**Geometric Polar Plane**

**Geometric Polar Plane**

Figure 10-12 shows a polar plane on which the radial scale is graduated geometrically. The point corresponding to 1 on the *r* axis is halfway between the origin and the outer periphery, which is labeled ∞ (the “infinity” symbol). Succeeding integer points are placed halfway between previous integer points and the outer periphery. In this way, the entire polar coordinate plane is depicted within a finite open circle.

The radial scale of this coordinate system can be expanded or compressed by multiplying all the values on the *r* axis by a constant. This allows various relations and functions to be plotted, minimizing distortion in particular regions of interest. Distortion relative to the conventional polar coordinate plane is greatest near the periphery, and is least near the origin.

This “geometric axis compression” scheme can also be used with the axes of rectangular coordinates in two or three dimensions. It is not often seen in the literature, but it can be interesting because it provides a “view to infinity” that other coordinate systems do not.

**Polar Vs Cartesian Conversions**

Figure 10-13 shows a point *P* = (x _{0} , *y* _{0} ) = ( *θ* _{0} / *r* _{0} ) graphed on superimposed Cartesian and polar coordinate systems. If we know the Cartesian coordinates, we can convert to polar coordinates using these formulas:

*θ* _{0} = arctan ( *y* _{0} /x _{0} ) if *x* _{0} > 0

*θ* _{0} = 180° + arctan ( *y* _{0} / *x* _{0} ) if *x* _{0} < 0 (for *θ* _{0} in degrees

*θ* _{0} = π + arctan ( *y* _{0} / *x* _{0} ) if *x* _{0} < 0 (for *θ* _{0} in radians)

*r* _{0} = ( *x* ^{2} _{0} + *y* ^{2} _{0} ) ^{1/2}

We can’t have *x* _{0} = 0 because that produces an undefined quotient. If a value of *θ* _{0} thus determined happens to be negative, add 360° or 2 *π* rad to get the “legitimate” value.

Polar coordinates are converted to Cartesian coordinates by the following formulas:

*x* _{0} = *r* _{0} cos *θ* _{0}

*y* _{0} = *r* _{0} sin *θ* _{0}

These same formulas can be used, by means of substitution, to convert Cartesian-coordinate relations to polar-coordinate relations, and vice versa. The generalized Cartesian-to-polar conversion formulas look like this:

*θ* = arctan ( *y* / *x* ) if *x* > 0

*θ* = 180° + arctan ( *y* / *x* ) if *x* < 0 (for *θ* in degrees)

*θ = π* + arctan ( *y* / *x* ) if *x* < 0 (for *θ* in radians)

*r* = ( *x* ^{2} + *y* ^{2} ) ^{1/2}

The generalized polar-to-Cartesian conversion formulas are:

*x* = *r* cos *θ*

*y* = *r* sin *θ*

When making a conversion from polar to Cartesian coordinates or vice versa, a relation that is a function in one system may be a function in the other, but this is not always true.

**PROBLEM 1**

Consider the point ( *θ* _{0} , *r* _{0} ) = (135°,2) in polar coordinates. What is the ( *x* _{0} , *y* _{0} ) representation of this point in Cartesian coordinates?

**SOLUTION 1**

Use the conversion formulas above:

*x* _{0} = *r* _{0} cos *θ* _{0}

*y* _{0} = *r* _{0} sin *θ* _{0}

Plugging in the numbers gives us these values, accurate to three decimal places:

*x* _{0} = 2 cos 135° = 2 × (−0.707) = −1.414

*y* _{0} = 2 sin 135° = 2 × 0.707 = 1.414

Thus, ( *x* _{0} , *y* _{0} ) = (−1.414,1.414).

Practice problems for these concepts can be found at: Polar Coordinates Practice Test.

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