Parabola

See also: Parabola (homonymy)

The parabola is the intersection of a plan with a cone when the plan is parallel with the one of the Génératrice S of the cone. It is a type of Courbe whose many geometrical properties interested the Mathématicien S as of the Antiquité and received varied technical applications.

Mathematics

Conic section

The parabolas belong to the family of the Conique S, i.e. curves which are obtained by the intersection of a cone of revolution with a plan; in fact, the parabola is obtained when the plan is parallel to the one of the generators of the cone.

Director, hearth and eccentricity

Are $D$ a line and $F$ a point not belonging to $D$, and is P the plan containing the line $D$ and the point $F$). One calls parabola of right director $D$ and hearth $F$ the whole of the points $M$ of the plan P checking:

$\backslash \; frac\; \{D\; (M,\; F)\}\{D\; (M,\; D)\}\; =\; 1$

where $d\; (M,\; F)$ measurement the Distance of the point M at the point F and $d\; (M,\; D)$ measures the Distance point M with the right-hand side D . It is thus conical whose eccentricity E is worth 1

Equations

Starting from the hearth and of the director

If the parabola is given by its hearth F and its director $\backslash \; mathcal\; D$, one calls O the projected orthogonal one of F on $\backslash \; mathcal\; D$, one calls OFF p (parameter of the parabola) the distance and one calls S the medium of ''. Then, in the reference mark orthonormé $(S,\; \backslash \; vec\; I,\; \backslash \; vec\; J)$ where $\backslash \; vec\; j$ has even direction and direction that $\backslash \; overrightarrow\; \{OFF\}$, the equation of the parabola is:

$y\; =\; \backslash \; frac\; \{x^2\}\; \{2p\}$

Starting from the function of the second degree

See also: Function of the second degree

The curve representative of a Function polynomial of the second degree of equation

there = ax ² + bx + C

where has , B and C is real constants (not no one has) is a parabola. In the case has = 1 , B = 0 , and C = 0 one obtains a simple expression for a parabola: y  =  X ².

The top S of a parabola is the point of coordinates $\backslash \; left\; (-\; \backslash \; tfrac\; \{B\}\; \{2a\};\; -\; \backslash \; tfrac\; \{b^2\; -\; 4ac\}\; \{4a\}\; \backslash \; right)$. Its axis of symmetry is the axis $(S\; \backslash \; vec\; J)$. In the reference mark $(S,\; \backslash \; vec\; I,\; \backslash \; vec\; J)$, its equation is

$Y\; =\; aX^2$

Its hearth is the point $F\; \backslash \; left\; (0;\; \backslash \; tfrac\; \{1\}\; \{4a\}\; \backslash \; right)$ and its director is the line $\backslash \; mathcal\; D$ of equation $Y\; =\; -\; \backslash \; frac\; \{1\}\; \{4a\}$

Starting from the general equation

That is to say the equation $Ax^2\; +\; 2Bxy\; +\; Cy^2\; +\; 2Dx\; +\; 2Ey\; +F\; =\; 0$, in an orthonormal reference mark. If $B^2\; -\; AC\; =\; 0$ then this equation is that of a parabola or two parallel straight lines.

That is to say the equation $Ax^2\; +\; Cy^2\; +\; 2Dx\; +\; 2Ey\; +\; F\; =\; 0$, in an orthonormal reference mark. If AC = 0 with AE or cd. not no one then this equation is that of a parabola.

Lastly, in any orthonormal reference mark, the equation of a parabola is form

$Ax^2\; +\; 2Bxy\; +\; Cy^2\; +\; 2Dx+2Ey+F=\; 0$ with $B^2\; -\; AC\; =\; 0$.

Parameterization

In the reference mark $(O,\; \backslash \; vec\; I,\; \backslash \; vec\; J)$ where O is the point located in the middle of the segment made up of the hearth F and its projection H on the director and where $\backslash \; vec\; i$ is a Unit vector directed O towards F, one can consider several parameterizations parabola:

1. a Cartesian parameterization by the ordinate: $\backslash \; overrightarrow\; \{COp\}\; (there)\; =\; \backslash \; frac\; \{y^2\}\; \{2p\}\; \backslash \; overrightarrow\; \{I\}\; +y\; \backslash \; overrightarrow\; \{J\}$.
2. a Cartesian parameterization by the X-coordinate:

$\backslash \; overrightarrow\; \{COp\}\; (X)\; =x\; \backslash \; overrightarrow\; \{I\}\; +\; \backslash \; sqrt\; \{2px\}\; \backslash \; overrightarrow\; \{J\}$.

1. parameterization: $\backslash \; begin\; \{boxes\}$

x= \ frac 12 pt^2 \ \ y=pt \ end {boxes} , for all $t\; \backslash \; in\; \backslash \; R$ This parameterization is regular (i.e the derived vector is not cancelled). The vector $(T,\; 1)$ then directs the tangent at the point of parameter $t$.

Some geometrical properties of the parabola

Parallel cords

All the parallel cords have their medium located on a line perpendicular to the director. The tangent parallel with this direction has its contact point on this line. The two tangents with the parabola at the ends of such a cord are cut on this line.

Property relative to the orthoptic one

Are $M$ and $M\text{'}$ the points of intersection an unspecified line passing by the hearth of the parabola with the parabola. The two tangents of the parabola passing by $M$ and $M\text{'}$ are cut on the director by forming a right angle between them. Moreover, by calling $H$ and $H\text{'}$ the projected respective ones of $M$ and $M\text{'}$ on the director and $O$ the point of intersection of the two tangents and the director, one has that $O$ and the medium of $$.

While moving along its director, the parabola is always seen under a right angle.

Applications

One uses the parabolas to concentrate waves, or rays in a point, the hearth of the parabola. The parabolas are also used to concentrate the solar rays in a point. For example, one can make pass from water in a pipe which passes by the hearth of a solar concentrator, this water goes up then very quickly in temperature, even vaporizes. Who says vaporization, known as increase in pressure. One can then use this pressure to make turn an alternator to produce electricity.

Physics

The parabola is the trajectory described by an object which one launches if one can neglect the curve of the Ground, the friction of the air (wind, deceleration of the object) and the variation of the Gravité with the height.

The mechanical energy for an object describing a parabola is always null.

Hertzian waves

By Métonymie, a parabola indicates a parabolic Aerial. It is more exactly about an application of the properties of named surface paraboloid of revolution.

See too

Related articles

• Conical S
• Hyperbole
• Ellipse (mathematics)

External bonds

• Course of geometry of Mr. Gerhard Wanner of the university of Geneva, section of mathematics

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