Geometric Polar Plane
Figure 512 shows a variant of the polar coordinate plane on which the radial scale is graduated geometrically, rather than in linear fashion. 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, in effect, portrayed inside an open circle having a finite radius.
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.
Mathematician’s Polar Vs Cartesian
Figure 513 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)
We can’t have x _{0} = 0 because that produces an undefined quotient in the conversion formula to θ _{0} . If a value of θ _{0} thus determined happens to be negative, you can add 360° or 2π 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 Cartesiancoordinate relations to polarcoordinate relations, and vice versa. The general Cartesiantopolar 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 general polartoCartesian 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 is sometimes a function in the other system, but that is not always the case.
Compression and Conversion Practice Problems

1
 2
Ask a Question
Have questions about this article or topic? AskRelated Questions
See More QuestionsPopular Articles
 Kindergarten Sight Words List
 First Grade Sight Words List
 10 Fun Activities for Children with Autism
 Signs Your Child Might Have Asperger's Syndrome
 Theories of Learning
 A Teacher's Guide to Differentiating Instruction
 Child Development Theories
 Social Cognitive Theory
 Curriculum Definition
 Why is Play Important? Social and Emotional Development, Physical Development, Creative Development