School of Mathematics and Statistics, The University of Sydney
 6. Cartesian and polar coordinates in two dimensions
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Polar representations

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If you are looking at an aeroplane on a radar screen you could determine its position by giving its distance from you and a direction or angle, say northwest.

Let us see how to do this in terms of coordinates. Imagine that the point Q represents the position of the aeroplane, and you are at the origin.

  Y   S                Q = (x, y)   y      r        y                          X O        x     R

The length r of the line segment OQ is calculated by applying Pythagoras’ theorem to the right angled triangle ORQ. This gives

     V~ ------------    V~ -------- r =   OR2  + RQ2  =    x2 + y2.

This formula for r is valid for both positive and negative values of x and y.

To measure an angle or direction for Q we have to measure it starting from somewhere. By convention, all angles are measured starting from the positive OX axis, increasing in the anti-clockwise direction. Now we have the following diagram, in which h denotes the angle /QOX and the vector r represents the position vector of the aeroplane relative to the origin, that is, r = ---> OQ.

Y             rcos h        Q = (x, y) S    j                      rsinh   O hr           i           R       X

Elementary trigonometry and a comparison with the previous diagram show that

x = r cosh,     y = rsinh,

where r is the length of OQ and h is the angle /-QOX. The position of the point Q can now be described in two ways: either by giving its Cartesian coordinates (x,y) or by giving what are called its polar coordinates (r,h), where the scalars x, y, r, h are linked by the equations

x = r cosh,     y = rsinh.

Thus given the polar coordinates (r,h) of a point Q, we can calculate the Cartesian coordinates (x,y). Conversely, given the Cartesian coordinates (x,y), we can calculate the polar coordinates (r,h), since

     V~ -2----2-          x-          y- r =   x  + y ,  cos h = r ,  sin h = r .

The actual value of h is obtained in practice by using the inverse cosine or sine functions on a calculator, and by knowing the quadrant in which h lies.

As r = - --> OQ = ---> OR + ---> OS, we obtain what is known as the polar representation of the vector r, or alternatively the polar form of r,

r = r cosh i + r sin hj.

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