Michael Thomason's write-up for assignment ten, exploration three.
(click here to view this page as a Maple worksheet that you can manipulate to see what happens with the graphs.)

For various a and b , I will investigate the graphs of these parametric equations:

> x:=t->a*cos(t);
y:=t->b*sin(t);

Leaving them both equal to 1 seems like a good place to start.

> a:=1;b:=1;plot([x(t),y(t),t=0..2*Pi],scaling=constrained);

What happens when I increase either a or b ? I'll go with a.

> a:=2;b:=1;plot([x(t),y(t),t=0..2*Pi],scaling=constrained);a:=3;b:=1;plot([x(t),y(t),t=0..2*Pi],scaling=constrained);a:=4;b:=1;plot([x(t),y(t),t=0..2*Pi],scaling=constrained);

Increasing the coefficient a in x ( t ) = a sin( t ) stretches the graph along the x -axis. Increasing the coefficient y in x ( t ) = b cos( t ) will stretch the graph along the y -axis.

> a:=1;b:=5;plot([x(t),y(t),t=0..2*Pi],scaling=constrained);

What if I increase both a and b ? I predict that the graph will stretch along both axes proportionally to a and b .

> a:=3;b:=5;plot([x(t),y(t),t=0..2*Pi],scaling=constrained);

The graph did stretch to 3 along the x -axis to correspond with the change in a . The y -direction also behaves as expected. What if a is a function of t ? Here's a graph for a ( t ) = t .

> x:=t->a(t)*cos(t);y:=t->b*sin(t);
a:=t->t;
b:=1;plot([x(t),y(t),t=0..2*Pi],scaling=constrained);

What's happening here? Each of the previous graphs was an ellipse, but here we have a spiral. The graph has the same range as when b = 1 earlier because the y parameter has remained the same, but the x parameter increases as t (the counter-clockwise angle from the x -axis to the graph) increases. This is because a ( t ) = t . I want to draw a graph that sprials in the opposite direction, so I'm going to need to make a or b negative. I'll let a ( t ) = - t .

> x:=t->a(t)*cos(t);y:=t->b*sin(t);
a:=t->-t;
b:=1;plot([x(t),y(t),t=0..2*Pi],scaling=constrained);

If I let both a and b be functions of t , the graph will spiral outward with the x direction growing proportionally to a ( t ) and the y direction growing proportionally to b( t ).

> x:=t->a(t)*cos(t);y:=t->b(t)*sin(t);
a:=t->t;
b:=t->t;
plot([x(t),y(t),t=0..2*Pi],scaling=constrained);

As we see here, letting a ( t ) = b ( t ) = t , creates a smoother spiral than before because the x and y directions grow outward at an equal pace. Here are a few more pictures for different a ( t )'s and b ( t )'s.

> x:=t->a(t)*cos(t);y:=t->b(t)*sin(t);
a:=t->3*t;
b:=t->t;
plot([x(t),y(t),t=0..2*Pi],scaling=constrained);
a:=t->t+2;
b:=t->t;
plot([x(t),y(t),t=0..2*Pi],scaling=constrained);

Finally, let's see what happens when t runs through a greater range of values. The graph should continue spiraling outward, creating a nice picture.

> a:=t->t;
b:=t->t;
plot([x(t),y(t),t=0..40*Pi],scaling=constrained);