Encyclopedia
- For the song by Tool, see Parabola .
The
parabola is a
conic section generated by the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. A parabola can also be defined as locus of points which are equi
distant from a given point and a given line .
A particular case arises when the plane is tangent to the conical surface. In that case the intersection is a degenerate parabola consisting of a straight line.
Definitions and overview
Analytic geometry equations
In
Cartesian coordinates, a parabola with an axis parallel to the
y axis with vertex , focus , and directrix
y =
k -
p, with
p being the distance from the vertex to the focus, has the equation
or, alternatively
More generally, a parabola is a curve in the
Cartesian plane defined by an irreducible equation of the form
such that , where all of the coefficients are real, where A and/or C is non-zero, and where more than one solution, defining a pair of points on the parabola, exists. That the equation is irreducible means it does not factor as a product of two not necessarily distinct linear factors.
Other geometric definitions
A parabola may also be characterized as a conic section with an eccentricity of 1. As a consequence of this, all parabolas are similar. A parabola can also be obtained as the limit of a sequence of
ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction. In this sense, a parabola may be considered an ellipse that has one focus at
infinity. The parabola is an
inverse transform of a
cardioid.
A parabola has a single axis of reflective
symmetry, which passes through its focus and is perpendicular to its directrix. The point of intersection of this axis and the parabola is called the vertex. A parabola spun about this axis in three dimensions traces out a shape known as a
paraboloid of revolution.
The parabola is found in numerous situations in the physical world .
Equations
Cartesian
Vertical axis of symmetry
- .
Horizontal axis of symmetry
- .
Semi-latus rectum and polar coordinates
In polar coordinates, a parabola with the focus at the origin and the top on the negative
x-axis, is given by the equation
-
where
l is the
semi-latus rectum: the distance from the focus to the parabola itself, measured along a line perpendicular to the axis. Note that this is twice the distance from the focus to the apex of the parabola or the perpendicular distance from the focus to the latus rectum.
Gauss-mapped form
A Gauss-mapped form:
has normal.
See also
Derivation of the focus
Given a parabola parallel to the
y-axis with vertex and with equation
then there is a point — the focus — such that any point
P on the parabola will be equidistant from both the focus and a line perpendicular to the axis of symmetry of the parabola , in this case parallel to the
x axis. Since the vertex is one of the possible points P, it follows that the linea directrix passes through the point . So for any point
P=, it will be equidistant from and . It is desired to find the value of
f which has this property.
Let
F denote the focus, and let
Q denote the point at . Line
FP has the same length as line
QP.
Square both sides,
Cancel out terms from both sides,
Cancel out the
x2 from both sides ,
Now let
p=f and the equation for the parabola becomes
Q.E.D.
Reflective property of the tangent
The tangent of the parabola described by equation has slope
This line intersects the
y-axis at the point = , and the
x-axis at the point . Let this point be called
G. Point
G is also the midpoint of points
F and
Q:
Since
G is the midpoint of line
FQ, this means that
and it is already known that
P is equidistant from both
F and
Q:
and, thirdly, line
GP is equal to itself, therefore:
It follows that
.
Line
QP can be extended beyond
P to some point
T, and line
GP can be extended beyond
P to some point
R. Then and are vertical, so they are equal . But is equal to . Therefore is equal to .
The line
RG is tangent to the parabola at
P, so any light beam bouncing off point
P will behave as if line
RG were a mirror and it were bouncing off that mirror.
Let a light beam travel down the vertical line
TP and bounce off from
P. The beam's angle of inclination from the mirror is , so when it bounces off, its angle of inclination must be equal to . But has been shown to be equal to . Therefore the beam bounces off along the line
FP: directly towards the focus.
Conclusion: Any light beam moving vertically downwards in the concavity of the parabola will bounce off the parabola moving directly towards the focus.
Parabolae in the physical world
In nature, approximations of parabolae and paraboloids are found in many diverse situations. The most well-known instance of the parabola in the history of
physics is the trajectory of a particle or body in motion under the influence of a uniform
gravitational field without
air resistance . The parabolic trajectory of projectiles was discovered experimentally by
Galileo in the early 17th century, who performed experiments with balls rolling on inclined planes. The parabolic shape for projectiles was later proven
mathematically by
Isaac Newton. For objects extended in space, such as a diver jumping from a diving board, the object itself follows a complex motion as it rotates, but the
center of mass of the object nevertheless forms a parabola. As in all cases in the physical world, the trajectory is always an approximation of a parabola. The presence of air resistance always distorts the shape, for example, although at low speeds, the shape is a good approximation of a parabola. At higher speeds, such as in ballistics, the shape is highly distorted and does not resemble a parabola.
Another situation in which parabola may arise in nature is in two-body orbits, for example, of a small planetoid or other object under the influence of the gravitation of the sun. Such parabolic orbits are a special case that are rarely found in nature. Orbits that form a
hyperbola or an
ellipse are much more common. In fact, the parabolic orbit is the borderline case between those two types of orbit.
Approximations of parabolas are also found in the shape of cables of
suspension bridges. Freely hanging cables do not describe parabolas, but rather
catenary curves. Under the influence of a uniform load , however, the cable is deformed towards a parabola.
Paraboloids arise in several physical situations as well. The most well-known instance is the
parabolic reflector, which is a mirror or similar reflective device that concentrates light or other forms of
electromagnetic radiation to a common focal point. The principle of the parabolic reflector was discovered in the 3rd century BC by the geometer
Archimedes, who, according to a legend of debatable veracity, constructed parabolic mirrors to defend
Syracuse against the
Roman fleet, by concentrating the sun's rays to set fire to the decks of the Roman ships. The principle was applied to
telescopes in the 17th century. Today, paraboloid reflectors can be commonly observed throughout much of the world in
microwave and satellite dish antennas.
Paraboloids are also observed in the surface of a liquid confined to a container and rotated around the central axis. In this case, the
centrifugal force causes the liquid to climb the walls of the container, forming a parabolic surface. This is the principle behind the liquid mirror telescope.
See also
References
External links
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