Lemniscate

Definition

A lemniscate is a figure-eight or infinity-shaped curve that can be represented in polar coordinates as $r^2 = a^2 \cos(2\theta)$ or $r^2 = a^2 \sin(2\theta)$. It is an important shape in the study of polar graphs and trigonometric applications.

5 Must Know Facts For Your Next Test

1. The lemniscate has two main forms: the lemniscate of Bernoulli ($r^2 = a^2 \cos(2\theta)$) and the lemniscate of Gerono ($r^2 = a^2 \sin(2\theta)$).
2. In polar coordinates, $a$ represents the maximum distance from the origin to any point on the lemniscate.
3. The curve intersects itself at the origin (0,0), forming its characteristic figure-eight shape.
4. To graph a lemniscate, symmetry about both the x-axis and y-axis should be considered.
5. Lemniscates are used to model various physical scenarios such as orbital paths and electrical field distributions.

Review Questions

• What are the two main forms of a lemniscate in polar coordinates?
• How does changing the value of 'a' affect the size of the lemniscate?
• Explain why a lemniscate is symmetric about both axes.

Related terms

Polar Coordinates: A coordinate system where each point on a plane is determined by an angle and a distance from a reference point.

Rose Curve: A sinusoidal curve that appears flower-like with petals, represented in polar form as $r = a \cos(k\theta)$ or $r = a \sin(k\theta)$. The number of petals depends on 'k'.

Cardioid: $$ A heart-shaped curve represented in polar coordinates by $$ r = a(1 + \ cos \ theta) $$ or $$ r = e(1 - \ sin \ theta).$$