“Ellipse”的概念、定义、习题解题方法及翻译-数学辞典

单词

Ellipse

释义

Ellipse

A curve which is the Locus of all points in the Plane the Sum of whose distances and from two fixed points and (the Foci) separated by a distance of is a given Positiveconstant (left figure). This results in the two-center Bipolar Coordinate equation

(1)

where

is the Semimajor Axis and the Origin of the coordinate system is at one of theFoci. The ellipse can also be defined as the Locus of points whose distance from the Focusis proportional to the horizontal distance from a vertical line known as the Directrix (right figure).

The ellipse was first studied by Menaechmus, investigated by Euclid, and named by Apollonius. TheFocus and Directrix of an ellipse were considered by Pappus. In 1602,Kepler believed that the orbit of Mars was Oval; he later discovered that it was an ellipsewith the Sun at one Focus. In fact, Kepler introduced the word ``Focus'' and published hisdiscovery in 1609. In 1705 Halley showed that the comet which is now named after him moved in anelliptical orbit around the Sun (MacTutor Archive).

A ray passing through a Focus will pass through the other focus after a single bounce. Reflections not passingthrough a Focus will be tangent to a confocal Hyperbola or Ellipse, depending on whether the raypasses between the Foci or not. Let an ellipse lie along the x-Axis and find the equation ofthe figure (1) where and are at and . In Cartesian Coordinates,

(2)

Bring the second term to the right side and square both sides,

(3)

Now solve for the Square Root term and simplify



	(4)

Square one final time to clear the remaining Square Root,

(5)

Grouping the

terms then gives

(6)

which can be written in the simple form

(7)

Defining a new constant

(8)

puts the equation in the particularly simple form

(9)

The parameter

is called the Semiminor Axis by analogy with the parameter

, which is called theSemimajor Axis. The fact that

as defined above is actually the Semiminor Axis is easily shown by letting

and

be equal. Then two Right Triangles are produced, each with Hypotenuse

, base

, and height

. Since the largest distance along the Minor Axis will beachieved at this point,

is indeed the Semiminor Axis.

If, instead of being centered at (0, 0), the Center of the ellipse is at (, ), equation (9) becomes

(10)

As can be seen from the Cartesian Equation for the ellipse, the curve can also be given by asimple parametric form analogous to that of a Circle, but with the and coordinates having different scalings,

			(11)
			(12)

The unit Tangent Vector of the ellipse so parameterized is

			(13)
			(14)

A sequence of Normal and Tangent Vectors are plotted below forthe ellipse.

For an ellipse centered at the Origin but inclined at an arbitrary Angle to the x-Axis, theparametric equations are


			(15)

In Polar Coordinates, the Angle measured from the center of the ellipse iscalled the Eccentric Angle. Writing for the distance of a point from the ellipse center, the equation inPolar Coordinates is just given by the usual

			(16)
			(17)

Here, the coordinates

and

are written with primes to distinguish them from the more common polar coordinatesfor an ellipse which are centered on a focus. Plugging the polar equations into the Cartesian equation (9) andsolving for

gives

(18)

Define a new constant

called the Eccentricity (where

is the case of a Circle) to replace

(19)

from which it also follows from (8) that

			(20)
			(21)
			(22)

Therefore (18) can be written as

(23)

(24)

, then

(25)

(26)

If and are measured from a Focus instead of from the center, as they commonly are inorbital mechanics, then the equations of the ellipse are

			(27)
			(28)

and (9) becomes

(29)

Clearing the Denominators gives

(30)

(31)

Plugging in (21) and (22) to re-express

and

in terms of

and


	(32)

Simplifying,

(33)

(34)

The sign can be determined by requiring that

must be Positive. When

, (34) becomes

, butsince

is always Positive, we must take the Negative sign, so (34) becomes

(35)

(36)

(37)

The distance from a Focus to a point with horizontal coordinate

is found from

(38)

Plugging this into (37) yields

(39)

(40)

Summarizing relationships among the parameters characterizing an ellipse,

			(41)
			(42)
			(43)

The Eccentricity can therefore be interpreted as the position of the Focus as a fraction of theSemimajor Axis.

In Pedal Coordinates with the Pedal Point at the Focus, the equation of the ellipse is

(44)

To find the Radius of Curvature, return to the parametric coordinates centered at the center of theellipse and compute the first and second derivatives,

			(45)
			(46)

			(47)
			(48)

Therefore,




			(49)

Similarly, the unit Tangent Vector is given by

(50)

The Arc Length of the ellipse can be computed using




			(51)

where

is an incomplete Elliptic Integral of the Second Kind. Again, note that

is a parameter which does nothave a direct interpretation in terms of an Angle. However, the relationship between the polar angle from theellipse center

and the parameter

follows from

(52)

This function is illustrated above with shown as the solid curve and as the dashed, with . Care must betaken to make sure that the correct branch of the Inverse Tangent function is used. As can be seen, weaves backand forth around , with crossings occurring at multiples of .

The Curvature and Tangential Angle of the ellipse are given by

			(53)
			(54)

The entire Perimeter of the ellipse is given by setting (corresponding to ), which is equivalentto four times the length of one of the ellipse's Quadrants,

(55)

where

is a complete Elliptic Integral of the Second Kind with Modulus

.The Perimeter can be computed numerically by the rapidly converging Gauss-Kummer Series


			(56)

where

(57)

and

is a Binomial Coefficient. Approximations to the Perimeter include

			(58)
			(59)
			(60)

where the last two are due to Ramanujan (1913-14),

(61)

and (60) is accurate to within

The maximum and minimum distances from the Focus are called the Apoapsis and Periapsis, and aregiven by

			(62)
			(63)

The Area of an ellipse may be found by direct Integration




			(64)

The Area can also be computed more simply by making the change of coordinates

and

from theelliptical region

to the new region

. Then the equation becomes

(65)

, so

is a Circle of Radius

. Since

(66)

the Jacobian is

(67)

The Area is therefore


			(68)

as before. The Area of an arbitrary ellipse given by the Quadratic Equation

(69)

(70)

The Area of an Ellipse with semiaxes

and

with respect to a Pedal Point

(71)

The ellipse Inscribed in a given Triangle and tangent at its Midpoints is called theMidpoint Ellipse. The Locus of the centers of the ellipses Inscribed in a Triangle is the interiorof the Newton gave the solution to inscribing an ellipse in a convex Quadrilateral(Dörrie 1965, p. 217). The centers of the ellipses Inscribed in a Quadrilateral all lie on the straight linesegment joining the Midpoints of the Diagonals (Chakerian 1979,pp. 136-139).

The Area of an ellipse with Barycentric Coordinates Inscribed in aTriangle of unit Area is

(72)

(Chakerian 1979, pp. 142-145).

The Locus of the apex of a variable Cone containing an ellipse fixed in 3-space is a Hyperbola through theFoci of the ellipse. In addition, the Locus of the apex of a Cone containing thatHyperbola is the original ellipse. Furthermore, the Eccentricities of the ellipse andHyperbola are reciprocals. The Locus of centers of a Pappus Chain of Circles is anellipse. Surprisingly, the locus of the end of a garage door mounted on rollers along a vertical track but extending beyondthe track is a quadrant of an ellipse (the envelopes of positions is an Astroid).

See also Circle, Conic Section, Eccentric Anomaly, Eccentricity, Elliptic Cone,Elliptic Curve, Elliptic Cylinder, Hyperbola, Midpoint Ellipse, Parabola,Paraboloid, Quadratic Curve, Reflection Property, Salmon's Theorem, Steiner'sEllipse

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 126 and 198-199, 1987.

Casey, J. ``The Ellipse.'' Ch. 6 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 201-249, 1893.

Chakerian, G. D. ``A Distorted View of Geometry.'' Ch. 7 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., 1979.

Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 75, 1996.

Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, 1965.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 72-78, 1972.

Lee, X. ``Ellipse.''http://www.best.com/~xah/SpecialPlaneCurves_dir/Ellipse_dir/ellipse.html.

Lockwood, E. H. ``The Ellipse.'' Ch. 2 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 13-24, 1967.

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

Ramanujan, S. ``Modular Equations and Approximations to .'' Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914.

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