16-889 Assignment 1

1. Practicing with Cameras

1.1. 360-degree Renders (5 points)

1.2. Re-creating the Dolly Zoom (10 points)

2. Practicing with Meshes

2.1 Constructing a Tetrahedron (5 points)

The tetrahedron should have 4 vertices and 4 faces.

2.2 Constructing a Cube (5 points)

The cube should have 8 vertices and 12 faces.

3. Re-texturing a mesh (10 points)

color1 = (0, 1, 0) and color2 = (1, 0, 0)

4. Camera Transformations (20 points)

R_relative = [[0, 1, 0], [-1, 0, 0], [0, 0, 1]]

T_relative = [0, 0, 0]

R_relative = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]

T_relative = [0, 0, 3]

R_relative = [[0.9962, 0, 0.0872], [0, 1, 0], [-0.0872, 0, 0.9962]]

T_relative = [0.25, -0.5, 0]

R_relative = [[0, 0, 1], [0, 1, 0], [-1, 0, 0]]

T_relative = [-3, 0, 3]

5. Rendering Generic 3D Representations

5.1 Rendering Point Clouds from RGB-D Images (10 points)

5.2 Parametric Functions (10 points)

5.2 Implicit Surfaces (10 points)

When rendering an object as point cloud using its parametric function, we sample values from its parameters by pre-defined steps to generate corresponding 3D coordinates for point cloud. As generally the domain and range of the parameters is given for a parametric function, we can simply sample the value accordingly by a pre-defined step. And the time complexity of rendering using parametric function is \(O(n^k)\), where n is the number of samples for each parameters and k is the number of parameters.

While when rendering an object as mesh using its implicit function, we sample values from the whole 3D space, evaluate the implicit function and find a corresponding zero level-set. Different from the parametric function case, we usually don't know the domain or the range for a given implicit function (that is, how large the object is), making it harder to sample values from the whole 3D space, as its domain could be unlimited large. But if we do know the domain for its 3D coordinates, then the time complexity of rendering using implicit function is \(O(n^3)\), where n is the number of samples for each dimension. Besides, to render an object as mesh smoothly takes extra effort to build a smooth embedding array, making its time complexity higher.