Equality of Coordinate Systems

According to the literature of mathematics, right angular and polar coordinate systems are of equal value. This means that it is possible to transfer from one to the other without loss of data. The transfer is done with the application of a few simple formulas.

In the case of a plane,
– when transferring from right angular to polar:
r = SQR(x2 + y2), and Φ = ATN(y/x),
with SQR being a square root with a positive sign and ATN being the symbol of the principal value of the arctangent function;
– When transferring from polar to right angular:
x = r*COS(Φ), and y = r*SIN(Φ),
with * being the symbol of multiplication, SIN being the symbol of the principal value of the sine function and COS, that of the principal value of the cosine function.

Let us see whether or not this can be applied for each point of the plane.

With any point given with an x, y pair, there belongs one, and only one point, belonging with which, calculating with the first pair of formulas, there is one, and only one r, Φ pair. It does not make an exception to this if x or y is of zero value. However, if both are, we will get a 0/0 form for determining the value of Φ, that is, in the case of the origo of the right angular coordinate system, the undeterminable value of angle of the polar coordinate system will belong with r = 0.

In an inverse case, however, x = y = 0 will belong with every r = 0 value, regardless of the value of Φ, as neither SIN, nor COS can assume an infinite value, due to which the value of both multiplication products will be zero. As a result of this, the same point will belong with any Φ value.

To put it in another way:

One of the variables of the right angular coordinate system taken to be of zero value, we will get a geometrical shape of one grade lower dimension. In the case of a plane, for example, it is a straight line, one of the axes of the coordinate. The two variables behave in the same way.

In the case of the polar coordinate system,
belonging with the value Φ = 0 is x = r and y = 0, a shape of one grade lower dimension (in the case of a plane, it is a semi-straight line, the positive branch of the x axis),
while belonging with the value r = 0 in each case is the origo. Thus, the two variables do not behave in the same way.

This however does not incur a loss of data, as the prerequisite r = 0 is not fulfilled in any point besides the origo, and the other way round, if r = 0, it is the origo.

However if the point of departure is the polar coordinate system and r = 0, and different Φ values are assumed, belonging with each will be the value x = y = 0, that is, the origo. Thus, in this case the same x, y value pair belongs with an infinitely great number of data that are different in the value Φ. This does incur a loss of data.

Dr. Endre Simonyi