A parametric plot yields a visual description of a set of
parametric equations. If x and y are both functions of a variable t,
then they create a set of parametric equations. For example, the two equations 
and 
form a set of parametric equations
in which y and x are functions of t, the graph of which looks
like
If you substitute 
into 
for t,
the result is the familiar equation graphed above, 
.
The general idea is simple, for every value of t, evaluate the functions x and y, then plot the resulting values on a set of xy-axes as the point ( x, y ). Thought of in another way, given a value of t, evaluate the function x for that particular t. Move to the place on the x-axis that corresponds to the resulting value and draw a line perpendicular through the x-axis at that point. Do the analogous process for the y function. The resulting two lines intersect at a point that is on the graph of the parametric plot of x and y as functions of the parameter t.
Another classic example of a parametric plot is that of a circle with
radius 1. The trigonometric identity that describes this circle is 
.
The set of equations, { 
, 
},
produces the same graph parametrically. The sample point displayed in the graph
below is 
= 5.59 radians. Note that = 5.59/(2*Pi) radians evaluates to the
same point on the graph.
The parametric descriptions of the above two examples both translate
easily into non-parametric forms. In the first example, use substitution to easily
eliminate the parameter t from the x and y functions and
end up with the equation 
.
Then all pairs of real numbers (x, y) that satisfy the equation create the first graph.
For the circle, use substitution to eliminate the parameter and end
up with the equation 
.
Then all pairs of real numbers (x, y) that satisfy the equation create the circle. The
idea here is that for these two examples, simple non-parametric techniques exist to
describe the graphs.
So the obvious questions is why add another layer of complexity to the
situation? The answer is that with more complex equations than these, the parametric
description is much easier to work with, and many graphs described parametrically do not
have an algebraic expression without the parameter. For example, the set of
parametric equations, {
} (e is the base of the natural log), represents the
beautiful graph below for t = -1.3 to 1.3, but it is impossible to eliminate the
parameter t and express the points on the graph as an equation of just y
and x. Thus, without parametric descriptions, we could not express many
graphs in a manageable mathematical form. In addition, as discussed in Kinematics 2, parametric plots offer a more
intuitive description of certain physical situations.
The following groupings of command lines produce interesting plots of sets of parametric equations. Select and copy a whole group at once from you browser window, then paste the selection at a command prompt in your Maple worksheet and press enter. See Kinematics 2 for a detailed description of the plot command and some of its optional arguments. At this University of Pennsylvania site you'll find some other beautiful expamples of parametric plots produced by Maple.
A := 2*beta-2*sin(beta);
plot( A, beta=0..6*Pi, labels=[beta, "A"], scaling=constrained, title=`A as a
function of beta` );
B := 2-2*cos(beta);
plot( B, beta=0..6*Pi, scaling=constrained, labels=[beta,"B"], title=`B as a
function of beta`);
plot( [A, B, beta=0..6*Pi], labels=["A", "B"], scaling=constrained,
title=`parametric plot of A and B as functions of beta` );
h := 3*(cos(alpha))^3;
plot( h, alpha=0..6*Pi, labels=[alpha,"h"], scaling=constrained, title=`h as a
function of alpha` );
j:= 3*(sin(alpha))^3;
plot( j, alpha=0..6*Pi, labels=[alpha,"j"], scaling=constrained, title=`j as a
function of alpha` );
plot( [h, j, alpha=0..6*Pi], title=`parametric plot of h and j as functions of alpha`,
labels=["h","j"], scaling=constrained);
m := alpha+3*(cos(alpha))^3;
plot( m, alpha=0..6*Pi, labels=[alpha,"m"], scaling=constrained, title=`m as a
function of alpha` );
n:= alpha+3*(sin(alpha))^3;
plot( n, alpha=0..6*Pi, labels=[alpha,"n"], scaling=constrained, title=`n as a
function of alpha` );
plot( [m, n, alpha=0..6*Pi], title=`parametric plot of m and n as functions of alpha`,
labels=["m","n"], scaling=constrained);
f := t^3 + t;
plot( f, t=-1..1, scaling=constrained, labels=[t,"f"], title=`f as a function of
t` );
g := t^7 + t + 1;
plot( g, t=-1..1, title=`g as a function of t`, scaling=constrained,
labels=[t,"g"] );
plot( [f, g, t=-1..1], labels=["f","g"], title=`parametric plot of f
and g as functions of t`, scaling=constrained );
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