In this explorations we want to look at parametric curves but first let's look at the rational form of a circle.

So we see that this is a circle with a radius 1 where u represents out parameter (imagine the scale isn't there). So u is the value of the x-axis and for any value of u we will have a point (u,0). Obviously, we can see that the line that joins the point (0,-1) and (u,0) intersects the circle at two points (let the other point be (x,(u), y(u)). Therefore, we derive the following equation: x = uy + u and we know that the circle equation is , so using the circle equation we can plug our line equation. When we solve it for y, the results are two roots where one is -1 and the other is . When we plug this into our line equation we get , therefore we get the following equations for each u: and . From this we know that the trigonometric parametric form is the following (where t ranges from 0 to 2PI)

and

Now let's explore some basic equations first.

We will start off with

We see that we actually get the unit circle. what if we change the values of a and b.

We see that our results are just simple. a is the positive and negative value on the x-axis while b is the positive and negative value on the y-axis. What if we have different values for a and b? What kind of shape would we get? Ellipse maybe?

Yep, it is an ellipse. We know that an ellipse formula is for some a and b. How did we get our parametric equation from that ellipse formula? Remember in order to derive our parametric form, we replaced y with b sin(t) and solved for x.

Now, that we can clearly see when our vector z= 0 and a and b are equal, we get a circle and when a and b vary from each other then we get an ellipse. So an extension to this exploration could be...what would happen if we let z equal something, say where t ranges from 0 to 6pi?

OK...this is neat!!! And since we know that = , the curve must lie on the circular cylinder . As t increases from 0 to 6pi, the point (x, y, z) moves in an upward clockwise cylinder creating a circular curve that is called a helix.

What about this parametric equation:

Look HERE to see the graph. We have another circular curve. what does it resemble? A tornado possibly!!

from that graph we know that if we eliminate our t parameter, this curve will lie on the cone .

How about this parametric equation:

Look HERE to see graph! This curve lies on what?