CSCI 480 / 580 - Written Homework 1

Complete the following problems and submit your solutions to the HW1 assignment on Canvas. For all questions, justify your answers either or showing your work or giving a brief explanation. Please typeset your solutions using latex or similar¹. You may work with your classmates on these problems, but you must write up your own solutions individually without using notes or photos made during your collaborative discussions. As usual, this is new material, so please let me know if you think anything is unclear or ambiguous and I’ll make corrections or clarifications as needed.

1. Consider the geometry of a triangle mesh approximating a cylinder as implemented in A1 with \(n = 6\). Assume that each position coordinate value is stored as a 32-bit float and, where applicable, indices are stored as 32-bit integers. Calculate the number of bytes required to store the mesh’s geometry only (no texture coordinates or normals), stored using each of the following data structures. Assume there is no overhead required for distinguishing coordinates from indices, strips from fans, etc. Show your work:
   1. Triangle set
   2. Indexed mesh (as in A1)
   3. One indexed triangle strip for the shell and one triangle fan for each cap.

2. Consider a perspective camera in canonical pose (centered at the origin, looking down the -z axis). Assume the viewport is orthogonal to and centered on the \(-z\) axis. This camera’s horizontal field of view is the angle between the left-most viewing ray in the \(xz\) plane and the right-most viewing ray in the \(xz\) plane. Rather than specifying the viewport size and distance, a user might want to specify the viewport size and field of view. Derive a formula to set the value of \(d\) (the distance to the viewport), given desired viewport dimensions \(v_h, v_w\) and horizontal field of view \(\alpha\).

3. [This problem will help you with A2 TODO 9] In A1, we used a particular texture mapping function for the sphere. In my implementation, I mapped the parameter variables \((\phi, \theta)\) to texture coordinates \((u, v)\). In contrast, when raytracing an ideal sphere, we end up with the \(x,y,z\) coordinates of the intersection point on the surface of the sphere, but we don’t have easy access to the \(\phi\) and \(\theta\) coordinates of that point. To apply a texture to our raytraced sphere we need to be able to take the \((x,y,z)\) coordinates of a surface point and find its \((u,v)\) coordinates. Given a sphere centered at \((c_x, c_y, c_z)\) with radius \(r\), and a point \((x,y,z)\) on its surface, derive a formula for the \((u,v)\) coordinates of the point using the same texture mapping function as we used in A1. You may find Julia’s atan function helpful; note that contrary to what you might expect, atan is not exactly the arctangent function, but what is often known in other languages as atan2.

4. [This problem may help you with A2 TODO 8. The camera in A2 is Perspective not orthographic, but several steps are the same.]In class we derived viewing ray generation for general perspective cameras. We also talked about cameras that perform a parallel projection, but didn’t derive viewing rays. Remember that in a parallel (sometimes called orthographic projection), the “viewport” lies in the \(uv\) plane of the camera’s local coordinate system, and each pixel’s ray begins at the center of the pixel and points in the view direction. Assume a square viewport with \(vh = vw = 1\). Write a view ray generation procedure (in pseudocode or similar - mathy pseudocode is fine) that takes the following inputs and returns the viewing ray for pixel \((i, j)\) of an orthographic camera:

   eye - the 3D position of the center of the viewport
   view - the view direction of the camera
   up - a vector pointing "up" in the scene
   img_h, img_w, the integer pixel dimensions of the image
   i, j, the pixel coordinates (Julia-style, 1-indexed from the top left)

5. Consider the plane in 3D that contains the \(z\) axis and the vector \((1,1,0)\).

   • Write a parametric equation for the plane.
   • Write an implicit equation of the plane, with zero on one side of the equals sign. Hint: think of points in the plane as vectors; what property do they all have?

6. Consider a triangle in 3D, specified by the 3 points \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\). In each of the following, \(\mathbf{x}\) is a point in the plane occupied by the triangle. For each of the following scenarios, what you can be determined about the barycentric coordinates (\(\alpha,\beta, \gamma\)) of the point \(\mathbf{x}\) with respect to the triangle? If the coordinates can be determined exactly, give their values. If not, give the most strict inequality possible about each. If the situation does not constrain a given coordinate at all, you can simply say (for example) \(\alpha \in \mathbb{R}\). Hint: Draw a picture and/or refer to Figure 2.36 in the book.

   1. \(\mathbf{x} = \mathbf{c}\)
   2. \(\mathbf{x} = \mathbf{b}\)
   3. \(\mathbf{x}\) lies on the edge between \(\mathbf{b}\) and \(\mathbf{c}\).
   4. \(\mathbf{x}\) is outside the triangle
   5. \((\mathbf{x} - \mathbf{a}) \cdot (\mathbf{c} - \mathbf{a}) < 0\)