Archimedean spiral

Definition

An Archimedean spiral is a type of spiral defined in polar coordinates by the equation $r = a + b\theta$, where $a$ and $b$ are real numbers. The distance between consecutive turns of the spiral remains constant.

5 Must Know Facts For Your Next Test

1. The general equation for an Archimedean spiral is $r = a + b\theta$.
2. In this spiral, $a$ adjusts the starting radius, while $b$ determines the distance between turns.
3. The Archimedean spiral has applications in various fields such as physics, engineering, and computer graphics.
4. As $\theta$ increases, the radius $r$ increases linearly if $b \neq 0$.
5. The curve intersects each radial line from the origin at equally spaced intervals.

Review Questions

• What is the general form of the equation for an Archimedean spiral?
• How do parameters $a$ and $b$ affect the shape of an Archimedean spiral?
• Describe how the radius changes as $\theta$ increases in an Archimedean spiral.

Related terms

Polar Coordinates:

A coordinate system where each point on a plane is determined by its distance from a reference point and its angle from a reference direction.

Parametric Equations:

Equations that express coordinates of points as functions of one or more independent parameters.

Lemniscate:

A figure-eight-shaped curve defined in polar coordinates typically by equations like $(r^2 = a^2 \cos(2\theta))$.