1.7 precalc

Limaçon

A polar curve of the form a + b cos(θ) or sin(θ), characterized by different shapes based on the ratio a/b.

Inner loop

Occurs when a/b < 1 in a Limaçon, resulting in a curve that loops inside of itself.

Cardioid

Occurs when a/b = 1 in a Limaçon, resulting in a heart-shaped curve.

Dimple

Occurs when 1 < a/b < 2 in a Limaçon, resulting in a shape with a dimple or indentation.

Convex Limaçon

Occurs when a/b ≥ 2 in a Limaçon, resulting in a shape that is entirely outward curving.

Rose Curve

A polar curve described by a cos(nθ) or sin(nθ), creating a petal-like shape.

Number of petals in a rose curve

If n is odd, the curve has n petals; if n is even, it has 2n petals.

Circle in polar coordinates

Represented by r = constant for a circle centered at the pole; r = a cos(θ) or r = a sin(θ) for circles centered along axes.

Lemniscate

A polar curve given by r² = a² cos(2θ) or sin(2θ), visually resembling a figure 8.

Figure 8

The shape produced by the lemniscate curve in polar coordinates.