Polar Coordinate Equations

Vocabulary flashcards covering equations and characteristics of polar curves like Rose Curves, Lemniscates, Limaçons, and Circles.

Rose Curve

A polar graph with the equation $r=a\cdot{cos}(ntheta)$ or $r=a\cdot{sin}(ntheta)$ , where the sine vs. cosine only changes the orientation (rotation).

Rose Curve Petals ($n$ is odd)

If $n$ is odd, the rose curve has $n$ petals.

Rose Curve Petals ($n$ is even)

If $n$ is even, the rose curve has $2n$ petals.

Lemniscate

A graph with the equation $r^2 = a^2 ext{cos}(2 heta)$ or $r^2 = a^2 ext{sin}(2 heta)$ that forms a figure-eight ($ext{∞}$ shape).

Lemniscate Orientation

The cosine version is oriented horizontally, while the sine version is oriented diagonally.

Limaçon

A polar graph defined by the equation $r = a + b ext{cos}( heta)$ or $r = a + b ext{sin}( heta)$, where sine vs. cosine changes orientation.

Inner Loop Limaçon

A limaçon shape that occurs when a < b.

Cardioid

A specific type of limaçon that occurs when $a = b$.

Dimpled or Convex Limaçon

A limaçon shape that occurs when a > b.

Circle (Centered at the origin)

A polar curve represented by the equation $r = a$, where $a$ is the radius.

Circle (Alternative Form)

Represented by $r = 2a ext{cos}( heta)$ (shifted horizontally) or $r = 2a ext{sin}( heta)$ (shifted vertically), these circles are not centered at the origin.