# Epitrochoid

## Plane Curves

#### Description

An Epitrochoid is a plane curve generated by the motion of a fixed point on the radius or extension of the radius of a circle that rolls externally, without slipping, on a fixed circle. The epitrochoid is a generalization of the epicycloid. The Epitrochoid is a special case of the Limacon of Pascal with R = r.

### Mapping

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

### Constants

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

### Cite this as:

Plane Curves: Epitrochoid from Differential Geometry Library. http://digi-area.com/DifferentialGeometryLibrary/PlaneCurves/Epitrochoid.php

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