixel_size, 10):
mb = Metaball(r, 1.0, 0.0)
coeffs = mb.get_coeffs(True)
interval = (mb.lower, mb.upper)
self._test_result(coeffs, interval, -r, r)
# One root left
mb = Metaball(1.0, 2.0, 0.0)
coeffs = mb.get_coeffs(True)
interval = mb.lower, mb.center - mb.r / 2
self._test_result(coeffs, interval, -1.0, np.nan)
# One root right
interval = mb.center + mb.r / 2, mb.upper
self._test_result(coeffs, interval, 1, np.nan)
# Make interval end points stationary.
coeffs = mb.get_coeffs(True)
interval = (mb.lower, 0)
self._test_result(coeffs, interval, -1, np.nan)
interval = (0, mb.upper)
self._test_result(coeffs, interval, 1, np.nan)
# Put interval end points to roots.
# In right endpoint.
interval = mb.lower, -1.0
self._test_result(coeffs, interval, -1.0)
# In left endpoint.
interval = -1.0, 0.0
self._test_result(coeffs, interval)
# By epsilon too far -> no root expected
interval = mb.lower, - 1 - self.pixel_size
self._test_result(coeffs, interval)
interval = 1.0, mb.upper
# By epsilon too far -> no root expected
interval = 1.0 + self.pixel_size, mb.upper
self._test_result(coeffs, interval, np.nan, np.nan)
# Left end point is a root.
coeffs = coeffs - np.array([0, 0, 0, 0, f(coeffs, mb.center)])
interval = 0.0, mb.upper
self._test_result(coeffs, interval, 0.0)
# Right end point is a root.
interval = mb.lower, 0.0
self._test_result(coeffs, interval, 0.0)
# No roots
coeffs = mb.get_coeffs(True)
coeffs = coeffs - np.array([0, 0, 0, 0, f(coeffs, mb.center) + 1])
interval = mb.lower, mb.upper
self._test_result(coeffs, interval, np.nan, np.nan)
# No roots on the left from the stationary point
coeffs = mb.get_coeffs(True)
interval = mb.lower, (mb.lower - mb.r) / 2
self._test_result(coeffs, interval, np.nan, np.nan)
# No roots on the right from the stationary point
coeffs = mb.get_coeffs(True)
interval = mb.upper - (mb.upper - mb.r) / 2, mb.upper
self._test_result(coeffs, interval, np.nan, np.nan)
# No roots because the metaball does not reach y = 1.
mb = Metaball(0.0, 0.5, 0.0)
coeffs = mb.get_coeffs(True)
interval = mb.lower, mb.upper
self._test_result(coeffs, interval, np.nan, np.nan)