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=a⁢cos⁡θ creates a circle of diameter a centered at the point r,θ = a2,0. The equation r=a⁢sinθ creates a circle of diameter a centered at the point r,θ = a2,π2.
Cardioid: The equations r = a + a⁢cos⁡θ and r = a + a⁢sinθ create horizontal and vertical heart-like shapes called cardioids.
Archimedean Spiral: any equation of the form r=a⁢θ1n 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=a⁢cosk⁢θcreate 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 2⁢k 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=l1+e⁢cos⁡θ, 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 θ = 0. The special case where e = 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 q. 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 q grows from 0 to 2⁢π. Select the check box to extend the animation past the default stop value of θ=2 π. Click "Reset" to reset both plots.
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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