The goal of this homework become familiar with two important methods of defining curves in computer graphics: Bézier curves and cubic splines. We will be using Adobe Illustrator for Bézier curves since it is widely used and helps gain an understanding of how the control points can be manipulated to create the desired effect.

Part I: Your name in Bézier curves (8 pts)

This part uses Adobe Illustrator. I would recommend using the lab machines since it shouldn't take too much time once you get the hang of it. If you'd prefer to use your own computer, you can download a 30-day free trial here:

Adobe Illustrator

It's not required, but I would recommend first doing the pen tool tutorial:

The Bézier Game

There are also several YouTube tutorials for the pen tool, one example is here:

Pen Tool Tutorial

Okay, now you are ready to use the pen tool in Adobe Illustrator to write your name in Bézier curves. You can write your first name, last name, middle name, initials, etc, as long as it is at least three letters long and uses some Bézier curves (not just lines!)

Save your work and submit on Moodle.

Part II: Cubic splines (7 pts)

For this part, we will define the equations for a cubic spline with 3 control points and compare that to a Bézier curve with 3 control points.

1. Given the three control points p₀ = (1,4), p₁ = (6,0), p₂ = (10,5), first write out the form of the two (n=2) cubic equations that will make up the cubic splines connecting them.
2. How many unknowns are there to solve for? Let this number be k. Write out k equations using the cubic spline procedure discussed in class (for each dimension, x and y).
3. Simplify these equations as much as possible, then write out the resulting system of equations in matrix form. Use software (Mathematica, online equation solver, etc) to solve this system (or you can do it by hand). Write out the resulting two cubic equations (two for x and two for y).
4. Write out the equations for the Bézier curve with these three control points.
5. Finally, sketch what these two curves would look like (doesn't have to be exact, just a rough idea of how the Bézier curve differs from the cubic spline). Submit your work on Moodle.

COLLABORATION:

For this assignment, you are welcome to work together, but everyone should turn in their own letters/math that they have written and understood. Please cite anyone you worked with at the top of your assignment.