Parabola

In mathematics, a parabola is (from Latin parabola to ancient Greek παραβολή Parabole, juxtaposition, comparison, simile '; due to παρά pará, next ' and βάλλειν ballein, throw ') is a second order curve. In addition to the circle, the ellipse and the hyperbola it is one of the conic sections: It arises at the intersection of a right circular cone with a plane that is parallel to a surface line and does not go through the cone tip. A parabola can be considered as an ellipse, in which one of the two focal points at infinity.

The parable was discovered by Menaechmus and named by Apollonius of Perga, the term refers to their eccentricity as a conic section, which is equal to 1 for the parabola.

Although the parable is a special case of conic sections, but it plays in daily life a big role: Satellite antennas and parabolic mirrors have great industrial importance. This is based on the geometric property of the parabola to collect parallel to its axis incident beam at the focal point (see below). A diagonally thrown up stone moves approximately on a parabolic path, the parabola (see bouncing ball, fountain). In an airplane, which moves along a parabolic trajectory, there is a weightless state. Such parabolic flights are used to train astronauts. Parabolas are often used in mathematics to approximate more complicated functions, as they claimed the straight line ( equation ): own and making their cling as straight lines on a function graph, the simplest curved graphs ( equation). In the field of CAD (Computer Aided Design) parabolas appear as Bezier curves. A great advantage of the parabolas (as opposed to a circle, ellipse and hyperbola ) is the ability to describe them with the help of polynomials ( 2nd degree ).

• 9.1 peripheral angle theorem for parabolas
• 9.2 3 -point shape of a parabola

Definition with guideline

A parabola can be geometrically described as locus:

A parabola is the locus of all points whose distance from a specific fixed point - the focal point - equal to a specific line - Guideline - is.

As a set of points listed:

The point situated in the center between the focal point and directrix, ie the vertex of the parabola. The connecting line of focus and vertex is also called the axis of the parabola. It is the only axis of symmetry of the parabola.

Does it coordinates so that is the guideline and has the equation, it follows from the equation for a parabola open upward.

Half the length of the parabola at the height of the focal point arises from to

Is like ellipse ( the main peak ) and the hyperbola apex curvature radius. In a parable applies in addition

The equation of the parabola can thus be written:.

Interchanging x and y, we obtain with

Open to the right ( ) or left () parabolas.

Parabola as a function graph

Any parabola with vertex at the origin (0,0) of the y- axis as an axis and up and down is opened ( in Cartesian coordinates) by an equation

Described. your

For one obtains the standard parabola. Your focal point is the half- parameters and the guideline has the equation

For a < 0 the parabola open downwards (see picture).

After a shift we obtain the vertex form of any upward or downward parabola that opens:

Multiplying out gives the general equation of a parabola that opens up or down:

It is the graph of the quadratic function

If the function is given, we find the vertex by completing the square:

Parable as a special case of conic sections

The flock of conics, the axis of the x - axis, a vertex of the point (0,0), can be expressed by the equation

Describe. for

The general equation for conics is

In order to identify which conic section is described by a specific equation, one has to perform a principal axis transformation ( rotation and subsequent displacement of the coordinate system ). See conic.

Parabola as a conic

If you cut a vertical circular cone with a plane whose inclination is equal to the inclination of the lateral lines of the cone, the result is a parabolic curve of intersection (see picture, red curve ). The proof of the defining property with respect to the focal point and guideline (see above) is carried by a Dandelin'schen ball, i.e., a sphere that touches the cone in a circle and the parabola plane at one point. It turns out that the focal point of the parabolic section, and the intersection of the plane with the plane of the Berührkreises the guideline.

Steiner generating a parabola

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

For the creation of single points of the parabola we go from the pencil of lines in the vertex and the parallel clump of parallels to the y- axis ( i.e., the pencil of the farthest point of the y -axis). Now Be a point of the parabola, and, . We divide the range into n equal pieces and transfer this division means of a parallel projection in the direction of the track (see picture). The used parallel projection gives the required projective transformation of the tuft and in the parallel bundle. The intersections of the associated line and the i-th parallel to the y- axis are then uniquely determined by the requirements parabola ( see picture).

The proof follows by a simple calculation. See also: projective conic.

Note: The left half of the parabola is obtained by reflection in the y- axis.

Comment:

Parabola as affine image of the standard parabola

Another definition of the parabola used a special geometric figure, namely the affinity. Here is a parable is defined as affine image of the standard parabola. 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 parabola is normal to the parabola

Mapped. is a point of the parabola and the tangent vector at that point. are not perpendicular to each other i A.. This means that if A., not the vertex of the parabola. But: the parabola axis ( symmetry axis through the vertex ) must be parallel. This definition of a parabola provides a simple parametric representation of any parabola.

As the tangent to the parabola in the vertex axis is vertical and the tangent direction is a parable in point, the parameter of the vertex resulting from the equation

The vertex form of the parametric equation of the parabola

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

Affine self-maps of the parabola y = x ²

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

These are the only affine transformations that leave invariant the parabola.

To prove: Set and turn on the first binomial formula.

Special cases:

Note: Supplementing by a line at infinity and their distant points to a projective plane, adding to the Parable of the far point of the y- axis added, the result is a non-degenerate projective conic and has more pictures, projective collineations available the real affine plane. For example, can the projective collineation with

So the extended parabolic invariant. This figure is in involution, the parabolic points can be fixed and reversed the parabolic point to the farthest point of the y -axis.

Properties

Focus

If a beam incident parallel to the axis of the parabola - that is, at its tangent - mirrored, the mirrored beam passes through the focal point. This reflected beam is also called focal line or focal beam of the respective parabolic point. The corresponding property is a paraboloid of revolution, that is, the surface which is produced when rotating a parabola about its axis; It is often used in the art (see, parabolic ).

In order to prove this property of a parabola, we assume a parabolic shape. This is not a limitation since each parabola may be represented in a coordinate system as appropriate. Has the tangent at a point parabola equation ( the slope of the tangent is obtained from the derivative. ) The tangent intersects the y axis at the point. The focal point is. The nadir point of the perpendicular from the guideline is. For a parabola. From the picture you can see that is. Thus, the quadrilateral is a rhombus and the tangent is a diagonal of this rhombus and thus a bisector. It follows:

• The focal beam is the reflection of the incident beam at the tangent / parabola.

Midpoints of parallel chords

For each parabola applies:

• The midpoints of parallel chords (see picture) lie on a straight line. This line is parallel to the axis of the parabola.

That for every pair of points of a tendon there is an oblique reflection in a line that interchanges the points and maps the parable itself. Is understood as an oblique reflection is a generalization of an ordinary reflection in a straight line, where all routes point - pixel Although each other but not necessarily parallel perpendicular to the mirror axis. If the tendon perpendicular to the axis of the parabola, so the line crosses the axis of the parabola and the oblique mirror is an ordinary mirror.

The proof of this property is carried them most easily by the standard parabola. Since all parabolas affine images of the standard parabola are (see above) and go with an affine mapping midpoints of lines in the centers of the photo spreads, the above property is true for all parabolas.

Point construction

An arbitrary parabola may be described in a suitable coordinate system by an equation.

Another possibility parabola points to construct requires knowledge of three parabola points and the direction of the axis of the parabola advance:

For a parable applies: Are

• Four points of the parabola and
• The point of intersection of a secant with the straight line, and
• The intersection of the secant with the straight line (see picture)

Then, the secant is parallel to the straight line. and are parallel to the axis of the parabola.

Are the three points of a parabola given, (not parallel to the axis of the parabola, and no tangent ) can be constructed with this property of parabolic point on this line by specifying a line through.

For proof: Since only cutting, joining and concurrency play a role, you can give the proof of the affine equivalent standard parabola. A short calculation shows that the straight line parallel to the line.

Note: This property of a parabola is an affine version of the 5-point degeneracy of the set of Pascal.

Tangent construction

An arbitrary parabola may be described in a suitable coordinate system by an equation.

Method 1

For a parable applies:

• Are three points of the parabola and

This property can be used for the construction of the tangent at the point.

For proof: Since only cutting, joining and parallelism plays a role, one can carry out the proof of the affine equivalent standard parabola. A short calculation shows that the line has the slope. This is the slope of the tangent at the point.

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

Method 2

A second way of constructing the tangent at a point based on the following properties of a parabola:

• If two points of the parabola and

For proof: Since only cutting, joining and concurrency play a role, you can give the proof of the affine equivalent standard parabola.

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

Axis direction construction

At the point of the tangent and construction structure ( see above), the axial direction of the parabola is assumed to be known. The following characteristic of a parabola allowed to construct the axis direction from a knowledge of two points whose tangents parabola.

An arbitrary parabola may be described in a suitable coordinate system by an equation.

For a parable applies: Are

• Two points of the parabola,
• The associated tangents,
• The intersection of the two tangents,
• The intersection of the parallels through the point parallel to the (s. picture)

Then the straight line parallel to the axis of the parabola and the equation

For proof: As in the previous parable properties can be expected by the proof of the standard parabola.

Note: The property described here is an affine version of the theorem on perspective triangles of a non-degenerate conic section.

Pole - polar relationship

A parabola can always be described by an equation of the form in a suitable coordinate system. The equation of the line tangent to a parabola point. The mixture is allowed to the right side of the equation, that any point in the plane is then

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 parabola are the points of contact of the tangents through the pole to the parabola.

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

To prove: The determination of the intersections of the polars of a point with the parabola and the search for parabolic points whose tangents contain the point, lead to the same quadratic equation.

Comment:

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

Orthogonal tangents

A parabola has the following property:

• Mutually orthogonal tangents intersect on the guideline.

Proof: A parabola can be described by an equation in suitable coordinates. The tangents at two Prabelpunktenpunkten have the equations

By multiplying the first equation and the second equation with x can be eliminated. Obtained first. It follows

For orthogonal tangents the product of their slopes must be -1, ie it is. Thus follows from the equation:

This is the equation of the guide line ( supra).

Parabolas of the form y = ax ² bx c

Peripheral angle theorem for parabolas

Parabolas 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 parabolas.

To measure an angle between two chords we carry two lines that are not parallel to the y- axis, an angular one:

Two lines are parallel if and so is the angle measure = 0.

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

Peripheral angle theorem: ( f parabolas )

( Proof by recalculation. Use can for which presuppose a direction that the points lie on a parabola. )

3 -point shape of a parabola

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

3 -point form: ( f parabolas )

Parabola in polar coordinates

A parable, which is described in Cartesian coordinates by, met in polar coordinates the equation

Your focal point is. If one of the coordinate origin in its focus, it applies to the polar equation

Graphical multiplication

A standard parabola is a " multiplication machine": One can calculate the product of two numbers with their graphical methods. Do this, draw first a normal parabola in a Cartesian coordinate system. The factors to be multiplied to wear off to the axis, and determines for each value of a point on the parabola. Are the numbers and called, so get two points. And by the straight line intersecting the axis at a point whose coordinate is equal. In the limiting case, there is the straight line tangent to the parabola.

If and have the same sign, it is more practical to apply one of the factors in the negative direction rather than later reverse the sign of the result, as happened in the example with the values ​​and. This contributes to the factors as values ​​with different signs in the coordinate system, ie, as and. If you connect the points by a straight line, it can be seen that the intersection of the straight line with the axis equal to 6 = 2.3.

Parabola and catenary

Chain lines resemble parables, but they are not. The cable of a suspension bridge that sags under its own weight, describes a catenary. These will not be described by a quadratic function, but by the hyperbolic cosine. Mathematically expressed, the similarity by the fact that the hyperbolic cosine in the series

Can be developed. The first two terms (red ) describe a parabola and can be used as an approximation of the cosh function for small x.

Parables as a quadratic Bezier curves

A quadratic Bezier curve is a curve whose parametric representation is determined by three points, and:

This curve is a parabola (see section: parabola as affine image of the standard parabola ).

Parables and numerical integration

In the numerical integration one approaches the value of a definite integral of the fact that one approaches the graph of the function to be integrated by parabolic arcs and integrates them. This leads to Simpson's rule, see Fig.

The quality of the approximation is increased, that increases the partition and the graph is replaced by a corresponding number of parabolic arcs and integrates them.

Parabolas as plane sections of quadrics

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

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

Parabolic cylinder

Elliptical paraboloid

A hyperbolic paraboloid

Single-leaf hyperboloid

Two-shell hyperboloid

Laguerre plane: Geometry of the parabolas

A Laguerre plane is in the classical case, an incidence structure, which essentially describes the geometry of curves, which are parabolas and straight lines in the real plane view. As the compound curves are not only straight lines but also parabolas available here. For example, there is a Laguerre - level to three dots having different x coordinates exactly such a compound curve.

Parable " higher order "

Under a parabola of order n is defined as the graph of the polynomial (in contrast to the graph of e-function or square root function, ...). A parable 3rd order is also called cubic parabola.

So only in the case n = 2 is a parabola of a higher order is an ordinary parabola.

Neilsche parabola

The Neilsche parabola or parabolic semikubische is an algebraic curve of 3rd order:

• Cartesian coordinates equation with a real parameter
• Explicitly:

It is not a parable in the usual sense.

Parabola y = x ² over an arbitrary number field

Looking in an affine plane over an arbitrary ( commutative ) body, the set of points which satisfies the parabolic equation, so many characteristics of the real standard parabola, "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 parabola in at most two points.
• Through each parabolic point there (next to the line) is exactly one line that the parabola has in common only the point, the tangent. One is called straight line without intersection Passante, one with two intersections secant.

Differences to the real case: