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);**