Simulic
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Polar coordinates and polar graphs – conversion, roses, limaçons, spirals and area

MathematicsFunctions & GraphsAges 16–17

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Use with my class

Drag a point P to convert between polar coordinates (r, θ) and Cartesian coordinates (x, y), see the other names of the same point and the modulus–argument form of a complex number. Graph r = f(θ) (roses, limaçons, cardioids, circles, Archimedean spiral, lemniscate) as θ sweeps, linked to the graph of r against θ, with dr/dθ, the tangent gradient dy/dx and where r increases or decreases. The area screen finds ½∫r²dθ between two rays and compares it with a sum of circular sectors.

Lesson: Polar coordinates and polar curves

What it shows

In polar coordinates a point is given by its distance r from the pole O and the angle θ its ray makes with the polar axis, measured anticlockwise. Then x = r cos θ and y = r sin θ, and every point has many polar names, including ones with negative r. A polar graph r = f(θ) is traced as θ increases; reading it next to the ordinary graph of r against θ shows where r grows or shrinks and when the point moves towards or away from the pole. The slope of the curve is dy/dx = (r′ sin θ + r cos θ)/(r′ cos θ − r sin θ), and the area swept by the radius is ½∫r² dθ.

How to use

On Polar coordinates, drag P or use the r of P and θ of P sliders; tick Complex form to see z. Press Target point or Convert to practise. On Graph r = f(θ), choose a Curve, set a, b or k, then press Trace or drag along either graph to move θ. On Area, set the Start angle α, End angle β and Number of sectors. Angle unit switches between degrees and radians.

Parameters you can change

  • Screen Polar coordinates, Graph r = f(θ), Area
  • Curve Rose (cos), Rose (sin), Limaçon (cos), Limaçon (sin), Circle centred at O, Circle through O, Archimedean spiral, Lemniscate of Bernoulli
  • Parameter a 0.5–5
  • Parameter b (limaçon) -5–5
  • Parameter k (rose) 1–8
  • Angle θ on the graph 0–720 °
  • Angle unit Degrees, Radians
  • Start angle α -360–720 °
  • End angle β -360–720 °
  • Number of sectors 1–60
  • r of point P -6–6
  • θ of point P -360–360 °
  • Complex form
  • Snap to grid

Questions to explore

  1. Why does r = 3 cos 2θ have four petals while r = 3 cos 3θ has only three?
  2. When r is negative and decreasing, is the point moving towards or away from the pole?
  3. Why does the sum of sectors ½ r² Δθ get closer to the area as the number of sectors grows?

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