Parabola (mathematics)
A parabola is a specific type of curve classified as one of the four conic sections, which include hyperbolas, ellipses, and circles. It can be defined in several ways: as the intersection of a cone and a plane at a certain angle, as the locus of points equidistant from a fixed point (the focus) and a fixed line (the directrix), or through mathematical equations that describe its shape in a coordinate plane. Key components of a parabola include the vertex, axis of symmetry, focus, directrix, and latus rectum, each playing a vital role in its geometric properties.
Parabolas are significant in various physical contexts, such as the paths of objects influenced by gravity; a projectile that travels at escape velocity follows a parabolic trajectory. They are also important in optics and engineering, as the reflective properties of parabolas allow light rays to converge at a focus, making them ideal for telescope mirrors, satellite dishes, and headlights. Additionally, the shape of a rotating fluid can resemble a parabola, illustrating its prevalence in both natural and man-made systems. Understanding parabolas is essential for applications in science, technology, and mathematics.
Subject Terms
Parabola (mathematics)
A parabola is one of the four conic sections: hyperbola, parabola, ellipse, and circle. These are called conic because they are the result of cutting a double-cone with a plane.
Overview
There are five definitions of a parabola.
1) The intersection of a cone and a plane that meets the axis of the cone at the same angle as the cone makes with its axis.
2) The locus of all points whose distance p from a fixed point (the focus) equals their distance from a fixed line (the directrix).
3) The curve in the xy coordinate plane defined by 4py = x2 where p is as defined in 2, above.
4) The curve defined by the parametric equations x = 2pt and y = pt2 where p is as defined in 2, above.
5) The limiting case of an ellipse whose eccentricity ε approaches 1 (that is, whose foci are becoming infinitely far apart) or the limiting case of a hyperbola whose eccentricity approaches 1 (for an ellipse, 0 < ε < 1, for a hyperbola ε > 1, and for a parabola ε = 1).
Parts of the Parabola
The directrix and the focus are defined in definition 2. The line perpendicular to the directrix and running through the focus is the axis of symmetry. The point where the axis of symmetry meets the parabola is the vertex. The line through the focus perpendicular to the axis of symmetry is the latus rectum. The distance along the latus rectum from the focus to the directrix is 2p, and the distance along the axis from the focus to the vertex is p.
Uses of the Parabola
An object moving around a source of gravity (for example, the sun or the earth) will follow a path that is one of the conic sections. If the object is moving faster than the escape velocity of the gravity source, its path will be a hyperbola, and if it’s moving slower, its path will be an ellipse. If it is moving at exactly the escape velocity, its path will be a parabola. Similarly, a charged object moving around another charged object will follow a conic path.
When light rays parallel to the axis of a parabola meet the parabola, they are reflected to the focus. When an object is far away, light rays from that object can be considered parallel, and therefore a telescope mirror is shaped as a section of a parabola. Similarly, radio waves from a distant source are parallel, so a dish antenna is also shaped as a parabola.
Furthermore, light rays originating at the focus emerge as parallel rays. An automobile headlight, a flashlight, and a searchlight all have a light source at the focus of a parabola to produce a beam of parallel light rays. Similarly, a radar transmitter has a parabolic reflector with the radio wave source at the focus.
Generating a Parabola
If a fluid is rotated rapidly, its surface will take the shape of a parabola.
A telescope mirror is first ground to a section of a sphere. This is easy to do: If a glass mirror and a similar piece of glass are moved back and forth while one is being rotated, the only shape the two can assume are sections of a sphere. Once the proper spherical shape is achieved, a different stroke is used to deepen the mirror in the center until a parabolic shape is achieved.
Bibliography
Coxeter, H. S. M. Introduction to Geometry. New York: Wiley, 1989.
McKellar, Danica. Girls Get Curves. New York: Penguin, 2012.
Posamentier, Alfred S, and Robert L. Bannister. Geometry, Its Elements and Structure. Mineola, NY: Dover, 2014.
Stankowski, James F., ed. Geometry and Trigonometry. New York: Rosen, 2015.
Swokowski, Earl W., Jeffery A. Cole. Algebra and Trigonometry with Analytic Geometry. Belmont CA: Cengage, 2011.