Hyperbola

In the planar geometry is meant by a hyperbola, a special curve that consists of two mutually symmetric branches extending into infinity. It is one of the circle, the parabola and the ellipse to the conic sections, with a right circular cone arise (see picture) at the intersection of a plane.

How ellipse and parabola can be hyperbolas as loci in the plane define (see section definition).

The hyperbola was discovered by Menaechmus. The introduced by Apollonius of Perga name comes from the Greek and refers to the exaggeration ( ὑπερβολή, hyperbole, from Ancient Greek βάλλειν ballein "throw", ὑπερβάλλειν hyperballein " over the top throw " ) of the cutting angle ( or numerical eccentricity, see below) the conic: with increasing cutting angle, the circle (= 0) transforms first to ever more elongated ellipses, and then the parabola ( 1, and the cutting plane parallel to a tangent plane of the cone) hyperbolas with > 1

• 8.1 Tangent construction
• 8.2 point construction
• 8.3 tangents asymptotes Triangle

Definition of a hyperbola as a locus

A hyperbola is defined as the set of all points of the plane of the drawing, for which the absolute difference of the distances from two given points, the so-called hot spots, and is constant and equal to:

The center of the foci is called the center of the hyperbola. The straight line joining the foci is the major axis of the hyperbola. The two Hyperbelpunkte on the major axis, the vertex, and have the distance from the center. The distance of the focal points from the center is the focal length or linear eccentricity and is commonly referred to. The mentioned in the introduction, dimensionless numerical eccentricity.

That the intersection of a right circular cone with a plane, a) is steeper than the generatrices of the cone, and b) does not include the conical tip is actually a hyperbola showing one by detecting the above defining characteristic using Dandelinschen balls ( S. hyperbola as a conic section ).

Note: The equation can also be interpreted as:

Hyperbole in 1 main location

Equation

The equation of the hyperbola is replaced by an especially simple form when it is in " the first main layer ", which means that the two focal points are symmetrical to the origin on the axis; at a hyperbola in 1 main sheet ie the focal points have the coordinates and, and the vertices have the coordinates and.

For any point in the plane is equal to the distance to the focal point and the other focal point. The point is exactly then on the hyperbola, if the difference of these two expressions is equal or the same.

To the equation

Equivalent. The latter equation is called the equation of the hyperbola in 1 main sheet.

Vertex

A hyperbola has two vertices. In contrast to the ellipse are no curve points here. The latter are therefore also called imaginary co-vertex. The straight line through the secondary vertex is called the minor axis. The hyperbola is symmetrical to the major and minor axis.

Asymptotes

Solving the hyperbola equation for y, we obtain

Here you can see that the hyperbola for large magnitude x of the straight lines

Arbitrarily close approaches. These straight lines passing through the center and are called the asymptotes of the hyperbola.

Parameter p

Half the length of a Hyperbelsehne that goes through a focal point and is perpendicular to the major axis (sometimes transverse dimension or only parameter ) is called the semi- parameter of the hyperbola. It can be calculated by

Further meaning of p:

Ie, p is the radius of the circle by an apex that best conforms to the hyperbola at the apex. ( See: formulary / vertex equation)

Tangent

The equation of the tangent at a Hyperbelpunkt can be found most easily by implicit differentiation of the hyperbola:

Taking into account the results

Equilateral hyperbola

A hyperbola, applies to the, ie equilateral hyperbola. Your asymptotes are perpendicular. The linear eccentricity, eccentricity, and the semi- parameter.

Parameter representation with hyperbolic

With the hyperbolic functions results in a ( analogous to the ellipse) parametric representation of the hyperbola:

Hyperbola in 2 main location

Interchanging x and y, we obtain hyperbolas in 2 main location:

Hyperbola as conic

If you cut a vertical circular cone with a plane whose inclination is greater than the slope of the generating lines of the cone and not passing through the cone tip, the result is a hyperbolic curve of intersection (see picture, red curve ). The proof of the defining property with respect to the focal points ( see above) is carried by two Dandelin'schen balls, which are balls that the cone in circles, respectively, and the hyperbolic plane in points or touch. It turns out that the focal points of the Schnitthyperbel are.

Tangent as a bisector

For a hyperbola applies:

• The tangent at a point is the angle bisector of the internal radiation

For the proof we use the auxiliary point on the focal beam emitted from the distance has (see image, a is the semi-major axis of the hyperbola ). The straight line is the bisector of the internal rays. To prove that the tangent at the point is, we show that each can not lie on the hyperbola from different point of. So the hyperbola can only intersect at the point and is therefore the tangent. It can be seen from the drawing ( triangle ) that is, that it is. If a Hyperbelpunkt would, the difference would have to be the same.

Guidelines property

The term directrix or guideline refers to the two parallels to the minor axis at a distance. For any point on the hyperbola is the ratio between the distances to one focus and the associated Guideline equal to the numerical eccentricity:

For proof, one can show that for and the equation

Conversely, it is a point ( a focal point ) and a straight line pretending ( as a guideline ) and a real number and define a hyperbola as

• The set of all points in the plane, for which the ratio of the distances to the point and the straight line is the same.

( If you choose, you get a parabola. For results in an ellipse. )

For the proof goes by and the requirement that a curve point is made ​​. The guideline is then described by the equation. For follows

With the abbreviation is obtained

This is the vertex equation of an ellipse (), a parabola () or a hyperbola (). See section formulary. Is introduced in the case of new constants so that, as the vertex goes into equation

This is the equation of a hyperbola with a center x-axis as the main shaft and half-axles.

Steiner generation of a hyperbola

The following idea to construct single points of a hyperbola is based on the Steiner- generation of a conic section (after the Swiss mathematician Jakob Steiner):

For the creation of single points of the hyperbola, we assume that the straight tufts in the vertices. Now Be a point of hyperbole and. We divide the side of the rectangle into n equal pieces and transfer this division means of a parallel projection in the direction of the diagonals on the track (see picture). The used parallel projection gives the required projective transformation of the tufts in and. The intersections of the associated line and then lie on the uniquely determined by the requirements hyperbola.

Note: The partitions can be beyond the points and continue to construct more points. But since then rubbing cuts and a very unequal distribution of points occur, it is better to transfer the construction of the above points symmetrically on the other Hyperbelteile (see animation).

Comment:

Hyperbola as affine image of Einheitshyperbel

Another definition of the hyperbola uses a special geometric figure, namely the affinity. Here the hyperbola is defined as affine image of Einheitshyperbel. An affine transformation on the real plane, the shape with a regular matrix ( determinant is not 0 ), and an arbitrary vector. Are the column vectors of the matrix, the Einheitshyperbel the hyperbola

Mapped. is the center of the hyperbola and a tangent vector at that point. i.a. are not perpendicular. Ie are i.a. not the vertices of the hyperbola. But are the direction vectors of the asymptotes. This definition of a hyperbola provides a simple parametric representation of any hyperbola.

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

( It formulas were used.)

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

The two vertices of the hyperbola are

From

And the addition theorems for the hyperbolic functions yields the vertex form of the parametric equation of the hyperbola:

Examples:

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

Hyperbola as affine image of the hyperbola y = 1 / x

Since the Einheitshyperbel hyperbole is equivalent (see above), one can also be regarded as an arbitrary hyperbolic affine image of the hyperbola:

Is the center of the hyperbola, point in the direction of the asymptotes and a point on the hyperbola.

For the tangent vector results

A vertex, the tangent to the corresponding Hyperbeldurchmesser is perpendicular, that is, it is

So is the peak parameters

Tangent construction

The tangent vector can be written by factoring out like this:

That is, in the parallelogram is the diagonal parallel to the tangent at Hyperbelpunkt (see picture). This property provides an easy way to construct the tangent in a Hyperbelpunkt.

Note: This property of a hyperbola is an affine version of the 3-point degeneracy of the set of Pascal.

Point construction

Another feature allows the construction of a hyperbola Hyperbelpunkten if the asymptotes and a point of the hyperbola are known:

For a hyperbola with the parametric representation ( The center of the simplicity was assumed for the sake than zero) applies:

If two Hyperbelpunkte, so the points

The simple proof is given by:.

Note: This property of a hyperbola is an affine version of the 4-point degeneracy of the set of Pascal.

Tangents asymptotes Triangle

For the following considerations, we assume for simplicity that the center is located at the origin (0,0) and that the vectors have the same length. If the latter is not the case, the parameter representation is first brought into vertex form (see above). This has the consequence that the crown and the secondary crown are. So and.

If one calculates the points of intersection of the tangent in the Hyperbelpunkt with the asymptotes, we obtain the two points

The area of ​​the triangle can be expressed with the help of a 2x2 determinant:

(see calculation rules for determinants. ) is the area of ​​the plane defined by diamond. The area of ​​a rhombus is equal to half the diagonal of the product. The diagonals of the rhombus are the semi-axes. So the following applies:

Affine self-maps of the hyperbola y = 1 / x

Not every affine transformation of the real affine plane (see previous section ) is the hyperbola from a different hyperbola. The following affine transformations leave invariant the hyperbola as a whole:

Special cases:

Midpoints of parallel chords

For each hyperbola applies:

• The midpoints of parallel chords (see picture) lie on a line through the center of the hyperbola.

That is, for every pair of points of a tendon there is an oblique reflection in a. Line through the center of the hyperbola, which swaps the points and maps the hyperbola to be Is understood as an oblique reflection is a generalization of an ordinary reflection in a straight line, where all routes point - image point though but not necessarily parallel perpendicular to the mirror axis.

The proof of this property is carried them most easily by the hyperbola. Since all hyperbolas are affine images of Einheitshyperbel and thus also of the hyperbola and pass in an affine mapping midpoints of lines in the centers of the photo spreads, the above property is true for all hyperbolas.

Note: The points of the tendon may lie on different branches of the hyperbola.

One implication of this symmetry is that the asymptotes of the hyperbola are reversed in the oblique reflection and the center of a Hyperbelsehne also halved the associated distance between the asymptotes, ie it is. This property can be used to construct any size at a point known asymptotes and many other Hyperbelpunkte by using each route for the construction of.

The touch degenerate the tendon to a tangent, so halved the section between the asymptotes.

Pole - polar relationship

A hyperbola can always be described by an equation of the form in a suitable coordinate system. The equation of the tangent in a Hyperbelpunkt is Leaving to in this equation that any different from the zero point in the plane, then it will

This straight line does not pass through the center of the hyperbola.

Conversely, one can

Such a mapping point < - > Straight called a polarity or pole - polar relationship. The pole is the point that is the corresponding polar line.

The importance of this pole - polar relationship is that the possible intersections of the polar of a point with the hyperbola are the points of contact of the tangents through the pole of the hyperbola.

• If the point (pole ) on the hyperbola, its polar is the tangent line at this point (see picture: ).
• If the pole outside the hyperbola, so are the intersection points of the polar with the hyperbola, the points of contact of the tangents through the pole of the hyperbola (see picture: ).
• If the point lies within the hyperbola, so its polar has no intersection with the hyperbola (see picture: ).

To prove: The determination of the intersections of the polars of a point with the hyperbola and the search for Hyperbelpunkten whose tangents contain the point, lead to the same system of equations.

Comment:

Note: pole - polar relationships also exist for ellipses and parabolas. See also projective conic.

Hyperbolas of the form y = a / (x -b ) c

Peripheral angle theorem for hyperbolas

Hyperbolas of the form are graphs of functions, which are uniquely determined by the three parameters. Thus, one needs three points to determine these parameters. A fast method is based on the peripheral angle theorem for hyperbolas.

To measure an angle between two chords we carry two lines, the x - nor are parallel to the y - axis either to a square one:

Similar to the peripheral angle theorem for circles is considered here the

Peripheral angle theorem: ( f hyperbolas )

( Proof by recalculation. Here there is one direction assume that the points lie on a hyperbola y = a / x. )

3 -point shape of a hyperbola

( Measured with the slope inclination angle) Analogous to the two - point form of a straight line follows from the peripheral angle theorem for the hyperbolas

3 -point form: ( f hyperbolas )

Formulary

Hyperbola

A hyperbola with the center (0 | 0) and the x-axis as the principal axis satisfying the equation

The asymptotes of the hyperbola are the corresponding straight lines:

Focal points are:

A hyperbola with center point and the straight line as the main axis satisfies the equation

Vertex equation

The crowd of hyperbolas whose axis is the x - axis, a vertex of the point (0,0) and the center (-a, 0), can be expressed by the equation

Describe.

For hyperbolas. Substituting in this equation

The conics have at the same half- parameters all the same curvature radius at the apex

Parametric equations

The center (0 | 0), x-axis as the main axis:

In polar coordinates,

Note

Angle to the principal axis, the center pole (0,0):

Angle to the principal axis, pole at a focal point:

Tangent equation

Center ( 0 | 0), major axis as the x axis, the contact point

Center, major axis parallel to the x - axis, the contact point

Curvature radius

The curvature radius of the hyperbola in the two vertices is:

Hyperbolas as plane sections of quadrics

The following second-order ( quadric ) surfaces have hyperbolas as plane sections:

• Elliptic cone ( see also Conic )
• Hyperbolic cylinder
• A hyperbolic paraboloid
• Single-leaf hyperboloid
• Two-shell hyperboloid

Hyperbolic cylinder

A hyperbolic paraboloid

Single-leaf hyperboloid

Two-shell hyperboloid

Hyperbola y = 1 / x over an arbitrary number field

Looking in an affine plane over a arbitrary ( commutative ) body, the set of points satisfying the hyperbola, as many characteristics of the real case, "cut" with the stay " connect ", and formulated "parallel" and their evidence only multiplication / division and addition / subtraction using will receive. For example:

• A straight line intersects the hyperbola in at most two points.
• Through each Hyperbelpunkt there except the axis parallel lines exactly a straight line with the hyperbola only point in common, the tangent. One is called straight line without intersection Passante, one with two intersections secant.

Differences to the real case: