Question 1

1.1 360-degree Renders

1.2 Re-creating the Dolly Zoom

Question 2

2.1 Constructing a Tetrahedron

The tetrahedron mesh should have 4 vertices and 4 triangular faces.

2.2 Constructing a Cube

The cube mesh should have 8 vertices and 12 triangular faces.

Question 3

For color1 (front of the cow), I chose green [0, 1, 0] while for color2 (back of the cow) I chose a mix of red and green [1, 1, 0] which is yellowish.

Question 4

No Transform

Without any rotation or translation of the camera, we get the same cow image.

Transform 1

R_relative: We want to rotate the image clockwise. We do this by rotating the camera counter-clockwise: interchange x and y axes and reverse the direction of the y-axis.

T_relative: We do not translate the camera.

Transform 2

R_relative: We don't want to rotate the image.

T_relative: We want to move the camera further from the object by roughly 2 units perpendicular to the image plane.

Transform 3

R_relative: We keep the orientation of the camera fixed.

T_relative: We keep the camera at the same distance (perpendicular to the image plane) from the object. But, we move the camera a little to the right and a little up parallel to the image plane.

Transform 4

R_relative: We look at the cow from the side, meaning we interchange the x and z axes while flipping the z axis.

T_relative: We adjust the position of the camera to get the cow at a particular size: we move 3 units away from the cow and 3 units to the left (after rotation).

Question 5

5.1 Point Clouds

First Camera	Second Camera	Combined

5.2 Parametric Function

5.3 Implicit Surfaces

Rendering Mesh vs Point Cloud: Tradeoffs

1. Rendering speed: rendering mesh is faster than rendering point cloud.
2. Ease of use: Point clouds are easier to use since they are simply a list of 3D points.
3. Memory usage: Point clouds have lower memory usage since we only need one list of 3D points, whereas for mesh we need the list of vertices as well as the list of faces.
4. Rendering quality: Mesh covers the overall surface and produces a "whole" looking rendering, even with relatively few faces. With few point clouds, we get holes in the structure which is not very satisfying to render.

Question 6

This is a point cloud of 500 points which parametrically morphs itself from a cylinder to a Möbius strip with increasing number of loops while the camera rotates around it to give us a 360-degree view of the metamorphosis.

Question 7

10 Points	100 Points	1000 Points	10000 Points	Original Cow Mesh