Polar Graphing

A polar graph represents points in a plane using a distance from a reference point (pole) and an angle from a reference direction (polar axis).

The general form of a polar equation is $$r = f(θ)$$ , where $$r$$ is the radius and $$θ$$ is the angle.

A limaçon is a type of polar curve that can have a dimple or a loop, represented by equations of the form $$r = a + b \, \text{cos} \, θ$$ or $$r = a + b \, \text{sin} \, θ$$ .

A cardioid is a special case of a limaçon where $$a = b$$ , resulting in a heart-shaped curve.

A rose curve is a polar graph of the form $$r = a \, \text{cos}(nθ)$$ or $$r = a \, \text{sin}(nθ)$$ , where $$n$$ determines the number of petals.

In a rose curve, if $$n$$ is odd, the curve has $$n$$ petals; if $$n$$ is even, it has $$2n$$ petals.

The angle in polar coordinates determines the direction of the point from the pole, measured from the polar axis.

To convert polar coordinates (r, θ) to Cartesian coordinates (x, y), use the formulas: $$x = r \, \text{cos}(θ)$$ and $$y = r \, \text{sin}(θ)$$ .

The polar form of a circle centered at the pole with radius 'a' is $$r = a$$ .

The polar equation for a line through the pole at an angle θ is $$θ = k$$ , where k is a constant.