1.3 POLAR COORDINATE SYSTEM

Using a two-dimensional Cartesian coordinate system, we represent a point, say, A( x ₁, y ₁) as shown in Fig. 1 6. To locate this point, one has to travel the distance x ₁ along the x-axis and then y ₁ in the direction parallel to the y-axis. The other way round, if we consider a line and then the intersection of these two lines is the point A( x ₁, y ₁) in the plane. The polar coordinate system is a derived coordinate system, where the required point is at a distance r ₁ from the origin and the position vector of the point makes an angle with the x-axis. (See Figure 1.7.)

FIG. 1 6: Cartesian and polar representation of a point.

FIG. 1 7: Unit vectors in polar coordinates.

From the geometry in Fig. 1 6, it is clear that x ₁= ? ₁ cos and y ₁= ? ₁ sin .

This set of equations is called scalar transformation equations. While in the reverse transformations, ? ₁ is obtained by simply squaring and adding the two equations, and , by dividing the latter by the first. Thus, the scalar transformation equations are given as:

Thus, if the Cartesian coordinates are known, the polar coordinates for the same point can be obtained or vice versa. Note that the point represented in the polar coordinate system can be obtained by...