# Project 1: Vectors, Matrices, and Coordinate Transformations

Your project should build on the provided base code. Before you start with this project, you will need to look through the base code and get familiar with it.

This project has three parts, but only the first two are mandatory to get full credit (100 points). The number of points you can get for each part is indicated below. In the first part you will create classes for vectors and matrices. You will use this functionality in the remaining projects throughout the quarter. In the second part of this project you will construct an animated scene with moving objects whose motion depends on one another.

The third part is optional. We make suggestions for how to extend your solution with some additional features. You will get extra credit for completing those.

Note that in this project you are not allowed to use OpenGL or GLUT matrix routines, such as glRotate, glTranslate, or glScale, because we want you to use your own matrix routines.

This project is due on Friday, October 5th. You will need to present your project in the lab starting at 1:30pm that day, or during one of the TA/tutors' office hours before that day. For this project you are allowed to work in teams of two. In this case both team members need to be present during the homework presentation.

The TA, Sid Vijay, is going to introduce this homework assignment in lab 260 on Monday, October 1st at 2:30pm in lab 260.

## 1. Vectors and Matrices (45 points)

Implement a basic matrix and vector library by extending the classes in the base code. You can use the C++ class shown below as a basis for 3-component vectors. Create separate classes for a Vector3 and a Vector4 type, with three or four components, respectively. The Vector4 type includes the homogeneous coordinate (w component), which helps distinguish points from vectors. The matrix class should be for 4x4 matrices.

### 1a. Vectors (One point for each method for a total of 25 points)

Implement the following methods in your Vector3 class:

• A constructor with three parameters for the vector coordinates
• Element access 'set': set the vector coordinates
• Element access 'get': return a specific coordinate of the vector
• Overload operator '[]' for 'get' access
• Vector subtraction
• Overload operator '-' for subtraction
• Negation
• Scale (multiplication with scalar value)
• Dot product
• Cross product
• Magnitude (length of vector)
• Normalize
• Print (display the vector's components numerically on the screen)

Implement the following methods in your Vector4 class:

• A constructor with three (or four, optionally) parameters for the point coordinates
• Element access 'set': set the (four) point coordinates
• Element access 'get': return one of the four point coordinates
• Overload operator '[]' for 'get' access
• Vector subtraction
• Overload operator '-' for subtraction
• Dehomogenize (make fourth component equal to 1)
• Print (display the point's components numerically on the screen)

### 1b. Matrices (Two points for each method for a total of 20 points)

Implement the following methods in your Matrix4 class:

• Constructor with 16 parameters to set the values of the matrix
• Multiply (matrix-times-matrix)
• Multiply (matrix-times-vector): if multiplying with a point, dehomogenize the result after the multiplication
• Make rotation matrix about x axis
• Make rotation matrix about y axis
• Make rotation matrix about z axis
• Make rotation matrix about arbitrary (unit) axis
• Make non-uniform scaling matrix
• Make translation matrix
• Print (display all 16 matrix components numerically on the screen in a 4x4 array)

## 2. Animated Scene (55 points)

Construct an animated 3D scene with at least two separate moving objects, where one object's motion is dependent on the other. You can use simple rectangular boxes and transform them, or you can use OpenGL and GLUT geometry creation functions, such as gluSphere, glutSolidCone, glutSolidTorus, glutSolidSphere, or glutWireSphere. For example:

• A car with spinning "wheels" (made of boxes), moving in a circle on a ground plane.
• A helicopter with spinning rotor blades, flying in a circle.
• An industrial robot moving its arm which holds a rotating saw blade.
• A moon rotating about a planet, which rotates about its sun.

You will get full credit if you have a moving object (10 points), which changes location and does not just rotate about its center (5 points), and another object (10 points) whose motion depends on the first one (20 points).

To make the shapes more interesting and distinguishable, use different colors for them (10 points).

## 3. Tessellated Object (optional, 10 points)

The base OpenGL library only provides support for modeling and rendering simple points, lines, and polygons. Neither 3D objects, nor commonly used 2D objects such as circles, are directly available.

OpenGL and GLUT offer utility functions to allow you to render such shapes, which you might have done in part 2 of the project. In part 3 you are not allowed to use these utility functions.

To get the optional credit, you will have to implement your own tessellation of a rotationally symmetrical object, for example a cylinder, sphere, cone or torus. Your object should use base OpenGL functions to draw a number of triangles or quads that constitute the 3D object. You need to compute the vertices and build the quads/triangles yourself. (7 points)

Your algorithm needs to have at least one parameter to control the tessellation level (proportional to the number of quads/triangles the object consists of), and you need to be able to change this parameter (in source code is okay) during your homework presentation. (2 points)

Your quads/triangles need to be either wire framed, have different colors, or be shaded so that they can be easily distinguished. (1 point)

Below are images of a tessellated sphere (using quads) and a torus (using triangles) for inspiration: