Math 3298  -  Hampton  -  Lab 1  -  
        Hypocycloids and Bezier Curves -  Due 9/28

(This lab is based on the lab exercises from the text,  
      pages 695 and 705. )

Hypocycloids
A hypocycloid is the curve traced out by a fixed point P on a circle C of radius b as C rolls on the inside of a circle with center O and radius a.

The parametric equations of the hypocycloid are:

x[t_] := (a - b)*Cos[t] + 
          b*Cos[((a - b)/b)*t];
y[t_] := (a - b)*Sin[t] - b*Sin[((a - b)/b)*t];

Here is an example of a hypocycloid when the larger circle has radius 3 and the smaller circle has radius 1.

a = 3;
b = 1;
ParametricPlot[{x[t], y[t]}, {t, 0, 2  π}, 
      AspectRatio -> Automatic]

Hypocycloid exercises

0.  Draw in the two circles in the sketch above to get a feeling for how the hypocycloid is formed.  You can do this by hand or using Mathematica's Circle command (combining graphics with Show and Graphics).

1.  Try different values for a (keeping b fixed) and determine how a affects the graph of the hypocycloid.  To get an accurate graph you may need to increase the number of points plotted by using the PlotPoints->n option within Parametric Plot; you may also want to change the t-interval that is plotted.

2.   Next, keep b = 1 fixed and let   a=(n/d) where n and d are integers with no common \(factors . \ 
\ \ \ \ \ \ \ \ \ \ a . \ \ \ First\) keep n fixed at   1and see what  happens when d takes on different \(values . \ \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ How\) does the shape of the hypocycloid  change as d \(changes?
\ \ \ \ \ \ b . \ \ Then\) keep d fixed and let n vary .   What is happening to the graph?
      c .    What  happens when n = d+ 1?
d.  How do the values of the parameter t change the shape of the curve?

3.  What happens to the hypocycloid if b = 1 and a is irrational?  Try a variety of irrational numbers such as (Sqrt[2]), e, 5e  - 7, and π + 2.
      	a.  How do the values of the parameter t change the shape of the curve?
      	b.  What happens to the curve as t becomes large when b = 1 and a is irrational?
      	c.  When b = 1 and a is irrational, what would the graph look like if you could let t take on all real values?

Bezier curves

The parametric equations for some simple Bezier curves are below, with t running between 0 and 1.

x[t_] :=  
        x0 (1 - t)^3 + 3 x1 (1 - t)^2 t  +  
          3 x2 (1 - t) t^2  +  x3 t^3;
y[t_] :=  
        y0 (1 - t)^3 + 3 y1 (1 - t)^2 t  +  
          3 y2 (1 - t) t^2  +  y3 t^3;

The points (x0, y0), (x1, y1), (x2, y2), (x3, y3) are called the control points.  Here is an example:

plt1 = 
      ParametricPlot[{x[t], y[t]} /. {x0 -> 4, y0 -> 1, 
            x1 -> 28, y1 -> 48, x2 -> 50, y2 -> 42, 
            x3 -> 40, y3 -> 5}, {t, 0, 1}]

plt2 = 
        Show[Graphics[{Line[{{4, 1}, {28, 48}, {50, 42}, {40, 
                    5}}]}]];

Show[plt1, plt2, PlotRange -> All]

Bezier exercises

1. It looks like the tangents at (x0, y0) and (x3, y3) pass through (x1, y1) and (x2, y2), respectively.  Prove it!

2. Try to produce a Bezier curve with a loop by changing (x2, y2).