Deltoid curve
Encyclopedia
In geometry
, a deltoid, also known as a tricuspoid or Steiner curve, is a hypocycloid
of three cusp
s. 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 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. http://www.btinternet.com/~se16/js/halfarea.htm
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.
where again a is the radius of the rolling circle; thus the area of the deltoid is twice that of the rolling circle.
The perimeter (total arc length) of the deltoid is 16a.
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.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a deltoid, also known as a tricuspoid or Steiner curve, is a hypocycloid
Hypocycloid
In geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle...
of three cusp
Cusp (singularity)
In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....
s. In other words, it is the roulette
Roulette (curve)
In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, and involutes....
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 times that of the rolling circle. It is named after the Greek letter delta
Delta (letter)
Delta is the fourth letter of the Greek alphabet. In the system of Greek numerals it has a value of 4. It was derived from the Phoenician letter Dalet...
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.btinternet.com/~se16/js/halfarea.htm
Equations
A deltoid can be represented (up to rotation and translation) by the following parametric equationswhere 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
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
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 genusGeometric genus
In algebraic geometry, the geometric genus is a basic birational invariant pg of algebraic varieties and complex manifolds.-Definition:...
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
Dual curve
In projective geometry, a dual curve of a given plane curve C is a curve in the dual projective plane consisting of the set of lines tangent to C. There is a map from a curve to its dual, sending each point to the point dual to its tangent line. If C is algebraic then so is its dual and the degree...
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.The perimeter (total arc length) of the deltoid is 16a.
History
Ordinary cycloidCycloid
A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line.It is an example of a roulette, a curve generated by a curve rolling on another curve....
s were studied by Galileo Galilei
Galileo Galilei
Galileo Galilei , was an Italian physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations and support for Copernicanism...
and Marin Mersenne
Marin Mersenne
Marin Mersenne, Marin Mersennus or le Père Mersenne was a French theologian, philosopher, mathematician and music theorist, often referred to as the "father of acoustics"...
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
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
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 groupGroup (mathematics)In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
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 lineSimson lineIn geometry, given a triangle ABC and a point P on its circumcircle, the three closest points to P on lines AB, AC, and BC are collinear. The line through these points is the Simson line of P, named for Robert Simson...
s of given triangle, form an envelopeEnvelope (mathematics)In geometry, an envelope of a family of curves in the plane is a curve that is tangent to each member of the family at some point. Classically, a point on the envelope can be thought of as the intersection of two "adjacent" curves, meaning the limit of intersections of nearby curves...
in the shape of a deltoid. This is known as the Steiner deltoid or Steiner's hypocycloid after Jakob SteinerJakob SteinerJakob Steiner was a Swiss mathematician who worked primarily in geometry.-Personal and professional life:...
who described the shape and symmetry of the curve in 1856. - The envelopeEnvelope (mathematics)In geometry, an envelope of a family of curves in the plane is a curve that is tangent to each member of the family at some point. Classically, a point on the envelope can be thought of as the intersection of two "adjacent" curves, meaning the limit of intersections of nearby curves...
of the area bisectors of a triangleTriangleA triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
is a deltoid (in the broader sense defined above) with vertices at the midpoints of the mediansMedian (geometry)In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposing side. Every triangle has exactly three medians; one running from each vertex to the opposite side...
. The sides of the deltoid are arcs of hyperbolaHyperbolaIn mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
s that are asymptoticAsymptoteIn analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors...
to the triangle's sides. http://www.btinternet.com/~se16/js/halfarea.htm