To tell high-school polar graphs apart, focus on their equation formats and key visual traits:
The limaçon family has the form r = a ± b cos(θ) or r = a ± b sin(θ). Their shapes depend entirely on the ratio of a to b:
- Inner loop: a/b < 1 (the curve passes through the origin to make a small loop inside).
- Cardioid: a/b = 1 (heart-shaped with a sharp point or cusp at the origin).
- Dimpled limaçon: 1 < a/b < 2 (rounded with an inward dent, but never touches the origin).
- Convex limaçon: a/b ≥ 2 (an oval or slightly flattened circle with no dent). (Cosine graphs point horizontally along the x-axis; sine graphs point vertically along the y-axis.)
Rose curves look like flower petals and follow r = a cos(nθ) or r = a sin(nθ). The petal length is always |a|. If n is odd, there are n petals; if n is even, there are 2n petals.
Lemniscates look like a figure-eight or propeller and are easy to spot because of the squared term: r² = a² cos(2θ) (horizontal) or r² = a² sin(2θ) (diagonal).
Circles and lines are the simplest forms. r = a is a circle centered at the origin, while r = a cos(θ) or r = a sin(θ) forms a circle sitting on an axis and passing through the origin. A constant angle θ = c simply forms a straight line through the center.
