1 Rotation
1.1 Your answer here
1.2.a your answer here
1.2.b your answer here
1.2.c
In [1]:
# Note Matplotlib is only suitable for simple 3D visualization.
# For later problems, you should not use Matplotlib to do the plotting
from mpl_toolkits.mplot3d import Axes3D
import numpy as np
import matplotlib.pyplot as plt
def show_points(points):
fig = plt.figure()
ax = fig.gca(projection=’3d’)
ax.set_xlim3d([-2, 2])
ax.set_ylim3d([-2, 2])
ax.set_zlim3d([0, 4])
ax.scatter(points[0], points[2], points[1])
def compare_points(points1, points2):
fig = plt.figure()
ax = fig.gca(projection=’3d’)
ax.set_xlim3d([-2, 2])
ax.set_ylim3d([-2, 2])
ax.set_zlim3d([0, 4])
ax.scatter(points1[0], points1[2], points1[1])
ax.scatter(points2[0], points2[2], points2[1])
In [2]:
npz = np.load(‘HW1_P1.npz’)
X = npz[‘X’]
Y = npz[‘Y’]
compare_points(X, Y) # noisy teapotsand

In [3]:
# copy-paste your hw0 solve module here
def hw0_solve(A, b, eps):
x = np.zeros(A.shape[1])
return x
In [4]:
R1 = np.eye(3)
# solve this problem here, and store your final results in R1
for __ in range(100):
pass
In [5]:
# Testing code, you should see the points of the 2 teapots roughly overlap
compare_points(R1@X, Y)
R1.T@R1
Out[5]:
array([[1., 0., 0.],
[0., 1., 0.],
[0., 0., 1.]])

1.4.a your solution here
1.4.b your solution here
2 Geometry
2.1 your solution here
In [13]:
a, b, c = 1, 1, 0.5
In [20]:
# These are some convenient functions to create open3d geometries and plot them
# The viewing direction is fine-tuned for this problem, you should not change them
import open3d
import numpy as np
import matplotlib.pyplot as plt
vis = open3d.visualization.Visualizer()
vis.create_window(visible = False)
def draw_geometries(geoms):
for g in geoms:
vis.add_geometry(g)
view_ctl = vis.get_view_control()
view_ctl.set_up((0, 1e-4, 1))
view_ctl.set_front((0, 0.5, 2))
view_ctl.set_lookat((0, 0, 0))
# do not change this view point
vis.update_renderer()
img = vis.capture_screen_float_buffer(True)
plt.figure(figsize=(8,6))
plt.imshow(np.asarray(img)[::-1, ::-1])
for g in geoms:
vis.remove_geometry(g)
def create_arrow_from_vector(origin, vector):
”’
origin: origin of the arrow
vector: direction of the arrow
”’
v = np.array(vector)
v /= np.linalg.norm(v)
z = np.array([0,0,1])
angle = np.arccos(z@v)
arrow = open3d.geometry.TriangleMesh.create_arrow(0.05, 0.1, 0.25, 0.2)
arrow.paint_uniform_color([1,0,1])
T = np.eye(4)
T[:3, 3] = np.array(origin)
T[:3,:3] = open3d.geometry.get_rotation_matrix_from_axis_angle(np.cross(z, v) * angle)
arrow.transform(T)
return arrow
def create_ellipsoid(a,b,c):
sphere = open3d.geometry.TriangleMesh.create_sphere()
sphere.transform(np.diag([a,b,c,1]))
sphere.compute_vertex_normals()
return sphere
def create_lines(points):
lines = []
for p1, p2 in zip(points[:-1], points[1:]):
height = np.linalg.norm(p2-p1)
center = (p1+p2) / 2
d = p2-p1
d /= np.linalg.norm(d)
axis = np.cross(np.array([0,0,1]), d)
axis /= np.linalg.norm(axis)
angle = np.arccos(np.array([0,0,1]) @ d)
R = open3d.geometry.get_rotation_matrix_from_axis_angle(axis * angle)
T = np.eye(4)
T[:3,:3]=R
T[:3,3] = center
cylinder = open3d.geometry.TriangleMesh.create_cylinder(0.02, height)
cylinder.transform(T)
cylinder.paint_uniform_color([1,0,0])
lines.append(cylinder)
return lines
In [24]:
# exapmle code to draw ellipsoid, curve, and arrows
arrow = create_arrow_from_vector([0.,0.,1.], [1.,1.,0.])
ellipsoid = create_ellipsoid(a, b, c)
cf = open3d.geometry.TriangleMesh.create_coordinate_frame()
cf.scale(1.5, (0,0,0))
curve = create_lines(np.array([[1,1,1], [-1,1,1], [-1,-1,1], [1,-1,1], [1,1,1]], dtype=np.float64))
draw_geometries([ellipsoid, cf, arrow] + curve)

2.2 your solution here
2.3.a your computation here
2.3.b
2.3.c
2.3.d
2.3.e
2.4.a
2.4.b
2.4.c
2.5.a
2.5.b
2.5.c
2.5.d
3 Mesh
3.1 your proof here
3.2 your proof here
3.3 your solution here
In [1]:
# You may want to restart your notebook here, to reinitialize Open3D
import open3d
import numpy as np
import matplotlib.pyplot as plt
vis = open3d.visualization.Visualizer()
vis.create_window(visible = False)
# Make sure you call this function to draw the points for proper viewing direction
def draw_geometries(geoms):
for g in geoms:
vis.add_geometry(g)
view_ctl = vis.get_view_control()
view_ctl.set_up((0, 1, 0))
view_ctl.set_front((0, 2, 1))
view_ctl.set_lookat((0, 0, 0))
view_ctl.set_zoom(1)
# do not change this view point
vis.update_renderer()
img = vis.capture_screen_float_buffer(True)
plt.figure(figsize=(8,6))
plt.imshow(np.asarray(img))
for g in geoms:
vis.remove_geometry(g)
In [7]:
import trimesh
mesh = trimesh.load(‘sievert.obj’)
pcd = open3d.geometry.PointCloud()
pcd.points = open3d.utility.Vector3dVector(mesh.vertices)
draw_geometries([pcd])
plt.title(“Sievert’s surface”)
Out[7]:
Text(0.5, 1.0, “Sievert’s surface”)

3.4
4 Point Cloud
4.1 your solution here
4.2 your solution here
4.3 your solution here
4.4 your solution here