16-825 A1 — Rendering Basics with PyTorch3D

1. Practicing with Cameras

1.1 — Cow 360° turntable — **1.1** 360° cow turntable (GIF).

1.2 — Dolly Zoom effect — **1.2** Dolly Zoom effect (GIF).

2. Practicing with Meshes

2.1 — Tetrahedron 360° — **2.1** Tetrahedron — 360° GIF. (4 verts / 4 faces)

2.2 — Cube 360° — **2.2** Cube — 360° GIF. (8 verts / 12 tri faces)

3. Re-texturing a Mesh

3 — Cow front→back color gradient — **3.1** Cow front→back gradient (GIF).

Color mapping (swapped): front (z_min) uses color2, back (z_max) uses color1.

4. Camera Transformations

New extrinsics are formed as R = R_relative @ R_0, T = R_relative @ T_0 + T_relative. R_relative rotates the original camera frame (yaw/pitch/roll); T_relative then translates the camera in that rotated frame (dolly/slide) to keep framing sensible.

4.0 — Base camera view — **4.0** Base view (R_rel=I, T_rel=0).

4.1 — Roll clockwise 90° — **4.1** Roll clockwise 90°.

4.2 — Dolly out — **4.2** Dolly out (translate farther along view direction).

4.3 — Yaw left 10° — **4.3** Yaw left 10°.

4.4 — Right-side view — **4.4** Right-side view (rotate + translate to reframe).

Chosen R_relative / T_relative (values)

# Base
R_rel = I_3
T_rel = [0, 0, 0]

# 4.1 Roll clockwise 90° about +Z
R_rel = [[ 0,  1, 0],
         [-1,  0, 0],
         [ 0,  0, 1]]
T_rel = [0, 0, 0]

# 4.2 Dolly out (farther)
R_rel = I_3
T_rel = [0, 0, +2.0]

# 4.3 Yaw left 10° about +Y
# c = cos(10°), s = sin(10°)
R_rel = [[  c, 0,  s],
         [  0, 1,  0],
         [ -s, 0,  c]]
T_rel = [0, 0, 0]

# 4.4 Right-side view (quarter turn right) + reframe
R_rel = [[0, 0, 1],
         [0, 1, 0],
         [-1,0, 0]]
T_rel = [-3, 0, 3]

5. Generic 3D Representations

5.1 RGB-D → Point Clouds

5.1 — Plant point clouds side-by-side — **5.1** View1 / View2 / Union — side-by-side 360° GIF.

5.2 Parametric Functions

5.2 — Torus still — **Torus (still)** — parametric point cloud.

5.2 — Torus 360° GIF — **Torus (360° GIF)** — hole visible.

5.2 extra — Vase surface of revolution — **Extra** — surface of revolution (vase) 360° GIF.

5.3 Implicit Surfaces

5.3 — Torus mesh still — **Torus mesh (still)** — implicit → marching cubes.

5.3 — Torus mesh 360° GIF — **Torus mesh (360° GIF)** — hole visible.

5.3 — Heart mesh still — **New object** — heart mesh (still).

5.3 — Heart mesh 360° GIF — **New object** — heart mesh (360° GIF).

Discussion (5.1–5.3)

RGB-D unprojection: For pixel (u,v) with depth d and intrinsics (f_x,f_y,c_x,c_y): X=((u-c_x)/f_x)·d, Y=((v-c_y)/f_y)·d, Z=d; convert via extrinsics to world. Concatenating two views (union) densifies coverage and fills occlusions.

Parametric torus: x=(R+r\cos v)\cos u, y=r\sin v, z=(R+r\cos v)\sin u. Tilt camera (e.g., elev≈30°) so the hole is visible; procedural color for “chocolate glaze”.

Implicit torus mesh: Evaluate F(x,y,z)=(\sqrt{x^2+z^2}-R)^2 + y^2 - r^2 on a voxel grid and extract the 0-level via marching cubes. Higher voxel resolution → smoother mesh but more time/memory.

Point cloud vs. mesh: Points render fast and are easy to generate; silhouettes can look fuzzy unless point radius is tuned. Meshes give crisp silhouettes and standard shading; require topology; performance scales with face count.

16-825 Assignment 1 — Rendering Basics with PyTorch3D