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Polar Plots

Main Concept

Plotting polar equations requires the use of polar coordinates, in which points have the form r, θ, where r  measures the radial distance from the pole O to a point P and θ measures the counterclockwise angle from the positive polar axis to the line segment OP.

When plotting a polar function, rθ, it is often helpful to first plot the function on a rectilinear grid, treating θ, r as Cartesian coordinates, with θ being plotted along the horizontal axis and r being plotted along the vertical axis. From this Cartesian plot, you can transfer critical points, such as minima, maxima, roots, and endpoints to a polar grid using the coordinates r, θ, and then fill in the behavior of r between the critical values of θ to create the final polar graph.

Common Figures in Polar Plots


Circle: The equation r = a creates a circle of radius a centered at the pole. The equation r=acosθ creates a circle of diameter a centered at the point r,θ = a2,0. The equation r=asinθ creates a circle of diameter a centered at the point r,θ = a2,π2.


Cardioid: The equations r = a + acosθ and r = a + asinθ create horizontal and vertical heart-like shapes called cardioids.


Archimedean Spiral: any equation of the form r=aθ1n creates a spiral, with the constant n determining how tightly the spiral winds around the pole. Special cases of Archimedean spirals include: Archimedes' Spiral when n = 1, Fermat's Spiral when n=2, a hyperbolic spiral when n = 1, and a lituus when n = 2.


Polar Rose: Equations of the form r=acoskθcreate curves which look like petaled flowers, where a represents the length of each petal and k determines the number of petals. If k is an odd integer, the rose will have k petals; if k is an even integer, the rose will have 2k petals; and if k is rational, but not an integer, a rose-like shape may form with overlapping petals.


Ellipse: Equations of the form r&equals;l1&plus;ecosθ, where 1 < e < 1 is the eccentricity of the curve (a measure of how much a conic section deviates from being circular) and l is the semi-latus rectum (half the chord parallel to the directrix passing through a focus), create ellipses for which one focus is the pole and the other lies somewhere along the line &theta; &equals; 0. The special case where e &equals; 0 creates a circle of radius l.

Choose a polar function from the drop-down menu or enter one in the text area. When entering your own function, type "theta" for the symbol θ. Click "Show Function" to see the function plotted on both a Cartesian and a polar grid. Click "Animate" to watch these two graphs being plotted simultaneously to see how r changes as θ grows from 0 to 2&pi;. Select the check box to extend the animation past the default stop value of &theta;&equals;2 &pi;. Click "Reset" to reset both plots.



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