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Boundless Algebra
Complex Numbers and Polar Coordinates
The Polar Coordinate System
Algebra Textbooks Boundless Algebra Complex Numbers and Polar Coordinates The Polar Coordinate System
Algebra Textbooks Boundless Algebra Complex Numbers and Polar Coordinates
Algebra Textbooks Boundless Algebra
Algebra Textbooks
Algebra
Concept Version 6
Created by Boundless

Other Curves in Polar Coordinates

Some curves have a simple expression in polar coordinates, whereas they would be very complex to represent in Cartesian coordinates.

Learning Objective

  • Recognize the equations for spirals and roses in polar coordinates


Key Points

    • The formulas that generate the graph of a rose curve are given by: $r=a\:\cos n\theta$  and $r=a\:\sin n\theta$  where $a≠0$.  If  $n$ is even, the curve has $2n$ petals. If  $n$ is odd, the curve has $n$ petals.
    • The formula that generates the graph of the Archimedes’ spiral is given by: $r=θ$ for  $θ≥0$.  As  $\theta$ increases,  $r$ increases at a constant rate in an ever-widening, never-ending, spiraling path.

Full Text

To graph in the rectangular coordinate system we construct a table of $x$ and $y$  values. To graph in the polar coordinate system we construct a table of $r$ and $\theta$ values. We enter values of $\theta$ into a polar equation and calculate $r$. However, using the properties of symmetry and finding key values of $\theta$ and $r$ means fewer calculations will be needed.

Investigating Rose Curves

Polar equations can be used to generate unique graphs. The following type of polar equation produces a petal-like shape called a rose curve. Although the graphs look complex, a simple polar equation generates the pattern. The formulas that generate the graph of a rose curve are given by:

 $\displaystyle r=a\cdot\cos \left( n\theta \right) \qquad \text{and} \qquad r=a\cdot\sin \left( n\theta \right) \qquad \text{where} \qquad a\ne 0$  

If $n$ is even, the curve has $2n$ petals. If $n$ is odd, the curve has $n$ petals.

Rose Curves

Complex graphs generated by the simple polar formulas that generate rose curves:$r=a\:\cos n\theta$ and $r=a\:\sin n\theta$ where $a≠0$.

Investigating the Archimedes' Spiral

Archimedes’ spiral is named for its discoverer, the Greek mathematician Archimedes ($c. 287 BCE - c. 212 BCE$), who is credited with numerous discoveries in the fields of geometry and mechanics.

The formula that generates the graph of the Archimedes’ spiral is given by:

$\displaystyle r=θ \qquad \text{for} \qquad \theta\geq 0$ 

As $\theta$ increases, $r$ increases at a constant rate in an ever-widening, never-ending, spiraling path.

Archimedes' Spiral

The formula that generates the graph of a spiral is $r=θ$ for $θ≥0$.

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