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[[File:Deltoid2.gif|right|thumb|500px|The red curve is a deltoid.]] | |||
In [[geometry]], a '''deltoid''', also known as a '''tricuspoid''' or '''Steiner curve''', is a [[hypocycloid]] of three [[cusp (singularity)|cusp]]s. In other words, it is the [[Roulette (curve)|roulette]] created by a point on the circumference of a circle as it rolls without slipping along the inside of a circle with three times its radius. It can also be defined as a similar roulette where the radius of the outer circle is three times that of the rolling circle. It is named after the Greek letter [[Delta (letter)|delta]] which it resembles. | |||
More broadly, a deltoid can refer to any closed figure with three vertices connected by curves that are concave to the exterior, making the interior points a non-convex set. [http://www.se16.info/js/halfarea.htm] | |||
==Equations== | |||
A deltoid can be represented (up to rotation and translation) by the following [[parametric equation]]s | |||
:<math>x=2a\cos(t)+a\cos(2t) \,</math> | |||
:<math>y=2a\sin(t)-a\sin(2t)\,</math> | |||
where ''a'' is the radius of the rolling circle. | |||
In complex coordinates this becomes | |||
:<math>z=2ae^{it}+ae^{-2it}</math>. | |||
The variable ''t'' can be eliminated from these equations to give the Cartesian equation | |||
:<math>(x^2+y^2)^2+18a^2(x^2+y^2)-27a^4 = 8a(x^3-3xy^2)\,</math> | |||
and is therefore a [[algebraic curve|plane algebraic curve]] of degree four. In [[polar coordinates]] this becomes | |||
:<math>r^4+18a^2r^2-27a^4=8ar^3\cos 3\theta\,.</math> | |||
The curve has three singularities, cusps corresponding to <math>t=0,\, \pm\tfrac{2\pi}{3}</math>. The parameterization above implies that the curve is rational which implies it has [[geometric genus|genus]] zero. | |||
A line segment can slide with each end on the deltoid and remain tangent to the deltoid. The point of tangency travels around the deltoid twice while each end travels around it once. | |||
The [[dual curve]] of the deltoid is | |||
:<math>x^3-x^2-(3x+1)y^2=0,\,</math> | |||
which has a double point at the origin which can be made visible for plotting by an imaginary rotation y ↦ iy, giving the curve | |||
:<math>x^3-x^2+(3x+1)y^2=0\,</math> | |||
with a double point at the origin of the real plane. | |||
==Area and perimeter== | |||
The area of the deltoid is <math>2\pi a^2</math> where again ''a'' is the radius of the rolling circle; thus the area of the deltoid is twice that of the rolling circle.<ref name=Weisstein>Weisstein, Eric W. "Deltoid." From [[MathWorld]]--A Wolfram Web Resource. http://mathworld.wolfram.com/Deltoid.html </ref> | |||
The perimeter (total arc length) of the deltoid is 16''a''.<ref name=Weisstein/> | |||
== History == | |||
Ordinary [[cycloid]]s were studied by [[Galileo Galilei]] and [[Marin Mersenne]] as early as 1599 but cycloidal curves were first conceived by [[Ole Rømer]] in 1674 while studying the best form for gear teeth. [[Leonhard Euler]] claims first consideration of the actual deltoid in 1745 in connection with an optical problem. | |||
== Applications == | |||
Deltoids arise in several fields of mathematics. For instance: | |||
* The set of complex eigenvalues of [[unistochastic]] matrices of order three forms a deltoid. | |||
* A cross-section of the set of [[unistochastic]] matrices of order three forms a deltoid. | |||
* The set of possible traces of unitary matrices belonging to the [[group (mathematics)|group]] SU(3) forms a deltoid. | |||
* The intersection of two deltoids parametrizes a family of [[Complex Hadamard matrix|Complex Hadamard matrices]] of order six. | |||
* The set of all [[Simson line]]s of given triangle, form an [[Envelope (mathematics)|envelope]] in the shape of a deltoid. This is known as the Steiner deltoid or Steiner's hypocycloid after [[Jakob Steiner]] who described the shape and symmetry of the curve in 1856.<ref>Lockwood</ref> | |||
* The [[Envelope (mathematics)|envelope]] of the [[Bisection#Triangle area bisectors|area bisectors]] of a [[triangle]] is a deltoid (in the broader sense defined above) with vertices at the midpoints of the [[Median (geometry)|medians]]. The sides of the deltoid are arcs of [[hyperbola]]s that are [[Asymptote|asymptotic]] to the triangle's sides.<ref>Dunn, J. A., and Pretty, J. A., "Halving a triangle," ''[[Mathematical Gazette]]'' 56, May 1972, 105-108.</ref> [http://www.se16.info/js/halfarea.htm] | |||
== See also == | |||
* [[Astroid]], a curve with four cusps | |||
* [[Reuleaux triangle]] | |||
* [[Superellipse]] | |||
==References== | |||
{{reflist}} | |||
* {{cite book | author=E. H. Lockwood| title=A Book of Curves | publisher=Cambridge University Press | year=1961| | chapter=Chapter 8: The Deltoid }} | |||
* {{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=131–134 }} | |||
* {{cite book | author = Wells D | year = 1991 | title = The Penguin Dictionary of Curious and Interesting Geometry | publisher = Penguin Books | location = New York | isbn = 0-14-011813-6 | pages = 52}} | |||
* [http://www-history.mcs.st-andrews.ac.uk/history/Curves/Tricuspoid.html "Tricuspoid" at MacTutor's Famous Curves Index] | |||
* [http://www.mathcurve.com/courbes2d/deltoid/deltoid.shtml "Deltoïde" at Encyclopédie des Formes Mathématiques Remarquables] (in French) | |||
*{{springer|title=Steiner curve|id=S/s087650|last=Sokolov|first=D.D.}} | |||
[[Category:Curves]] | |||
[[Category:Algebraic curves]] |
Revision as of 01:49, 8 January 2014
In geometry, a deltoid, also known as a tricuspoid or Steiner curve, is a hypocycloid of three cusps. In other words, it is the roulette created by a point on the circumference of a circle as it rolls without slipping along the inside of a circle with three times its radius. It can also be defined as a similar roulette where the radius of the outer circle is three times that of the rolling circle. It is named after the Greek letter delta which it resembles.
More broadly, a deltoid can refer to any closed figure with three vertices connected by curves that are concave to the exterior, making the interior points a non-convex set. [1]
Equations
A deltoid can be represented (up to rotation and translation) by the following parametric equations
where a is the radius of the rolling circle.
In complex coordinates this becomes
The variable t can be eliminated from these equations to give the Cartesian equation
and is therefore a plane algebraic curve of degree four. In polar coordinates this becomes
The curve has three singularities, cusps corresponding to . The parameterization above implies that the curve is rational which implies it has genus zero.
A line segment can slide with each end on the deltoid and remain tangent to the deltoid. The point of tangency travels around the deltoid twice while each end travels around it once.
The dual curve of the deltoid is
which has a double point at the origin which can be made visible for plotting by an imaginary rotation y ↦ iy, giving the curve
with a double point at the origin of the real plane.
Area and perimeter
The area of the deltoid is where again a is the radius of the rolling circle; thus the area of the deltoid is twice that of the rolling circle.[1]
The perimeter (total arc length) of the deltoid is 16a.[1]
History
Ordinary cycloids were studied by Galileo Galilei and Marin Mersenne as early as 1599 but cycloidal curves were first conceived by Ole Rømer in 1674 while studying the best form for gear teeth. Leonhard Euler claims first consideration of the actual deltoid in 1745 in connection with an optical problem.
Applications
Deltoids arise in several fields of mathematics. For instance:
- The set of complex eigenvalues of unistochastic matrices of order three forms a deltoid.
- A cross-section of the set of unistochastic matrices of order three forms a deltoid.
- The set of possible traces of unitary matrices belonging to the group SU(3) forms a deltoid.
- The intersection of two deltoids parametrizes a family of Complex Hadamard matrices of order six.
- The set of all Simson lines of given triangle, form an envelope in the shape of a deltoid. This is known as the Steiner deltoid or Steiner's hypocycloid after Jakob Steiner who described the shape and symmetry of the curve in 1856.[2]
- The envelope of the area bisectors of a triangle is a deltoid (in the broader sense defined above) with vertices at the midpoints of the medians. The sides of the deltoid are arcs of hyperbolas that are asymptotic to the triangle's sides.[3] [2]
See also
- Astroid, a curve with four cusps
- Reuleaux triangle
- Superellipse
References
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - "Tricuspoid" at MacTutor's Famous Curves Index
- "Deltoïde" at Encyclopédie des Formes Mathématiques Remarquables (in French)
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- ↑ 1.0 1.1 Weisstein, Eric W. "Deltoid." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Deltoid.html
- ↑ Lockwood
- ↑ Dunn, J. A., and Pretty, J. A., "Halving a triangle," Mathematical Gazette 56, May 1972, 105-108.