Math 3298 - Hampton - Lab 2 - Due 9/18

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[{RGBColor[1,0,0],Line[{{4,1},{28,48},{50,42},{40,5}}]}]];
Show[plt1,plt2,PlotRange->All]

Bezier exercises

1. Use mathematica to find the slope of the Bezier curve at each control point.

2. It looks like the tangents at (x0, y0) and (x3, y3) pass through (x1, y1) and (x2, y2), respectively.  Prove it! (You do not need mathematica to prove this result, but it can help in simplifying long expressions - use Simplify. The previous exercise should help.)

3. Produce a Bezier curve with a loop by changing (x2, y2).