# Spiral of Archimedes

## Plane Curves

#### Description

The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd century BC Greek mathematician Archimedes. It is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity.

### Mapping

\{x\to a t \cos (t),y\to a t \sin (t)\}
$<mrow> <mo>{</mo> <mrow> <mrow> <mi>x</mi> <semantics> <mo>&#8594;</mo> <annotation encoding='Mathematica'>&quot;\[Rule]&quot;</annotation> </semantics> <mrow> <mi>a</mi> <mo>&#8290;</mo> <mi>t</mi> <mo>&#8290;</mo> <mrow> <mi>cos</mi> <mo>&#8289;</mo> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mrow> <mo>,</mo> <mrow> <mi>y</mi> <semantics> <mo>&#8594;</mo> <annotation encoding='Mathematica'>&quot;\[Rule]&quot;</annotation> </semantics> <mrow> <mi>a</mi> <mo>&#8290;</mo> <mi>t</mi> <mo>&#8290;</mo> <mrow> <mi>sin</mi> <mo>&#8289;</mo> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mrow> </mrow> <mo>}</mo> </mrow>$
{x -> a*t*Cos[t], y -> a*t*Sin[t]}
[x = a*t*cos(t), y = a*t*sin(t)]

### Constants

\{a\}
$<mrow> <mo>{</mo> <mi>a</mi> <mo>}</mo> </mrow>$
{a}
[a]

### Cite this as:

Plane Curves: Spiral of Archimedes from Differential Geometry Library. http://digi-area.com/DifferentialGeometryLibrary/PlaneCurves/Spiral-Of-Archimedes.php