Geometry

Definition

The coordinates of can be written as in polar form where is the distance from the origin and is the angle of rotation counter-clockwise from the positive -axis. This is very similar to the Modulus-Argument Form of Complex Numbers


Conversion to Cartesian

To convert from polar to Cartesian you can use the following:


Sketching curves

Common curves are: , a circle with centre origin radius , a half line making angle , a spiral starting at origin

, They are called Cardioids They are defined for all if so to make - this can be defined in some contexts but for A-level this is undefined)

Cardioids fall into 2 categories “egg” shaped and those with a “dimple”, egg shaped are Convex curves, dimples are Concave at . They are egg shaped when and dimple shaped if - This is proved by considering the number of tangents that are perpendicular to the initial line.


Area enclosed by a Polar Curve

The area enclosed by a polar curve and half lines where is measured in radians is given as:

Very useful trig identities for Integration


Tangent to Polar Curve

You can differentiate parametrically to get:

To find a tangent line to the parallel to the initial set: To find a tangent perpendicular to the initial line set: