Polar Plots - Maple Programming Help

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Polar Plots

Main Concept

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

When plotting a polar function, $r\left(\mathrm{θ}\right)$, it is often helpful to first plot the function on a rectilinear grid, treating  as Cartesian coordinates, with $\mathrm{θ}$ 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 , and then fill in the behavior of r between the critical values of $\mathrm{θ}$ to create the final polar graph.

Common Figures in Polar Plots

 • Circle: The equation  creates a circle of radius a centered at the pole. The equation $r=a\mathrm{cos}\left(\mathrm{\theta }\right)$ creates a circle of diameter a centered at the point . The equation creates a circle of diameter a centered at the point .
 • Cardioid: The equations  and  create horizontal and vertical heart-like shapes called cardioids.
 • Archimedean Spiral: any equation of the form $r=a{\mathrm{θ}}^{\frac{1}{n}}$ 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 , Fermat's Spiral when $n=2$, a hyperbolic spiral when , and a lituus when .
 • Polar Rose: Equations of the form $r=a\mathrm{cos}\left(k\mathrm{θ}\right)$create 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 $r=\frac{l}{1+e\mathrm{cos}\left(\mathrm{\theta }\right)}$, where  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 . The special case where  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 $\mathit{2}\mathrm{π}$. Select the check box to extend the animation past the default stop value of . Click "Reset" to reset both plots.

 $r\left(\mathrm{θ}\right)$$=$ Choose Function1cos(theta)1 - sin(theta)thetaln(theta)2 + 3*cos(theta)sin(2*theta)3*cos(4*theta)1/(1+0.5*cos(theta))sqrt(theta)*sin(theta^2)+thetacos(exp(theta))theta^sin(theta)sin(theta/2)*cos(theta^2)abs(sin(theta)^3*ln(theta))2*cos(5*theta)-sin(10*theta)cos(2*sin(cos(5*theta)/3))sin(sqrt(3)*theta)

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