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.
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The general equation for an Archimedean spiral is $r = a + b\theta$.
In this spiral, $a$ adjusts the starting radius, while $b$ determines the distance between turns.
The Archimedean spiral has applications in various fields such as physics, engineering, and computer graphics.
As $\theta$ increases, the radius $r$ increases linearly if $b \neq 0$.
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 $(r^2 = a^2 \cos(2\theta))$.