1. To every "reasonable" function y = f (x) we can draw a curve as its graph. For example

2. Not every curve defines back a function y = f (x)
Not every curve defines back a function y = f (x). For example the curve y1 x y2
does not define a non-ambiguous mapping on any interval on the x-axix. We could define a function x = g (y), but, for some curves, even this is not possible.

3. Consider, for example, the following curves

4. The position on c of each point P is uniquely determined by its distance from a fixed reference point R covered by "walking" along c anti-clockwise. P c s
point of reference R
The x and y coordinates of P are then determined as functions

5. In the previous example, the curve c uniquely defines two functions.
Any two "reasonably tame" functions in turn define a curve. Then we say that the curve has a parametric definition or is defined parametrically.
Since any function y = f (x) defines a curve, this curve may also be defined parametrically, for example by putting
We then say that the function is defined parametrically or is parametric.

6. The polar coordinates P y P
radius angle
polar x
Cartesian

7. In polar coordinates, a curve may also be defined by viewing the radius  of a point on the curve as a function of its angle .
For example the curve to the right can be defined as

8. Straight line t
black box P
YES/NO
oracle
INPUT
OUTPUT
INPUT
OUTPUT

9. Circle
parametric equations
polar coordinates

10. Ellipse
parametric equations b a
polar coordinates

11. ASTROID or

12. CYCLOID
Cycloid is the curve generated by a point on the circumference of a circle that rolls along a straight line. If r is the radius of the circle and  is the angular displacement of the circle, then the polar equations of the curve are
Here r =3 and  runs from 0 to 4, that is, the circle revolves twice

13. CARDOID
implicit function
parametric equations
polar co-ordinates
a=2

14. FOLIUM OF DESCARTES

15. HELIX
Helix is the curve cutting the generators of a right circular cylinder under a constant angle .

16. Calculating the slope of a parametric curve
Provided that

17. Calculate the slope of the curve