Graph r = f(θ) interactively on a live polar grid. Explore cardioids, rose curves, limaçons, lemniscates, spirals, circles, and any custom expression — with symmetry analysis and curve identification.
Polar Equation
sin cos tan sqrt abs pow PI · use * for multiply
Curve Analysis
t for θ — no special characters needed3*cos(t) not 3cos(t)x = r·cos(θ) y = r·sin(θ)
Every polar point (r, θ) maps to a Cartesian point using the conversion formulas above. The polar graph plots these (x, y) points for each θ value in the range.
Negative r: When r < 0 at angle θ, the point is plotted at distance |r| in the direction θ + π — the opposite side of the pole.
Symmetry tests let you predict curve shape before graphing:
r = a + b·cos(θ) or r = a + b·sin(θ)
Circle: r = a (constant). A circle of radius |a| centered at the pole.
Cardioid: r = a ± a·cos(θ) or sin(θ) — a special limaçon where |a/b| = 1. Heart-shaped, passes through the pole.
Limaçon: r = a + b·cos(θ). When |a/b| < 1: inner loop. When |a/b| = 1: cardioid. When |a/b| > 1: dimpled or convex limaçon.
Rose curve: r = a·cos(nθ) or a·sin(nθ). Has n petals if n is odd; 2n petals if n is even.
Lemniscate: r² = a²·cos(2θ) or sin(2θ). Figure-8 shape; only exists where the right side is non-negative.
Archimedean Spiral: r = aθ. Each revolution increases the radius by the same amount.
Our tutors walk through r = f(θ) graphing, symmetry tests, and converting between polar and rectangular forms step by step.
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