Parametric Equations
As a point ( x, y) moves along a curve over a time interval, each coordinate x and y may be described as a function of a time variable t.
In this assignment we will be using Maple
to graph parametric equations. To plot the planar curve whose parametric
equations are x = x(
t ), y = y(
t ), 
using Maple, use the command
plot( [
x(
t ), y(
t ), t
= a..b]). To plot this curve on a set of coordinate aces whose
aspect ratio is 1, use the command plot(
[ x(
t ), y(
t ), t
= a..b], scaling = constrained).
The ability to graph parametric curves so easily in Maple, and the fact that there are few restrictions on what makes the graph o f a set of parametric equations interesting, moves graphing of parametric equation from a topic in Calculus to an art form. In general, the more complex your equations the more interesting its graph will likely be!
Part 1: Graphing Parametric Equations
1. Plot x = 8t - 3, y = 2 - t, 0 <
t < 1
Hint: Begin with the commands
> restart;
with(plottools): with(plots): with(student):
2. Eliminate the parameter in problem 1 and graph the resulting equation in x and y. Are the graphs of problems 1 and 2 the same?
3.
Plot x = 5 sin (t), y = 3 cos (t), 
. Be
sure to used the constrained scale so you see that these parametric equations
are an ellipse.
4.
Plot x = sin (3t), y = sin (4t), .
5.
Plot x = t + sin (3t), y = t + sin (4t), 
.
6.
Plot x = cos (t), y = sin(t + sin (5t)), .
Part II: Investigating Bezier Curves
The Bezier curves are used in computer-aided design and are named after a
mathematician working in the automotive industry. A cubic Bezier curve is
determined by four control points, P0(x0, y0),
P1(x1, y1), P2(x2, y2),
and P3(x3, y3), and is defined by the
parametric equations
7.
Graph the Bezier curve with control points P0(4,
1), P1(28, 48), P2(50, 42), and P3(40,
5).
Hint: To determine the x and y values use the command plot(
[ x(
t ), y(
t ), t
= a..b], x=r..s,
y=c..d,
scaling = constrained). In this
problem let 0 < x < 50 and 0 < y <
50.
8. By hand, plot the control points on this graph of problem 7, and draw in the line segments, P0P1, P1P2, and P2P3. Notice that P0 will be the starting point and will be the P3 ending point. If we were to move P1 to the right of P2 we would get a loop in our curve. Pick a point so that you create a loop in a Bezier curve.
9. Some laser printers use Bezier curves to represent letters and other symbols. Experiment with the control points until you find a Bezier curve that gives a reasonable representation of the letter C. Hint: Find 4 control points with the first one at the top of the C, the last one at the bottom of the C and the other two spaced to give the rounded shape of the C. Note that there is no loop in the C.
Part III: Creating Your Curves
10. Create two different graphs using parametric equations. Try to make them artistically interesting. See comment at beginning of assignment.
Your Assignment Report is a hare copy of your typed input and Maple's responses, and your text input and comments.