Ellipse

Ellipse (dr. Greek ἔλλειψις - omission, lack, in the sense of lack of eccentricity to 1) is the geometrical location of points M of the Euclidean plane, for which the sum of the distances to two given points and (called foci) is constant and greater than the distance between the foci, i.e.
and
The circle is a special case of an ellipse. Along with a hyperbola and a parabola, an ellipse is a conic section and a quadric.
An ellipse can also be described as the intersection of a plane and a circular cylinder or as an orthogonal projection of a circle onto a plane.
Contents
1 Related definitions
2 Properties
3 Relations between elements of an ellipse
4 Coordinate representation
4.1 Ellipse as a second-order curve
4.2 Canonical equation
4.3 Equations in parametric form
4.4 In polar coordinates
5 Arc length of an ellipse
5.1 Approximate formulas for the perimeter
6 Area el lime and its segment
7 Construction of an ellipse
8 See also
9 Notes
10 Literature
11 Links
Related definitions
The segment AB passing through the foci of the ellipse whose ends lie on an ellipse, called the major axis of the ellipse. The length of the major axis is 2a in the above equation.
The segment CD perpendicular to the major axis of the ellipse passing through the center point of the major axis, the ends of which lie on the ellipse, is called the minor axis of the ellipse.
The intersection point of the major and minor axes of the ellipse is called its center.
The segments drawn from the center of the ellipse to the vertices on the major and minor axes are called, respectively, the major axis and the minor axis of the ellipse, and are denoted by a and b.
The distances and from each of the foci to a given point on the ellipse are called focal The radius at this point.
distance is called the focal length.
magnitude is called the eccentricity.
diameter of the ellipse is called the arbitrary chord that passes through its center. The conjugated diameters of an ellipse are a pair of its diameters, which have the following property: the midpoints of the chords parallel to the first diameter lie on the second diameter. In this case, the midpoints of the chords parallel to the second diameter lie on the first diameter.
The radius of the ellipse at a given point (the distance from its center to a given point) is calculated by the formula, where is the angle between the radius vector of this point and the abscissa axis.
The focal parameter is called half the length of the chord passing through the focus and perpendicular to the major axis of the ellipse.
The ratio of the lengths of the minor and major semi-axes is called the compression coefficient of the ellipse or ellipticity: A value equal to is called compression of the ellipse. For a circle, the compression ratio is unity, compression is zero. The compression coefficient and eccentricity of the ellipse are related by the relation
For each of the tricks there is a line called the directrix, such that the ratio of the distance from an arbitrary point of the ellipse to its focus to the distance from this point to this line is equal to the eccentricity of the ellipse. The entire ellipse lies on the same side of such a straight line as the focus. The equations of the directrix of an ellipse in canonical form are written as for tricks, respectively. The distance between the focus and the director is equal to
Properties
Optical
Light from a source located in one of the foci is reflected by an ellipse so that the reflected rays intersect in the second focus.
Light from a source located outside any of of foci, is reflected by an ellipse so that the reflected rays do not intersect at any focus.
If and are the foci of the ellipse, then for any point X belonging to the ellipse, the angle between the tangent at this point and the line is equal to the angle between this tangent and the line.
A straight line drawn through the midpoints of two parallel lines intersecting the ellipse will always pass through the center of the ellipse. This allows the construction using a compass and a ruler to easily get the center of the ellipse, and then the axes, vertices and tricks.
The ellipse is the astroid, elongated along the vertical axis.
The intersection points of the ellipse with the axes are its vertices.
The eccentricity of the ellipse is equal to the ratio Eccentricity characterizes the elongation of the ellipse. The closer the eccentricity is to zero, the more the ellipse resembles a circle, and vice versa, the closer the eccentricity is to unity, the more elongated it is.
An ellipse can also be described as
a figure that can be obtained from a circle using the affine transformation
orthogonal projection of a circle onto a plane.
Intersection of a plane and a circular cylinder
Relations between elements of an ellipse
Parts of an ellipse (see the description in the “Related Definitions” section)
- major axis;
- minor axis ;
- focal distance (half-distance between do foci);
- focal parameter;
- focal length (minimum distance from focus to a point on an ellipse);
- apofocus distance (maximum distance from focus to a point on an ellipse);
.
- major axis
- minor axis
- focal distance
- focal parameter
- perifocal distance
- apofocus distance
Coordinate view
Ellipse as a second-order curve
The ellipse is a central nondegenerate curve of the second order and satisfies the general equation of the form
for invas and where:
Relations between second-order curve invariants and semi-axes of an ellipse (true only if the center of the ellipse coincides with the origin and):
Relations
If you rewrite the general equation in the form
then the coordinates the center of the ellipse:
the rotation angle is determined from the expression
The directions of the axes vectors:
from here
The axle lengths are determined by the expressions
The inverse relation - the coefficients of the general equation from the parameters of the ellipse - can be obtained by substituting in the canonical equation ( cm. section below) an expression for rotating the coordinate system by an angle Θ and transferring to a point:
After substituting and opening the brackets, we obtain the following expressions for the coefficients of the general equation:
It should be noted that in the equation of the general form of the ellipse defined in the Cartesian system coordinates, coefficients (or, which is the same thing) are determined up to an arbitrary constant factor, i.e. the above record and
where are equivalent. Therefore, one cannot expect that an expression of the form
will be satisfied for any.
The relationship between the invariant and the semi-axes in the general form is as follows:
where is the coefficient when the coordinate origin is transferred to the center of the ellipse when the equation is reduced to the form
Other invariants are in the following relationships:
Canonical equation
For any ellipse, you can find a Cartesian coordinate system such that the ellipse is described by the equation (canonical equation of the ellipse):
It describes an ellipse centered at the origin, axis otorrhea coincide with the coordinate axes. [1]
Relations
For definiteness, we assume that in this case the quantities and are, respectively, the major and minor semiaxes of the ellipse.
Knowing the semiaxes of the ellipse, you can calculate its focal distance and eccentricity:
Coordinates of the focuses of the ellipse:
The ellipse has two directrixes whose equations can be written as
The focal parameter (i.e., half the length of the chord passing through the focus and perpendicular to the axis of the ellipse) is equal to
Focal radii, i.e. the distance from the foci to arbitrary point of the curve
Equation of the diameter conjugated by chords with angle by the new coefficient:
The equation of a tangent to an ellipse at a point has the form
The condition of touching a line and an ellipse is written as a ratio
The equation of tangents passing through a point
The equation of tangents having a given angular coefficient:
points tangency of such a straight ellipse (or the same thing, the point of the ellipse where the tangent has an angle with a tangent):
The equation of the normal at the point
Equations in the parametric form
Geometric illustration of the parameterization of the ellipse (animation).
Canonical the ellipse equation can be pairs metrized:
where is the equation parameter.
In the case of a circle, the parameter is the angle between the radius vector of the given point and the positive direction of the abscissa axis.
In polar coordinates
If you take the focus of the ellipse as the pole and the major axis - for the polar axis, then its equation in polar coordinates will have the form
where e is the eccentricity and p is the focal parameter. With a negative sign in front of e, the second focus of the ellipse will be at the point and with a positive one at the point where the focal distance is
Conclusion
Let r1 and r2 be the distances to the given point of the ellipse from the first and second foci. Let also the pole of the coordinate system be in the first focus, and the angle be measured from the direction to the second focus. Then, from the definition of an ellipse,
From here,
On the other hand, from the cosine theorem
Excluding from the last two equations, we get
Given that
we get the desired equation.
If we take the center ellipse per pole, and the major axis - the polar axis, then its equation in polar coordinates will look like
Arc length of an ellipse
The length of an arc of a flat line is determined by the formula:
Using the parametric representation of the ellipse we get the following expression:
After replacing the expression for the length of the arc takes the final form:
Having received This integral belongs to a family of elliptic integrals that are not expressed in elementary functions, and reduces to an elliptic integral of the second kind. In particular, the perimeter of an ellipse is:
,
where is the complete elliptic integral of the second kind.
Approximate formulas for the perimeter
The maximum error of this formula is ~ 0.63% with an eccentricity of the ellipse ~ 0.988 (axis ratio ~ 1 / 6.5). The error is always positive.
Approximately two times smaller errors in a wide range of eccentricities are given by the formula:
, where
The maximum error of this formula is ~ 0.36% with an ellipse eccentricity of ~ 0.980 (axis ratio ~ 1/5) . The error is also always positive.
Significantly better accuracy is provided by the Ramanujan formula:
With an ellipse eccentricity of ~ 0.980 (axis ratio ~ 1/5), the error is ~ 0.02%. The error is always negative.
The area of the ellipse and its segment
The area of the ellipse is calculated by the formula
The area of the segment between an arc convex to the left and a chord passing through points and
[source not specified 1116 days]
If the ellipse is given by the equation, then the area can be determined by the formula
.
Building an ellipse
Ellipsograph in action
Building with needles, thread and a pencil.
Main article - article “Building an ellipse” at Wikibooks.
Tools for drawing an ellipse are:
an ellipsograph;
two needles stuck into the tricks of an ellipse and connected by a thread of length 2a, which is drawn with a pencil.
Using a compass or compass and a ruler, you can build any number of points that belong to the ellipse, but not the whole ellipse. See also
Second-order curve
Parabola
Caustics
Ellipsoid
Ellipsograph
Curve of constant difference of distances between two points - hyperbole,
constant ratio - Apollonius circle,
constant product - Cassini oval.
Notes
↑ If the unit with a minus sign is on the right side, the resulting equation:
describes an imaginary ellipse, it does not have points on the real plane. Literature
Corn G., Korn T. Properties of circles, ellipses, hyperbolas and parabolas // Mathematics Handbook. - 4th edition. - M .: Nauka, 1978. - S. 70-73.
Selivanov D.F.,. Ellipse // Brockhaus and Efron Encyclopedic Dictionary: in 86 volumes (82 volumes and 4 additional). - St. Petersburg, 1890-1907.
Links
Ellipse in Wiktionary?
Ellipse in Wikimedia Commons? A. V. Hakobyan, A.A. Zaslavsky. Geometric properties of second-order curves, - M.: MCCMO, 2007. - 136 pp.
I. Bronstein. Ellipse // Quantum, No. 9, 1970.
A. I. Markushevich. Wonderful Curves // Popular Math Lectures, Issue 4.
S. Sykora, Approximations of Ellipse Perimeters and of the Complete Elliptic Integral E (x). Review of known formulae
Grard P. Michon. Perimeter of an Ellipse (Final Answers), 2000-2005. - 20 p.
Video: How to draw an ellipse
Curves
Definitions
Analytical • Jordan • Kantorov • Uryson • Oval • Length • Radius of curvature
Transformed
Evolution • Involute • Podera • Antipodera • Parallel • Dual • Caustics - Not flat - Helix • Slope • Loxodrome • Orthodromy • Sponge - Algebraic
Conical sections - Hyperbola • Parabola • Ellipse (Circle)
3 order
Elliptic: Elliptic curve • Jacobi functions • Integral • Functions
Others: Verziers Agnese • Descarto leaf • Cube • Semi-cubic parabola • Strofoid • Cissoid Diocles
4th order
Kappa • Cardioid
Lemniscates
Bernoulli (Oval Cassini) • Buta • Gerono - Approximate
Spline (B Spline • Cubic • Monospline • Hermite) • Beziers - Cycloidal - Cardioid • Nephroid • Deltoid • Astroid • Snail Pascal - Flat transcendental - Spirals - Archimedes (Farm) • Galileo • Hyperbolic • “ Wand »• Clothoid • Logarithmic
Cycloidal
Cycloid • Epicycloid • Hypocycloid • Trochoid (Elongated + Shortened I am a cycloid) • Epitrochoid (Elongated + Shortened epicycloid • (“Rose”) • Hypotrochoid • Speedy descent (Brachistochrona, cycloid arc)
Other
Quadratrix • Cochleoid • Chases (Traktris) • Trochoid • Chain line (inverted arched) ) • Constant width • Sinusoid
Fractal
Simple
Koch • Levy • Minkowski • Peano
Topological
Napkin + Sierpinski carpet • Menger sponge
Conical sections
Main types
Ellipse • Hyperbola • Parabola
Degenerate
Point • Direct • A pair of lines
Special case ellipse and
Circle
Geometrical construction
conic sections • Balls Dandeli
See. also
Conic constant
Mathematics • Geometry


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