1. Equations Parametric

2. Why?
An object’s shape or the path of a particle moving in a plane need not trace out the graph of a function, hence we cannot always express y directly in terms of x.

4. But that won’t stop us!
Sometimes it is just more convenient to express x and y coordinates as functions of a third variable using a pair of equations: x = f(t),      y = g(t).
Equations of this form are called parametric equations for x and y; the unknown t is called the parametric variable.

5. We’ve already done some of this:
The ballistic equations are parametric in t: x = v0t and y = v0t - 1/2gt2

6. Circles
You have already seen a parametric representation of a curve in the equations for the points on a circle of radius r.
This is the basis of Polar coordinate systems.

7. Circles The parametric equations are x = r cos(q ),
y = r sin(q ). As q goes from 0 to 2p the corresponding points trace out the circle in a counter clockwise direction.

8. Ellipses The parametric equations are x = a cos(q ),
y = b sin(q ). 0 <= q <= 2p

9. Cycloids
A famous curve that was named by Galileo in 1599 is called a cycloid. It is the path traced out by a point on the circumference of a circle as the circle rolls (without slipping) along a straight line.

10. Cycloids
If the circle that is rolled has radius a, then the parametric equations of the cycloid are
x = a[q - sin(q)],     y = a[1 - cos(q)] where parameter q is the angle through which the circle was rolled.

11. Epicycloids
Take a large circle centered at the origin.  Place a smaller circle tangent to the original circle at the point where it crosses the positive x-axis and outside of the original circle.  Identify the point of tangency. 

12. Epicycloids
Roll the smaller circle around the larger circle and follow the path of the point of tangency. The resulting curve is called an epicycloid. The shape of the epicycloid depends on the relationship between the radius of the large circle and the small circle.

13. You can show:
The parametric equations of an epicycloid using a large circle of radius a and a small circle of radius b, where a > b, are
X = (a + b)cos(q ) – b cos[(a + b) q /b] ,    Y = (a + b)sin(q ) – b sin[(a + b) q /b] .
The epicycloid was studied by such luminaries as Leibniz, Euler, Halley, Newton and the Bernoullis. It This curve is of special interest to astronomers and the design of gears and cogs.

14. Other cycloids
The nephroid
The ranunculoid

15. Hypocycloids
Take a large circle centered at the origin. Place a smaller circle tangent to the original circle at the point where it crosses the positive x-axis and inside the original circle. Identify the point of tangency.

16. Hypocycloids
Roll the smaller circle around the larger circle and follow the path of the point of tangency. The resulting curve is called an hypocycloid. The shape of the hypocycloid depends on the relationship between the radius of the large circle and the small circle.

17. You can show
The parametric equations of an hypocycloid using a large circle of radius a and a small circle of radius b, where a > b, are
x = (a - b)cos(q) + b cos[(a - b) q /b] ,     y = (a - b)sin(q) – b sin[(a - b) q /b] .
Or you may just prefer to play with the spirograph!

18. Applications?
An interesting application of the cycloid is the concept of ‘square wheels’: If your road has the shape of a cycloid and your wheels are square, you get a smooth ride!

19. This guy’s not kidding!

20. Lissajous Figures
Lissajous simulator
Electronics