“Hypocycloid”的概念、定义、习题解题方法及翻译-数学辞典

单词

Hypocycloid

释义

Hypocycloid

The curve produced by a small Circle of Radius rolling around the inside of a large Circle ofRadius . A hypocycloid is a Hypotrochoid with . To derive the equations of the hypocycloid, call theAngle by which a point on the small Circle rotates about its center , and the Angle from thecenter of the large Circle to that of the small Circle . Then

(1)

(2)

Call

. If

, then the first point is at minimum radius, and the Cartesian parametric equations of the hypocycloid are


			(3)

			(4)

If instead so the first point is at maximum radius (on the Circle), then the equations of the hypocycloid are

			(5)
			(6)

An -cusped non-self-intersecting hypocycloid has . A 2-cusped hypocycloid is a Line Segment, as can be seenby setting in equations (3) and (4) and noting that the equations simplify to

			(7)
			(8)

A 3-cusped hypocycloid is called a Deltoid or Tricuspoid, and a 4-cusped hypocycloid is called anAstroid. If

is rational, the curve closes on itself and has

cusps. If

is Irrational, the curve never closes and fills the entire interior of the Circle.

-hypocycloids can also be constructed by beginning with the Diameter of a Circle, offsetting one end by aseries of steps while at the same time offsetting the other end by steps times as large in the opposite direction andextending beyond the edge of the Circle. After traveling around the Circle once, an -cusped hypocycloid isproduced, as illustrated above (Madachy 1979).

Let be the radial distance from a fixed point. For Radius of Torsion and Arc Length , a hypocycloidcan given by the equation

(9)

(Kreyszig 1991, pp. 63-64). A hypocycloid also satisfies

(10)

where

(11)

and

is the Angle between the Radius Vector and the Tangent to the curve.

The Arc Length of the hypocycloid can be computed as follows


			(12)

			(13)




	(14)

(15)

for

. Integrating,



			(16)

The length of a single cusp is then

(17)

is rational, then the curve closes on itself without intersecting after

cusps. For

and with

, the equations of the hypocycloid become

			(18)
			(19)

and

(20)

Compute



			(21)

The Area of one cusp is then




			(22)

is rational, then after

cusps,


			(23)

The equation of the hypocycloid can be put in a form which is useful in the solution of Calculus ofVariations problems with radial symmetry. Consider the case , then





			(24)

But

, so

, which gives




			(25)

			(26)

Now let

(27)

(28)

(29)

then


			(30)

The Polar Angle is

(31)

But

			(32)
			(33)
			(34)





			(35)

Computing



			(36)

then gives

(37)

Finally, plugging back in gives


			(38)

This form is useful in the solution of the Sphere with Tunnel problem, which is the generalization of theBrachistochrone Problem, to find the shape of a tunnel drilled through a Sphere (withgravity

varying according to Gauss's law for gravitation

) such that the travel time between two points on the surface of the gravity

is minimized.

See also Cycloid, Epicycloid

References

Bogomolny, A. ``Cycloids.'' http://www.cut-the-knot.com/pythagoras/cycloids.html.

Kreyszig, E. Differential Geometry. New York: Dover, 1991.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 171-173, 1972.

Lee, X. ``Epicycloid and Hypocycloid.''http://www.best.com/~xah/SpecialPlaneCurves_dir/EpiHypocycloid_dir/epiHypocycloid.html.

MacTutor History of Mathematics Archive. ``Hypocycloid.''http://www-groups.dcs.st-and.ac.uk/~history/Curves/Hypocycloid.html.

Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 225-231, 1979.

Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 50-52, 1991.

Yates, R. C. ``Epi- and Hypo-Cycloids.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 81-85, 1952.

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