Ellipse

An ellipse is a special closed oval curve. It is one next to the parabola and the hyperbola to the conic sections. A clear definition is the definition of the ellipse as a set of points.

In nature, ellipses in the form of undisturbed kepler between planetary orbits to the Sun. Even when drawing ellipses inclined images often required since a circle is represented by a parallel projection in general on an ellipse (see Ellipse ( Descriptive Geometry ) ).

The ellipse (from Greek ἔλλειψις élleipsis, lack ') was introduced and named by Apollonius of Perga, the term refers to the eccentricity.

• 2.1 Focus property 2.1.1 Natural occurrence and use in the technology

• 3.1 gardener construction
• 3.2 trammel
• 3.3 design by de la Hire
• 3.4 Parallelogrammmethode
• 3.5 paper strip method
• 3.6 On the basis of a circle
• 3.7 Rytzsche axle construction
• 3.8 approximation over circles of curvature

• 5.1 ellipse equation ( Cartesian coordinates) 5.1.1 derivation

• 5.3.1 derivation

• 5.4.1 derivation
• 5.4.2 Alternative derivation

• 6.1 tangent equation ( Cartesian coordinates)
• 6.2 tangent equation ( parameters form)
• 6.3 Relationship between Polar and normal angle 6.3.1 derivation

• 7.1 acreage 7.1.1 ellipse sector

• 7.2.1 series expansion
• 7.2.2 approximations

Definitions and Terms

There are various ways to define ellipse. In addition to the definition of certain distances of points it is also possible to refer to an ellipse in an oblique view a circle, or as a line of intersection between an inclined plane and a circle corresponding to the double cone.

Ellipse as a set of points

An ellipse may be defined as the set of all points P of the plane for which the sum of distances from two given points F1 and F2 are equal. The points F1 and F2 are called foci.

Vertices and axes

The axis passing through the two focal points is, the major axis and is divided by the center M in its two major axes MS1 and MS2. The points S1 and S2 are called main peak. The length of each of the two semi-major axes is referred to as a:

Similarly, we speak of the secondary vertices of S3 and S4 and the minor axis, consisting of the semi-minor axes MS3 and MS4. The length of the semi-minor axis is denoted by b:

Major and minor axis are perpendicular to each other and intersect at point M.

Specific distances

The defining equation together with symmetry considerations show that the distance of the side vertices S3 and S4 of the focal points F1 and F2 just equal to the size a in the definition:

• Applies after symmetry considerations

This means that the point set as specific

Can be specified.

Half the length p of an ellipse tendon passing through a focal point and perpendicular to the major axis is called the half- parameter, sometimes even parameter p or semi- latus rectum (half the latus rectum = 2 * P) of the ellipse:

Eccentricity

The distance of the focal points from the center is linear eccentricity and is denoted by s. The linear eccentricity is calculated over the right triangle Δ M F1 S3 with the Pythagorean theorem:

Apart from the linear eccentricity e, the dimensionless numerical eccentricity is often used:

It follows:

• If so, and the ellipse is a circle.
• If so, and it is called the ellipse an ellipse or ellipse equilateral most beautiful form.
• Is large compared so close to one and the ellipse is therefore a parabola very close.

Ellipse as a conic

An ellipse can also be regarded as the intersection of a plane with a cone, wherein the angle of intersection between the plane and the cone axis must be greater than half the opening angle of the double cone. The defining property ( " sum of the distances to two fixed points, ...", see above ) can be proved with the help of Dandelinschen balls. See also conic.

Main location and analytical definition

An ellipse whose center is at the origin and whose principal axis coincides with the X axis is called the ellipse in the first main pad. The equation applies

For the coordinates of the ellipse points of such an ellipse.

Ellipse as an affine image of the unit circle

Another definition of the ellipse used a special geometric figure, namely the affinity. Here the ellipse is defined as the affine image of the unit circle (see Leopold, C., p.55). An affine transformation on the real plane, the shape with a regular matrix ( determinant is not 0 ), and an arbitrary vector. If the column vectors of the matrix, the unit circle to the ellipse is

Mapped. is the center and two conjugate radius (see below) of the ellipse. i.a. are not perpendicular. That and are i.a. not the vertex of the ellipse. This definition of an ellipse provides a simple parametric representation (see below ) of an arbitrary ellipse.

Since a vertex, the tangent to the corresponding diameter of the ellipse is perpendicular to the tangent direction and in an ellipse point, the parameter of an apex obtained from the equation

And thus out. ( It formulas were used.)

If it is, is, and the parametric representation already in peak form!

4, the apex of the ellipse

The vertex form of the parametric representation of the ellipse

Examples:

Note: If the vectors from the, we obtain a parametric representation of an ellipse in space.

Properties

Focus property

The connecting line between a focal point and a point of the ellipse is called focal line, beacon, or thermal radiation. Your name were focal points and focal radiation due to the property that the angle between the two focal radiation is halved in a point of the ellipse by the normal at this point. Thus, the incident angle between the beam at an internal of the tangent line equal to the angle, formed by the tangent to the other fuel jet. A light beam emanating from a focal spot would therefore reflected on the ellipse tangent so that it meets the other focal point. For an ellipsoidal mirror all of a focal point outgoing light beams accordingly in the other focus.

Since the path from one to the other focus ( along two associated focal rays) is always the same length, for example, sound is not only "amplified" (see below ) is transferred from one to the other focal point, but is even time-and phase (ie, understandable and not interfering ) there.

Two ellipses with coincident foci are called confocal.

An ellipse whose two foci to a collapse is in this case the circle (corresponding to vanishing eccentricity, see above).

Natural occurrence and use in the technology

The ceilings of some caves resemble an ellipse half. If you find yourself - with your ears - at a focal point of this ellipse, you can hear every noise, whose origin is in the second focal point, reinforced ( " Flüstergewölbe "). This type of sound transmission works in some stations of the Paris Métro even from platform to platform. The same principle of sound focusing is now used to break up kidney stones with shock waves. In lamp-pumped Nd: YAG laser is used, a reflector in the form of an ellipse. The pump source - either a flash lamp or an arc lamp - is positioned in a focal point, and the doped crystal is placed in the other focal point.

Directrix

A parallel to the minor axis at a distance a ² / s is called the directrix or guideline. For any point P of the ellipse, the ratio of its distance from a focal point to the distance from the directrix D on the corresponding side of the minor axis is equal to the numerical eccentricity:

A given focal point F, a straight line D ( the directrix ), and a number

Define reversed an ellipse E the set of all points P for which the ratio of their distance

From the focal point to their distance

Is from the straight line d is equal to ε.

Conjugate diameter

Is considered to any ellipse diameter ( chord of an ellipse by the ellipse center ) PP ' all parallel chords, their centers lie on a likewise elliptical diameter QQ'. It's called QQ ' to the PP' conjugate diameter. If one forms the conjugate diameter again the conjugate diameter, we get back to the original ellipse diameter. In the drawing, which thus corresponds to the QQ ' conjugate diameters with the original diameter PP' match.

Construction

Gardener construction

An easy way to draw the ellipse exactly offers the gardener construction. It uses directly the ellipse Definition: To create an elliptical flower bed, you take two pegs in the foci and attached to the ends of a string of length 2a. Now one stretches the string and continues with a marking device along it. Because this method does require compass and straightedge additional tools, it is not a construction of classical geometry.

Trammel

Also ellipses with Frans van Schoo Tens trammel or based thereon replicas can be constructed. The hinge mechanism was invented by the Dutch mathematician in the 17th century. If you move to point E on the pin, this draws an ellipse. The mechanism is mounted at the focal points H and I on the drawing surface.

Construction according to de la Hire

By means of the construction according to the ellipse de la Hire (also construction according Proclus ) ellipse points can be constructed without the focal points must be known. Draw two concentric circles with radii b ( inner or secondary circuit ) and a ( external or main loop ), and additionally an outgoing line from the center of the slope, which intersects both circuits. The parallel to the X- axis through the point of intersection on the secondary circuit at point P meets the parallel to the Y axis through the intersection of the main circuit. Change of the polar angle P can be the contour of the ellipse with semi-axes a and b to follow.

Parallelogrammmethode

The sides of a rectangle with sides of length 2a × 4b are each divided into n and 2n sections. The ellipse with semi-axes a and b, then, point by point - as shown in the graph - constructs. This method is based on the set of Steiner on the production of a conic section. Analogous methods are also available for parabola and hyperbola.

Paper strip method

Done to a strip of paper the length, marks the dividing point that separates the strip into two stretches of length and, and leaves the ends of the paper strip to slide on the coordinate axes (see picture), so the point goes through an ellipse with semi-axes (see C. Leopold, p 60). The paper strip method is an aid to properly carry the main axes after a Rytz construction.

On the basis of a circle

Especially in computer graphics is worth the derivation of an ellipse from a circle shape. An axis-parallel ellipse is simply a circle, compressed or stretched in one of the coordinate directions in other words, was scaled differently. A general, rotated at any angle ellipse can be obtained from such an axis-aligned ellipse by shear, see a Bresenham algorithm.

Rytzsche axle construction

Two conjugate given diameter can be determined with the help of the axle construction Rytzschen primary and secondary vertex (and the axes).

Approximation over circles of curvature

Ellipses can be ( with ruler and compass ) construct only pointwise, that is, an exact construction such as the circle is impossible. With the help of circles of curvature at the vertices of a graph and ruler but also allows drawing a fairly accurate picture of the ellipse to create. ( A circle of curvature at a point on the curve is the circle that best conforms to the neighborhood of the point on the curve. ) The radii of curvature at the vertices are respectively (see formulary, below). The midpoints of the respective vertex curvature circles can be easily determined graphically ( see picture below ) and draw in the circles of curvature (see Leopold, C., p.64 ).

Examples

• If you look at an angle on a circle ( for example, on the top surface of a circular cylinder ), this circle appears as an ellipse. More precisely, A parallel projection maps circles from generally ellipses.

• In astronomy, ellipses often occur as orbits of celestial bodies. After the first Kepler 's law, each planet moves in an ellipse around the sun, which is in one of the two foci. The same applies to the orbits of recurring ( periodic ) comets, planets moons or double stars. Generally arise in each two-body problem of the gravitational force, depending on the energy ellipse, parabola or hyperbolic orbits.

• For each two - or three-dimensional harmonic oscillator, the movement takes place on an elliptical path. So swings of the pendulum about the body of a thread pendulum approximation in an elliptical orbit, if s not only carried the motion of the pendulum ( thread ) in a plane.

Formulary ellipse equations

Elliptic equation ( Cartesian coordinates)

Center,

Resolved by:

The last form is useful for displaying an ellipse using the two orbital elements, eccentricity, and semimajor axis.

Center, major axis parallel to the X -axis:

Derivation

To derive the equation of an ellipse (center ), one imagines, first with the help of the accompanying image following system of equations:

(1)

(2)

(3)

(4)

Formula (1 ) is as a direct result of the ellipse defined. This reshaped so that only quadratic terms occur:

Substituting ( 2) and (3) provides:

This yields together with ( 4) the desired relation:

Elliptic equation ( parameters form)

Center, major axis as the X axis:

Center, major axis parallel to the X -axis:

Center, major axis with respect to X - axis rotates:

It refers to the parameters of this representation. This does not correspond to the polar angle between the axis and the straight line passing through the origin and the respective ellipse point. In astronomy, Kepler ellipses of these parameters is called the eccentric anomaly at Meridian ellipses in geodesy it is called parametric or reduced width, see reference ellipsoid.

For non-rotated elliptical, so dependent is the polar angle, which is defined by, with the parameters together via:

This relationship allows a clear interpretation of the parameter: Stretches the one coordinate of a point of the ellipse by a factor, this new point is on a circle of radius and the same center as the ellipse. The parameter is then the angle between the axis and the line:

Elliptic equation ( polar coordinates with respect to the center point )

Major axis horizontal, as the center pole, the polar axis along the major axis to the right:

Expressed in Cartesian coordinates, parameterized by the angle of polar coordinates, wherein the center of the ellipse and with its major axis lying along the x -axis:

Derivation

Of the ellipse equation in Cartesian coordinates and the parameterization of the Cartesian to polar coordinates, and the following:

Moving and returns the square root radius as a function of the polar angle.

Elliptic equation ( polar coordinates with respect to a focal point )

Major axis horizontal, right focal point as a pole, the polar axis along the major axis to the right ( half- parameters):

Major axis horizontal, left focal point as a pole, the polar axis along the major axis to the right:

The range of radii extending from the Periapsisdistanz to Apoapsisdistanz that have the following values:

Expressed in Cartesian coordinates, and parameterized by the angle of the polar coordinates, with the right focal point of the ellipse in which the left focal point is:

The angle or, depending on which pole is the reference point, in astronomy is called the true anomaly.

Derivation

Considering a triangle that is defined by the two fixed points, and an arbitrary point on the ellipse.

The distances between these points are as follows: and and according to the definition of the ellipse. The angle at is. Using the cosine rule now applies:

Analog extends the derivation for the right pole. The distances and loud and. The angle at is, as defined, where the right main peak marked.

Alternative derivation

Equating the two equations one obtains

This corresponds with one hand and

And on the other hand and with

Formulary curve properties

Tangent equation ( Cartesian coordinates)

Center, major axis as the X axis, the contact point:

Center of the main axis parallel to the X-axis, the contact point:

Tangent equation ( parameters form)

A ( unnormalized ) tangent vector to the ellipse has the form:

The tangent equation is expressed as a vector centered at, the principal axis as the X axis and point of contact for:

Relationship between Polar and normal angle

The following relationship exists between polar angle and normal angle and ellipse parameters ( see chart )

Derivation

The relationship of the polar angle and the pitch angle of the normal (see chart at right ) can be found, for example like this:

Dissolving the tangent equation for

Gives the tangent slope as the coefficient of to

With one obtains the desired relation between and.

Normal equation ( Cartesian coordinates)

Center, major axis as the X axis, the contact point:

Or

Normal equation ( parameters form)

A ( unnormalized ) normal vector to the ellipse has the form:

The normal equation is expressed as a vector centered at, the principal axis as the X axis and point of contact for:

Radii of curvature

Radius of curvature at one of the two main peaks:

Radius of curvature at one of the two side vertices:

Radius of curvature at the point:

Formulary area and perimeter

Area

With the semi-axes and:

If the ellipse by an implicit equation

Given, then is her area

Ellipse sector

For an ellipse with semi-axes and and a sector which includes the angle with the semi-major axis, the following applies:

Describing the ellipse sector rather than by the polar angle by the parameters from the parameter representation, we obtain the formula

Scope

The circumference of an ellipse can not be exactly represented by elementary functions. It can be represented as an integral, which is therefore called elliptic integral. With the parameterization provides the scope of using the Pythagorean Theorem to

The last integral is obtained after the substitution. The amount depends on the numerical eccentricity and semi-major axis. is called elliptic integral. With the help of the diagram opposite can at a given eccentricity ε of the value of the factor for the product

Be read. is ( degenerate ellipse to the line ) for each ellipse between the extreme cases, and ( ellipse becomes a circle ).

Series expansion

For ε close to 1, this series expansion converges very slowly. It is therefore recommended a numerical integration, eg according to the Romberg method.

Approximations

The last approximation in a wide range of ε -

Very accurate. The relative error increases thereafter with increasing ε and is:

The converse, that is an image which ( for a given ellipse ) of the arc length of an angle maps is an elliptic function.

Character

Unicode contains in block Miscellaneous Symbols and Arrows four ellipse symbols in any text (including body) can be used as a graphic character or jewelry characters:

LaTeX knows also still a hollow horizontal ellipse with shadow right: \ ellipse Shadow.