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

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.

Consider, for example, the following curves

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

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.

The polar coordinates P y P radius angle polar x Cartesian

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

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

Circle parametric equations polar coordinates

Ellipse parametric equations b a polar coordinates

ASTROID or

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

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

FOLIUM OF DESCARTES

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

Calculating the slope of a parametric curve Provided that

Calculate the slope of the curve