Bezier Spline Sigmac Project

For your first project, you will generate a "sigmac" (signature machine) from your signature (or other word at least 6 distinct letters long), as well as carry out a few preliminary analyses.

Essentially you have to determine the so-called "control points" for a Bezier spline that will cause some piece of software (your choice) to draw your signature.

A Bezier spline is made up of cubic parametric curves of the form

\[ B(t)=(1-t)^3P_0+3(1-t)^2tP_1+3(1-t)t^2P_2+t^3P_3 \]

You can find a nice introduction to Bezier curves and splines here (although your text contains plenty to allow you to carry out this project).

1. Given points $P_0=(0,0)$ $P_1=(1,1)$ $P_3=(3,0)$, investigate graphically the range of motion of the Bezier cubic possible depending on the choice of control point $P_2$.

   Describe the kinds of qualitatively different curves you can generate.

   You may want to use my Mathematica code, or in Desmos.

2. Under what conditions will a Bezier cubic actually pass through one or both of the control points $P_1$ and \(P_2\)?

3. Since many letters contain a circle, create a Bezier cubic spline that does a nice job of approximating the unit circle. Describe your criterion for "nice".

4. Sample your signature for the interpolating points (best to do this on a piece of graph paper, so that you can get appropriate coordinates for control points), and then choose your control points so as to generate your sigmac. Print out a copy of your signature, as well as the output of your sigmac.

5. Discuss the success you had, and the problems and/or challenges you encountered.

Additional links:

• Check out this website for some preliminary info about Bezier splines; and
• You might check out some of the other links from that site, such as this page, which shows some control points in action.