def test_it(self):
import bezier
nodes1 = np.asfortranarray([[0.0, 1.0, 0.0], [0.0, 0.0, 1.0]])
surface1 = bezier.Surface(nodes1, degree=1, _copy=False)
nodes2 = np.asfortranarray([[0.25, -0.75, 0.25], [0.25, 0.25, -0.75]])
surface2 = bezier.Surface(nodes2, degree=1, _copy=False)
edge_nodes = tuple(edge._nodes for edge in surface1.edges) + tuple(
edge._nodes for edge in surface2.edges)
edge_info = (
(0, 0.0, 0.25),
(5, 0.75, 1.0),
(3, 0.0, 0.25),
(2, 0.75, 1.0),
)
result = self._call_function_under_test(edge_info, edge_nodes)
self.assertIsInstance(result, bezier.CurvedPolygon)
self.assertEqual(result._metadata, edge_info)
self.assertEqual(result.num_sides, 4)
# pylint: disable=unbalanced-tuple-unpacking
edge0, edge1, edge2, edge3 = result._edges
# pylint: enable=unbalanced-tuple-unpacking
# First edge.
expected = np.asfortranarray([[0.0, 0.25], [0.0, 0.0]])
self.assertEqual(edge0._nodes, expected)
# Second edge.
expected = np.asfortranarray([[0.25, 0.25], [0.0, 0.25]])
self.assertEqual(edge1._nodes, expected)
# Third edge.
expected = np.asfortranarray([[0.25, 0.0], [0.25, 0.25]])
self.assertEqual(edge2._nodes, expected)
# Fourth edge.
expected = np.asfortranarray([[0.0, 0.0], [0.25, 0.0]])
self.assertEqual(edge3._nodes, expected)
def unit_triangle():
"""Image for :class:`.surface.Surface` docstring."""
if NO_IMAGES:
return
nodes = np.asfortranarray([[0.0, 1.0, 0.0], [0.0, 0.0, 1.0]])
surface = bezier.Surface(nodes, degree=1)
ax = surface.plot(256)
ax.axis("scaled")
_plot_helpers.add_plot_boundary(ax)
save_image(ax.figure, "unit_triangle.png")
def image1():
figure, (ax1, ax2) = plt.subplots(1, 2)
nodes1 = np.asfortranarray([[0.0, 3.0, 7.0], [5.0, 0.0, 8.0]])
triangle1 = bezier.Surface(nodes1, degree=1)
triangle1.plot(256, ax=ax1)
nodes2 = np.asfortranarray([[0.0, 1.0, 2.0, 2.0, 2.0, 0.0],
[0.0, 0.0, 0.0, 1.0, 2.0, 2.0]])
triangle2 = bezier.Surface(nodes2, degree=2)
triangle2.plot(256, ax=ax2)
params = np.asfortranarray([[0.125, 0.125], [0.125, 0.75]])
points1 = triangle1.evaluate_cartesian_multi(params)
ax1.plot(points1[0, :], points1[1, :], marker="o", color="black")
points2 = triangle2.evaluate_cartesian_multi(params)
ax2.plot(points2[0, :], points2[1, :], marker="o", color="black")
for ax in (ax1, ax2):
ax.tick_params(labelsize=LABEL_SIZE, which="both")
ax.axis("equal")
ax1.set_title("Convex", fontsize=FONT_SIZE)
ax2.set_title("Not (Necessarily) Convex", fontsize=FONT_SIZE)
figure.set_size_inches(8.74, 4.8)
figure.subplots_adjust(left=0.05,
bottom=0.06,
right=0.99,
top=0.93,
wspace=0.12,
hspace=0.2)
filename = "not_convex.pdf"
path = plot_utils.get_path("slides", filename)
figure.savefig(path)
print("Saved {}".format(filename))
plt.close(figure)
def image1():
nodes = np.asfortranarray([[0.0, 1.5, 3.0, 0.75, 2.25, 0.0],
[0.0, -0.5, 0.0, 1.0, 1.5, 2.0]])
triangle = bezier.Surface(nodes, degree=2, _copy=False)
point = triangle.evaluate_cartesian(0.25, 0.125)
xv, yv = point.flatten()
sub_triangles = triangle.subdivide()
edges = triangle.edges
figure, all_axes = plt.subplots(2, 2, sharex=True, sharey=True)
all_axes = all_axes.flatten()
for ax, sub_triangle in zip(all_axes, sub_triangles):
# Add the bounding box for the sub triangle.
add_patch(sub_triangle, ax, plot_utils.GREEN)
# Add the triangle boundary to each subplot.
for edge in edges:
edge.plot(256, ax=ax, color=plot_utils.BLUE)
# Add the sub triangle.
sub_triangle.plot(256, ax=ax, color=plot_utils.BLUE)
# Add the point to be found.
ax.plot(
[xv],
[yv],
color="black",
marker="o",
markersize=3,
linestyle="none",
)
for ax in all_axes:
ax.set_aspect("equal")
# One axis sets all axis
all_axes[0].set_xticklabels([])
all_axes[0].set_yticklabels([])
figure.set_size_inches(4.0, 3.0)
figure.subplots_adjust(left=0.01,
bottom=0.01,
right=0.99,
top=0.99,
wspace=0.03,
hspace=0.04)
filename = "locate_in_triangle.pdf"
path = plot_utils.get_path("bezier-intersection", filename)
figure.savefig(path)
print("Saved {}".format(filename))
plt.close(figure)
def main():
nodes = np.asfortranarray(
[[1.0, 0.0, 1.0, 0.0, 0.0, 0.0], [0.0, 0.0, 1.0, 0.0, 0.0, 1.0]]
)
bez_triangle = bezier.Surface(nodes, degree=2, _copy=False)
# b(s, t) = [(1 - s - t)^2 + t^2, s^2 + t^2]
figure, ((ax1, ax2, ax3), (ax4, ax5, ax6)) = plt.subplots(
2, 3, sharex=True, sharey=True
)
for ax in (ax1, ax2, ax3):
ax.plot([0, 1, 0, 0], [0, 0, 1, 0], color="black")
# The "left" edge b(0, t) = [(1 - t)^2 + t^2, t^2] lies on the algebraic
# curve 4x = (1 + x - y)^2. Plugging b(s, t) into this algebraic curve
# we find interior points as well.
sv = np.linspace(0.0, 2.0 / 5.0, 250)
sqrt_part = np.sqrt((2.0 - 5.0 * sv) / (2.0 - sv))
poly1 = np.empty((501, 2))
poly1[:250, 0] = sv
poly1[:250, 1] = 0.5 * (1.0 - sv + sqrt_part)
poly1[250:499, 0] = sv[-2::-1]
poly1[250:499, 1] = 0.5 * (1.0 - sv[-2::-1] - sqrt_part[-2::-1])
poly1[499, :] = 1.0, 0.0
poly1[500, :] = 0.0, 1.0
patch = matplotlib.patches.PathPatch(
matplotlib.path.Path(poly1), color=plot_utils.GREEN
)
ax1.add_patch(patch)
for ax in (ax2, ax3):
patch = matplotlib.patches.PathPatch(
matplotlib.path.Path(poly1), color=plot_utils.GREEN, alpha=ALPHA
)
ax.add_patch(patch)
for ax in (ax4, ax5, ax6):
bez_triangle.plot(256, ax=ax, color=plot_utils.GREEN)
for edge in bez_triangle.edges:
edge.plot(256, ax=ax, color="black")
ax4.patches[-1].set_alpha(1.0)
ax5.patches[-1].set_alpha(ALPHA)
ax6.patches[-1].set_alpha(ALPHA)
# det(J) = 4[t^2 + (s - 1) t + (s - s^2)]
sv = np.linspace(0.0, 1.0 / 5.0, 250)
sqrt_part = np.sqrt((1.0 - sv) * (1.0 - 5.0 * sv))
poly2 = np.empty((500, 2))
poly2[:250, 0] = sv
poly2[:250, 1] = 0.5 * (1.0 - sv - sqrt_part)
poly2[250:499, 0] = sv[-2::-1]
poly2[250:499, 1] = 0.5 * (1.0 - sv[-2::-1] + sqrt_part[-2::-1])
poly2[499, :] = 0.0, 0.0
for ax in (ax1, ax2, ax3):
ax.plot(
poly2[:499, 0], poly2[:499, 1], color="black", linestyle="dashed"
)
patch = matplotlib.patches.PathPatch(
matplotlib.path.Path(poly2), color=plot_utils.RED
)
ax3.add_patch(patch)
for ax in (ax1, ax2):
patch = matplotlib.patches.PathPatch(
matplotlib.path.Path(poly2), color="black", alpha=ALPHA
)
ax.add_patch(patch)
# Shared slice in between two curves.
poly3 = np.empty((997, 2))
poly3[:499, :] = poly2[498::-1, :]
poly3[499:, :] = poly1[497::-1, :]
patch = matplotlib.patches.PathPatch(
matplotlib.path.Path(poly3), color=plot_utils.BLUE
)
ax2.add_patch(patch)
for ax in (ax1, ax3):
patch = matplotlib.patches.PathPatch(
matplotlib.path.Path(poly3), color="black", alpha=ALPHA
)
ax.add_patch(patch)
# Now, compute the image of the det(J) = 0 curve under b(s, t).
poly4 = np.empty((598, 2))
sv = poly2[:499, 0][::-1]
tv = poly2[:499, 1][::-1]
poly4[:499, 0] = (1.0 - sv - tv) * (1.0 - sv - tv) + sv * sv
poly4[:499, 1] = sv * sv + tv * tv
for ax in (ax4, ax5, ax6):
ax.plot(
poly4[:499, 0], poly4[:499, 1], color="black", linestyle="dashed"
)
# Combine this with the image b(0, t)
tv = np.linspace(0.0, 1.0, 100)[1:]
poly4[499:, 0] = (1.0 - tv) * (1.0 - tv)
poly4[499:, 1] = tv * tv
patch = matplotlib.patches.PathPatch(
matplotlib.path.Path(poly4), color="black", alpha=ALPHA
)
ax4.add_patch(patch)
for ax, color in ((ax5, plot_utils.BLUE), (ax6, plot_utils.RED)):
patch = matplotlib.patches.PathPatch(
matplotlib.path.Path(poly4), color=color
)
ax.add_patch(patch)
for ax in (ax1, ax2, ax3, ax4, ax5, ax6):
ax.set_aspect("equal")
# One axis sets all axis
ax1.set_xticklabels([])
ax1.set_yticklabels([])
ax2.text(
0.3,
0.35,
r"$+$",
horizontalalignment="center",
verticalalignment="center",
fontsize=16,
fontweight="bold",
color="black",
)
ax2.add_patch(
matplotlib.patches.Circle(
(0.3, 0.35),
radius=0.06,
fill=False,
edgecolor="black",
linewidth=1.0,
)
)
ax3.text(
0.1,
0.42,
r"$-$",
horizontalalignment="center",
verticalalignment="center",
fontsize=16,
fontweight="bold",
color="black",
)
ax3.add_patch(
matplotlib.patches.Circle(
(0.1, 0.422),
radius=0.06,
fill=False,
edgecolor="black",
linewidth=1.0,
)
)
figure.set_size_inches(4.8, 3.3)
figure.subplots_adjust(
left=0.01, bottom=0.01, right=0.99, top=0.99, wspace=0.04, hspace=0.0
)
filename = "inverted_element.pdf"
path = plot_utils.get_path("preliminaries", filename)
figure.savefig(path)
print("Saved {}".format(filename))
plt.close(figure)
def boundary_leak(slide_num):
bez_triangle = bezier.Surface(NODES, degree=2, _copy=False)
figure, ((ax1, ax2), (ax3, ax4)) = plt.subplots(2,
2,
sharex=True,
sharey=True)
# Add reference triangle to both top subplots.
for ax in (ax1, ax2):
if slide_num == 1:
ax.plot([0, 1], [0, 0], color=plot_utils.BLUE)
ax.plot([1, 0], [0, 1], color="black", alpha=ALPHA)
ax.plot([0, 0], [1, 0], color="black", alpha=ALPHA)
elif slide_num == 2:
ax.plot([0, 1], [0, 0], color="black", alpha=ALPHA)
ax.plot([1, 0], [0, 1], color=plot_utils.BLUE)
ax.plot([0, 0], [1, 0], color="black", alpha=ALPHA)
elif slide_num == 3:
ax.plot([0, 1], [0, 0], color="black", alpha=ALPHA)
ax.plot([1, 0], [0, 1], color="black", alpha=ALPHA)
ax.plot([0, 0], [1, 0], color=plot_utils.BLUE)
else:
ax.plot([0, 1], [0, 0], color="black", alpha=ALPHA)
ax.plot([1, 0], [0, 1], color="black", alpha=ALPHA)
ax.plot([0, 0], [1, 0], color="black", alpha=ALPHA)
# Add the "wrong" triangle to both bottom subplots.
edge1, edge2, edge3 = bez_triangle.edges
for ax in (ax3, ax4):
bez_triangle.plot(256, ax=ax, color="black")
ax.patches[-1].set_alpha(ALPHA)
# Remove the black boundary and add colored.
ax.lines.pop()
if slide_num == 1:
edge1.plot(256, ax=ax, color=plot_utils.BLUE)
elif slide_num == 2:
edge2.plot(256, ax=ax, color=plot_utils.BLUE)
elif slide_num == 3:
edge3.plot(256, ax=ax, color=plot_utils.BLUE)
# det(J) = 4[t^2 + (s - 1) t + (s - s^2)]
sv = np.linspace(0.0, 1.0 / 5.0, 250)
sqrt_part = np.sqrt((1.0 - sv) * (1.0 - 5.0 * sv))
jacobian_zero = np.empty((499, 2))
jacobian_zero[:250, 0] = sv
jacobian_zero[:250, 1] = 0.5 * (1.0 - sv - sqrt_part)
jacobian_zero[250:, 0] = sv[-2::-1]
jacobian_zero[250:, 1] = 0.5 * (1.0 - sv[-2::-1] + sqrt_part[-2::-1])
ax2.plot(
jacobian_zero[:, 0],
jacobian_zero[:, 1],
color="black",
linestyle="dashed",
)
# Now, compute the image of the det(J) = 0 curve under b(s, t).
poly4 = np.empty((598, 2))
sv = jacobian_zero[::-1, 0]
tv = jacobian_zero[::-1, 1]
poly4[:499, 0] = (1.0 - sv - tv) * (1.0 - sv - tv) + sv * sv
poly4[:499, 1] = sv * sv + tv * tv
ax4.plot(poly4[:499, 0], poly4[:499, 1], color="black", linestyle="dashed")
# Combine this with the image b(0, t)
tv = np.linspace(0.0, 1.0, 100)[1:]
poly4[499:, 0] = (1.0 - tv) * (1.0 - tv)
poly4[499:, 1] = tv * tv
if slide_num == 6:
patch = matplotlib.patches.PathPatch(matplotlib.path.Path(poly4),
color="black",
alpha=1.5 * ALPHA)
ax4.add_patch(patch)
for ax in (ax1, ax2, ax3, ax4):
ax.set_aspect("equal")
# One axis sets all axis
ax1.set_xticklabels([])
ax1.set_yticklabels([])
if slide_num < 5:
ax4.set_visible(False)
if slide_num < 4:
ax2.set_visible(False)
figure.set_size_inches(3.2, 3.3)
figure.subplots_adjust(left=0.01,
bottom=0.01,
right=0.99,
top=0.99,
wspace=0.04,
hspace=0.0)
filename = "boundary_leak{}.pdf".format(slide_num)
path = plot_utils.get_path("slides", filename)
figure.savefig(path)
print("Saved {}".format(filename))
plt.close(figure)