Common Polar Curves

Trigonometry · Axiom Academy

Explore how simple polar equations create beautiful geometric shapes: circles, cardioids, roses, and spirals. r = a: Circle centered at origin with radius a r = a cos θ: Circle tangent to y-axis, diameter a r = a sin θ: Circle tangent to x-axis, diameter a The simplest polar equation r = a creates a circle of radius a centered at the origin. But we can also create circles that pass through the origin using trigonometric functions! r = a(1 + cos θ): Cardioid pointing right r = a(1 - cos θ): Cardioid pointing left r = a(1 + sin θ): Cardioid pointing up r = a(1 - sin θ): Cardioid pointing down A cardioid (from the Greek word for "heart") is formed when the equation adds a constant to a trigonometric function. The result is a curve with a distinctive cusp at the origin. r = a cos(nθ) or r = a sin(nθ) If n is odd: The rose has n petals If n is even: The rose has 2n petals Rose curves are created by using a coefficient in the angle. The coefficient n determines how many "petals" the flower has. Let's see how different values of n create different roses! r = aθ: Distance increases linearly with angle Each complete rotation adds the same distance from center Unlike the closed curves we've seen, the Archimedean spiral continues outward forever. The distance from the origin grows proportionally to the angle, creating evenly-spaced loops.

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