A conic section (Latin: sectio conica, conic section english ) is a curve produced when cutting the surface of a double cone with a plane. Contains the section plane the cone tip, the result is as cut either a point or a line or a pair of lines to be cut. If the tip is not included, the result is an ellipse, parabola, or hyperbola.
Evidence that really caused these curves defined as loci in the plane, can be run without an invoice using the Dandelinschen balls. The mathematical proof is given here to the section plane cuts the cone unit.
A conic section can also be viewed as two-dimensional special case of a quadric and are described by an equation of second degree, the general conic equation.
Equations of conic sections
The conics can be described in a suitable xy coordinate system by 2nd degree equations:
- Ellipse having the center M in the point (0,0) and the main axis on the x -axis:
- Parabola with vertex in the point (0,0 ) and the axis on the y -axis:
- Hyperbole with midpoint M at the point ( 0,0) and the main axis on the x -axis:
- Yourself tapping Just married couple with intersection at point (0,0):
- Straight line through the point (0,0):
- Point, the point (0,0):
The sake of completeness, two other cases taken which will not act as a real conic sections, but also be described by equations of second degree:
- Parallel pair of straight lines:
- The empty set:
In the last two cases can occur as a planar sections of a straight circular cylinder. A circular cylinder can be regarded as the limiting case of a cone with apex at " infinity ". Therefore, we take these two cases with the conic sections.
Plane sections of the cone unit
To determine that the known as conic sections above curves / points actually occur at the intersection of a cone with a plane, here we cut the unit cone ( right circular cone ) with a plane that is parallel to the y- axis. This is not a limitation, as the cone is rotationally symmetrical. Any right circular cone is the affine image of the unit cone and ellipses / hyperbolas / parabolas / ... go with an affine mapping again in just such.
Given: level cone.
- Case I: In this case the plane is vertically and. After eliminating from the taper equation, we obtain. Case Ia: . In this case, the cutting is composed of the pair of lines.
- Case Ib: . Now, the above equation describes a hyperbola in the yz plane. So, the cutting curve itself is a hyperbola.
- Case IIa: For the plane passes through the apex and equation (1) now has the form.
- Case IIb: For not the plane passes through the apex and is not perpendicular.
Parametric representations of curves of intersection can be found in Weblink CDKG, pp. 106-107.
If the cutting plane does not contain the apex incurred by non-degenerate conic sections (see image to Ib, IIb ):
- An ellipse, the angle of inclination of the cutting plane ( with respect to the xy plane ) is smaller than the angle of inclination of the generatrices of the cone. If the plane is horizontal, the intersection curve is a circle.
- A parabola, when the inclination angle of the cutting plane ( with respect to the xy plane) is equal to the angle of inclination of the generatrices of the cone.
- A hyperbola, when the inclination angle of the cutting plane ( with respect to the xy plane ) is greater than the angle of inclination of the generatrices of the cone.
If the cutting plane passes through the apex, the degenerate conic sections (see image to Ia, IIa ) arise:
- A point when the cutting plane of the cone intersects only the top.
- A straight line, when the interface layer in contact with the cone along a generating line ( the cutting plane is tangential ).
- To be cut straight couple when the cutting plane contains two generating lines.
General conic equation
The general equation for conics is
The parameters a, b, c are not 0 all at the particular case a = b = c = 0, the equation describing a straight line or completely.
It will now be demonstrated that only the above 8 cases occur as sets of solutions of the general conic equation. We achieve our goal in two main steps, the principal axes transformation:
Step 1: If we perform the rotation
The conic equation then has the form
After these two steps, the conic section equation is (x 'and y' are again replaced by x, y) finally the mold
Only the above 8 cases occur:
In the transformations performed here ( rotation, translation ), the geometric form of the conic section described by the original equation is not changed. Parameters such as semi-axes for ellipses and hyperbola or focal length of the parabola or angle / distance between intersecting / parallel straight lines can be seen in the transformed conic.
Note: The square portion of the general conic equation can also be written using a 2x2 matrix:
As a rotation and a shift does not change the sign of the determinant of the 2x2 matrix, leading to the case of I: and the case II. Do we know that the original conic equation is a non-degenerate conic section, you can tell by the determinant of whether it is an ellipse () or a hyperbola () or a parabola ().
- Since the general conic equation is determined only up to a factor of 6 by the coefficient, 5 points ( equations) are needed for the determination of the coefficients. However, not every choice of five points determine a conic clearly. ( Counter-example:. 4 points on a straight line, one point not on the line) A nondegenerate conic ( ellipse, hyperbola, parabola) is by 5 points, with no three are collinear, uniquely determined. An elegant formula for the non-degenerate case uses a 6x6 determinant:
- A circle is already by 3 points (not on a straight line) is uniquely determined. The equation is obtained by the 4x4 determinant
For example, the conic through the five points (1,0 ), ( -1,0), (0,1 ), ( -1,0), (1,1) has claimed calculate the above determinant equation or after simplification: . The principal axis transformation is done to a turn. A shift is not necessary. Of the conical section is in the transformed equation and an ellipse.
Vertex equation of a conic - band
The crowd of non-degenerate conic sections whose axis is the x -axis and the point ( 0,0) have a vertex, can be represented by the equation
Describe (for proof see Guidance property of the hyperbola ). for
Is the numerical eccentricity.
Equivalence nondegenerate conic sections
- All ellipses are affine images of the unit circle (see ellipse),
- All parabolas are affine images of the standard parabola (see parabola) and
- All hyperbolas are affine images of Einheitshyperbel (see hyperbola ).
An ellipse is mapped but with an affine transformation does not (for example ) on a parabola. But it complemented the affine coordinate plane to a projective plane and adds a parabola the farthest point of its axis are added, can be an ellipse with a projective mapping to a so extended parabolic map. The same applies analogously for as supplemented by the two remote points of its asymptote hyperbola.
- From the projective point of view, ie all non-degenerate projective conics are equivalent to each other (see also Weblink CDKG, p 251).
Applications and Examples
An application can find the conic sections in astronomy, since the orbits of celestial bodies are approximated conic sections.
Also in the optics used - as a spheroid for car headlights, a paraboloid or hyperboloid for reflecting telescopes, etc.
In the Descriptive Geometry conics occur as images of circles in parallel and central projections. See ellipse ( Descriptive Geometry ).
The Greek mathematician Menaechmus examined at Plato's Academy, the conic sections using a cone model. He found out here that the Delian problem can be traced to the determination of the intersection point of two conics. Euclid wrote four books on conics which are but we do not receive. The entire knowledge of the ancient mathematicians about the conic sections summed Apollonius of Perga together in his eight -volume work Konika. The description of conic sections by coordinates equations was introduced by Fermat and Descartes.
Conics on any number bodies
Conic sections can be defined also over arbitrary fields. There remain surprisingly received many Incidence and symmetry properties. See Weblink Projective geometry, projective conic and conic sections over finite fields the article Quadratic amount.
Conic sections and Benz - levels
Conics play in the Benz - levels, which are Möbius planes ( geometry of circles), Laguerre planes ( geometry of parabolas ) and Minkowski planes ( geometry of hyperbolas ), an important role.