Graphing Polar Curves

Calculus 2 · Axiom Academy

Sweep the angle θ, let r = f(θ) set the distance, and watch circles, cardioids, and limaçons trace themselves out. 1. Plotting a Point by Sweeping θ Every polar point lives at an ordinary location (x,y) — you reach it by turning to angle θ , then walking distance r out along that ray. The bridge back to Cartesian coordinates is what makes a curve drawable: r and θ are the inputs — distance and direction x and y are where that point actually lands 2. When a Cosine Draws a Circle The plainest polar curve is r = a — every point sits the same distance out, so it is a circle centered at the origin. But is the surprise: the radius changes with θ, yet the trail is still a perfect circle — one that hugs the origin and is pushed off to the side. 3. Building a Cardioid Point by Point Add a constant to that cosine and the circle opens into a heart. The cardioid is the headline polar curve: the radius is largest dead ahead and squeezes to nothing behind, leaving a single sharp cusp at the origin. Radius at its max; the point reaches the rightmost tip at (2, 0) . Halfway around the radius is 1 ; the point sits straight up at (0, 1) . Radius collapses to 0 ; the curve pinches into the origin — the cusp. Coming back up the radius is 1 again; the lower half mirrors the top.

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