- Source: Hypotrochoid
In geometry, a hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the center of the interior circle.
The parametric equations for a hypotrochoid are:
x
(
θ
)
=
(
R
−
r
)
cos
θ
+
d
cos
(
R
−
r
r
θ
)
y
(
θ
)
=
(
R
−
r
)
sin
θ
+
d
sin
(
R
−
r
r
θ
)
{\displaystyle {\begin{aligned}&x(\theta )=(R-r)\cos \theta +d\cos \left({R-r \over r}\theta \right)\\&y(\theta )=(R-r)\sin \theta +d\sin \left({R-r \over r}\theta \right)\end{aligned}}}
where θ is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because θ is not the polar angle). When measured in radian, θ takes values from 0 to
2
π
×
LCM
(
r
,
R
)
R
{\displaystyle 2\pi \times {\tfrac {\operatorname {LCM} (r,R)}{R}}}
(where LCM is least common multiple).
Special cases include the hypocycloid with d = r and the ellipse with R = 2r and d ≠ r. The eccentricity of the ellipse is
e
=
2
d
/
r
1
+
(
d
/
r
)
{\displaystyle e={\frac {2{\sqrt {d/r}}}{1+(d/r)}}}
becoming 1 when
d
=
r
{\displaystyle d=r}
(see Tusi couple).
The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.
Hypotrochoids describe the support of the eigenvalues of some random matrices with cyclic correlations.
See also
Cycloid
Cyclogon
Epicycloid
Rosetta (orbit)
Apsidal precession
Spirograph
References
External links
Weisstein, Eric W. "Hypotrochoid". MathWorld.
Flash Animation of Hypocycloid
Hypotrochoid from Visual Dictionary of Special Plane Curves, Xah Lee
Interactive hypotrochoide animation
O'Connor, John J.; Robertson, Edmund F., "Hypotrochoid", MacTutor History of Mathematics Archive, University of St Andrews
Kata Kunci Pencarian:
- Daftar bentuk matematika
- Hypotrochoid
- Epitrochoid
- List of periodic functions
- Parametric equation
- Ellipse
- Tusi couple
- Trochoid
- Spirograph
- Roulette (curve)
- Guilloché
