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274:{\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}}}
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The red curve is a hypotrochoid drawn as the smaller black circle rolls around inside the larger blue circle (parameters are
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is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because
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Hypotrochoids describe the support of the eigenvalues of some random matrices with cyclic correlations.
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522:(drawn in red) may be expressed as a special case of the hypotrochoid, with
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684:"Universal hypotrochoidic law for random matrices with cyclic correlations"
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Aceituno, Pau
Vilimelis; Rogers, Tim; Schomerus, Henning (2019-07-16).
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Curve traced by a point outside a circle rolling within another circle
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Modern
Differential Geometry of Curves and Surfaces with Mathematica
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347:{\displaystyle 2\pi \times {\tfrac {\operatorname {LCM} (r,R)}{R}}}
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from Visual
Dictionary of Special Plane Curves, Xah Lee
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rolling around the inside of a fixed circle of radius
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471:{\displaystyle e={\frac {2{\sqrt {d/r}}}{1+(d/r)}}}
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292:is not the polar angle). When measured in radian,
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661:(Second ed.). CRC Press. p. 906.
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76:from the center of the interior circle.
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796:MacTutor History of Mathematics Archive
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403:. The eccentricity of the ellipse is
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780:Interactive hypotrochoide animation
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623:A catalog of special plane curves
559:toy traces out hypotrochoid and
54:traced by a point attached to a
627:. Dover Publications. pp.
769:Flash Animation of Hypocycloid
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718:10.1103/PhysRevE.100.010302
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365:Special cases include the
801:University of St Andrews
83:for a hypotrochoid are:
296:takes values from 0 to
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360:least common multiple
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81:parametric equations
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500:{\displaystyle d=r}
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596:Apsidal precession
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48:hypotrochoid
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561:epitrochoid
534:Tusi couple
509:Tusi couple
367:hypocycloid
701:1812.07055
607:References
601:Spirograph
586:Epicycloid
557:Spirograph
760:MathWorld
734:119325369
320:
310:×
307:π
260:θ
248:−
234:
225:−
222:θ
219:
207:−
192:θ
173:θ
161:−
147:
135:θ
132:
120:−
105:θ
811:Category
726:31499759
581:Cyclogon
570:See also
563:curves.
536:); here
379:and the
71:distance
52:roulette
44:geometry
706:Bibcode
629:165–168
576:Cycloid
520:ellipse
381:ellipse
354:(where
732:
724:
665:
635:
541:= 10,
284:where
60:radius
56:circle
730:S2CID
696:arXiv
545:= 5,
507:(see
383:with
369:with
50:is a
32:= 3,
28:= 5,
722:PMID
663:ISBN
633:ISBN
518:The
393:and
79:The
46:, a
714:doi
692:100
549:= 1
527:= 2
511:).
388:= 2
362:).
358:is
356:LCM
317:LCM
231:sin
216:sin
144:cos
129:cos
58:of
42:In
36:= 5
813::
799:,
793:,
789:,
757:.
728:.
720:.
712:.
704:.
690:.
686:.
631:.
398:≠
374:=
38:).
763:.
736:.
716::
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551:.
547:d
543:r
539:R
532:(
529:r
525:R
495:r
492:=
489:d
463:)
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456:/
452:d
449:(
446:+
443:1
436:r
432:/
428:d
423:2
417:=
414:e
400:r
396:d
390:r
386:R
376:r
372:d
339:R
335:)
332:R
329:,
326:r
323:(
304:2
294:θ
290:θ
286:θ
264:)
255:r
251:r
245:R
238:(
228:d
213:)
210:r
204:R
201:(
198:=
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189:(
186:y
177:)
168:r
164:r
158:R
151:(
141:d
138:+
126:)
123:r
117:R
114:(
111:=
108:)
102:(
99:x
74:d
67:R
63:r
34:d
30:r
26:R
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