Parabola

Drawing parabolas

Uses

Resources

A parabola is a type of conic section, which is an open curve formed by the intersection of a plane and a right circular cone. The word parabola is derived from the Greek word parabole, which means application or comparison. Italian astronomer and physicist Galileo Galilei (1564–1642) worked out parabolic trajectories of projectiles in the early part of the seventeenth century. English physicist and mathematician Sir Isaac Newton (1642–1727) later proved mathematically the parabolic shape of these projectiles.

A parabola occurs when the plane is parallel to one of the generatrices (geometric templates) of the cone (Figure 1).

A parabola can also be defined as the set of points which are equidistant from a fixed point (the focus) and a fixed line (the directrix) (Figure 2).

A third definition is the set of points (x,y) on the coordinate plane, which satisfy an equation of the form y = x², or, more usefully, 4ky = x². Other forms of equation are possible, but these are the simplest.

The axis of a parabola is the line which passes through the focus and is perpendicular to the directrix. The vertex is the point where the axis crosses the parabola. The latus rectum is the chord passing through the focus and perpendicular to the axis. Its length is four times the distance from the focus to the vertex.

When a parabola is described by the equation 4ky = x², the vertex is at the origin; the focus is at (o,k); the axis is the y-axis; the directrix is the line y = -k.

In spite of the infinitude of cones—from skinny ones to fat ones—that yield parabolas, all parabolas are geometrically similar. If one has two parabolas, one of them can always be enlarged, as with a photographic enlarger, so that it exactly matches the other. This can be shown algebraically with an example. If y = x² and y = 3x² are two parabolas, the transformation x = 3xy = 3y which enlarges a figure to three times its original size, transforms y = x² into 3y = (3x)², which can be simplified to y = 3x².

This reflects the fact that all parabolas have the same eccentricity, namely 1. The eccentricity of a conic section is the ratio of the distances point-to-focus divided by point-to-directrix, which is the same for all the points on the conic section. Since, for a parabola, these two distances are always equal, their ratio is always 1.

A parabola can be thought of as a kind of limiting shape for an ellipse, as its eccentricity approaches 1. Many of the properties of ellipses are shared, with

slight modifications, by parabolas. One such property is the way in which a line intersects it. In the case of an ellipse, any line that intersects it and is not simply tangent to it, intersects it in two points. So, surprisingly, does a line intersecting a parabola, with one exception. A line that is parallel to the parabola’s axis will intersect in a single point, but if it misses being parallel by any amount, however small, it will intersect the parabola a second time. The parabola continues to widen as it leaves the vertex, but it does so in this curious way.

A parabola’s shape is responsible for another curious property. If one draws a tangent to a parabola at any point P, a line FP from the focus to P and a line XP parallel to the axis, will make equal angles with the tangent. In Figure 3, – FPA = – XPB. This means that a ray of light parallel to the axis of a parabola would be reflected (if the parabola were reflective) through the focus, or a ray of light, originating at the focus, would be reflected along a line parallel to the axis.

A parabola, being an open curve, does not enclose an area. If one draws a chord between two points on the parabola, however, the parabolic segment formed does have an area. Consequently, this area is given by a remarkable formula discovered by ancient Greek mathematician Archimedes (287–212 BC) in the third century BC In Figure 4, M is the midpoint of the chord AB. C is the point where a line through M and parallel to the axis intersects the parabola. The area of the parabolic segment is 4/3 times the area of triangle ABC. For example, the area of the parabola y = x²and the line y = 9 is (4/3)(6× 9/2) or 36. What is particularly remarkable about this formula is that it does not involve the number π as the formulas for the areas of circles and ellipses do.

Drawing parabolas

Unlike ellipses, parabolas do not lend themselves to simple mechanical drawing aids. The ones occasionally described in texts work crudely. Templates are hard to find. The two best methods for drawing parabolas both involve locating points on the parabola and connecting those points either by the unaided eye, or with the help of a draftsman’s French curve.

The equation of 4ky = x² or y = x²/4k can be used to plot points on graph paper. The parameter K, which represents the distance from the focus to the vertex, should be chosen to make the parabola appropriately sharp or broad. A table of ordered pairs (x,y) will help in point plotting. Enough points should be plotted, especially near the vertex where the curvature of the parabola changes most rapidly, that a smooth, accurate curve can be sketched.

Uses

Parabolas show up in a variety of places. The path of a bomb dropped from an airplane is a section of a parabola. The cables of a well-designed suspension bridge follow a parabolic curve. The surface of the

water in a bowl that is rotating on a turntable will assume the shape of a parabola rotated around its axis. The area of a circle is a parabolic function of its radius. In fact, the graphs of all polynomial functions y=ax² + bx + c, of degree two are parabolic in shape.

Perhaps the most interesting application of a parabola is in the design of mirrors for astronomical telescopes. The rays of light from a star, a galaxy, or even such a nearby celestial object as a planet are essentially parallel. The reflective property of a parabola sends a ray that is parallel to the parabola’s axis through the focus. Therefore, if one grinds a mirror with its surface in the shape of a parabola rotated around

KEY TERMS

Directrix —The fixed line in the focus directrix definition of a conic section.

Focus —A point, or one of a pair of points, whose position determines the shape of a conic section.

Parabola —A set of points that are equidistant from a fixed point and a fixed line.

its axis and if one tilts such a mirror so that its axis points at a star, all the light from that star which strikes the mirror will be concentrated at the mirror’s focus.

Of course, such a mirror can be pointed at only one star at a time. Even so, the mirror will reflect rays from nearby stars through their own foci, which are near the real focus. It will bring into focus not only the one star at which it is pointed, but also the stars in the area around the star.

The process can be reversed. If the light source is placed at the focus, instead of concentrating the rays, the reflector will act to send them out in a bundle, parallel to the axis of the reflector. The large searchlights used during air raids in World War II (1939– 1945) were designed with parabolic reflectors.

Parabolic reflectors are used in other devices as well. Radar antennas, the dishes used to pick up satellitetelevision signals, and the reflectors used to concentrate sound from distant sources are all parabolic.

Resources

BOOKS

Ball, W.W. Rouse. A Short Account of the History of Mathematics. London: Sterling Publications, 2002.

Henle, Michael. Modern Geometries: Non-Euclidean, Projective, and Discrete. Upper Saddle River, NJ: Prentice Hall, 2001.

Hilbert, D. and S. Cohn-Vossen. Geometry and the Imagination. New York: Chelsea Publishing Co. 1952.

Noronha, Maria Helena. Euclidean and Non-Euclidean Geometries. Upper Saddle River, NJ: Prentice Hall, 2002.

Silvester, John R. Geometry: Ancient and Modern. Oxford, UK, and New York: Oxford University Press, 2001.

Slavin, Stephen L. Geometry: A Self-teaching Guide. Hoboken, NJ: John Wiley & Sons, 2005.

Zwikker, C. The Advanced Geometry of Plane Curves and Their Applications. New York: Dover Publications, Inc., 1963.

J. Paul Moulton

Parabola

A parabola is the open curve formed by the intersection of a plane and a right circular cone. It occurs when the plane is parallel to one of the generatrices of the cone (Figure 1).

A parabola can also be defined as the set of points which are equidistant from a fixed point (the "focus") and a fixed line (the "directrix") (Figure 2).

A third definition is the set of points (x,y) on the coordinate plane which satisfy an equation of the form y = x2, or, more usefully, 4ky = x2. Other forms of equation are possible, but these are the simplest.

The "axis" of a parabola is the line which passes through the focus and is perpendicular to the directrix. The "vertex" is the point where the axis crosses the parabola. The "latus rectum" is the chord passing through the focus and perpendicular to the axis. Its length is four times the distance from the focus to the vertex.

When a parabola is described by the equation 4ky = x2, the vertex is at the origin; the focus is at (o,k); the axis is the y-axis; the directrix is the line y = -k.

In spite of the infinitude of cones—from skinny ones to fat ones—that yield parabolas, all parabolas are geometrically similar. If one has two parabolas, one of them can always be enlarged, as with a photographic enlarger, so that it exactly matches the other. This can be shown algebraically with an example. If y = x2 and y = 3x2 are two parabolas, the transformation x = 3xy = 3y which enlarges a figure to three times its original size, transforms y = x2 into 3y = (3x)2, which can be simplified to y = 3x2.

This reflects the fact that all parabolas have the same eccentricity, namely 1. The eccentricity of a conic section is the ratio of the distances point-to-focus divided by point-to-directrix, which is the same for all the points on the conic section. Since, for a parabola, these two distances are always equal, their ratio is always 1.

A parabola can be thought of as a kind of limiting shape for an ellipse , as its eccentricity approaches 1. Many of the properties of ellipses are shared, with slight modifications, by parabolas. One such property is the way in which a line intersects it. In the case of an ellipse, any line which intersects it and is not simply tangent to it, intersects it in two points. So, surprisingly, does a line intersecting a parabola, with one exception. A line which is parallel to the parabola's axis will intersect in a single point, but if it misses being parallel by any amount, however small, it will intersect the parabola a second time. The parabola continues to widen as it leaves the vertex, but it does so in this curious way.

A parabola's shape is responsible for another curious property. If one draws a tangent to a parabola at any point P, a line FP from the focus to P and a line XP parallel to the axis, will make equal angles with the tangent. In Figure 3, - FPA =- XPB. This means that a ray of light parallel to the axis of a parabola would be reflected (if the parabola were reflective) through the focus, or a ray of light, originating at the focus, would be reflected along a line parallel to the axis.

A parabola, being an open curve, does not enclose an area. If one draws a chord between two points on the parabola, however, the parabolic segment formed does have an area, and this area is given by a remarkable formula discovered by Archimedes in the third century b.c. In Figure 4, M is the midpoint of the chord AB. C is the point where a line through M and parallel to the axis intersects the parabola. The area of the parabolic segment is 4/3 times the area of triangle ABC. For example, the area of the parabola y = x2 and the line y = 9 is (4/3)(6 × 9/2) or 36. What is particularly remarkable about this formula is that it does not involve the number π as the formulas for the areas of circles and ellipses do.

Drawing parabolas

Unlike ellipses, parabolas do not lend themselves to simple mechanical drawing aids. The ones occasionally described in texts work crudely. Templates are hard to find. The two best methods for drawing parabolas both involve locating points on the parabola and connecting those points either by eye, or with the help of a draftsman's french curve.

The equation of 4ky = x2 or y = x2/4k can be used to plot points on graph paper . The parameter K, which represents the distance from the focus to the vertex, should be chosen to make the parabola appropriately "sharp" or broad. A table of ordered paris (x,y) will help in point plotting. Enough points should be plotted, especially near the vertex where the curvature of the parabola changes most rapidly, that a smooth, accurate curve can be sketched.

Uses

Parabolas show up in a variety of places. The path of a bomb dropped from an airplane is a section of a parabola. The cables of a well-designed suspension bridge follow a parabolic curve. The surface of the water in a bowl that is rotating on a turntable will assume the shape of a parabola rotated around its axis. The area of a circle is a parabolic function of its radius. In fact, the graphs of all polynomial functions y = ax2 +bx + c, of degree two are parabolic in shape.

Perhaps the most interesting application of a parabola is in the design of mirrors for astronomical telescopes. The rays of light from a star , a galaxy , or even such a nearby celestial object as a planet are essentially parallel. The reflective property of a parabola sends a ray that is parallel to the parabola's axis through the focus. Therefore, if one grinds a mirror with its surface in the shape of a parabola rotated around its axis and if one tilts such a mirror so that its axis points at a star, all the light from that star which strikes the mirror will be concentrated at the mirror's focus.

Of course, such a mirror can be pointed at only one star at a time. Even so, the mirror will reflect rays from nearby stars through their own "foci" which are near the real focus. It will bring into focus not only the one star at which it is pointed, but also the stars in the area around the star.

The process can be reversed. If the light source is placed at the focus, instead of concentrating the rays, the

reflector will act to send them out in a bundle, parallel to the axis of the reflector. The large searchlights used during air raids in World War II were designed with parabolic reflectors.

Parabolic reflectors are used in other devices as well. Radar antennas, the "dishes" used to pick up satellite television signals, and the reflectors used to concentrate sound from distant sources are all parabolic.

Resources

books

Ball, W.W. Rouse. A Short Account of the History of Mathematics. London: Sterling Publications, 2002.

Finney, Thomas, Demana, and Waits. Calculus: Graphical, Numerical, Algebraic. Reading, MA: Addison Wesley Publishing Co., 1994.

Hilbert, D., and S. Cohn-Vossen. Geometry and the Imagination. New York: Chelsea Publishing Co. 1952.

Zwikker, C. The Advanced Geometry of Plane Curves and Their Applications. New York: Dover Publications, Inc., 1963.

J. Paul Moulton

KEY TERMS

Directrix

—The fixed line in the focus directrix definition of a conic section.

Focus

—A point, or one of a pair of points, whose position determines the shape of a conic section.

Parabola

—A set of points which are equidistant from a fixed point and a fixed line.

pa·rab·o·la / pəˈrabələ/ • n. (pl. -las or -lae / -lē/ ) a symmetrical open plane curve formed by the intersection of a cone with a plane parallel to its side. The path of a projectile under the influence of gravity ideally follows a curve of this shape.

Parabola (Journal)

Journal of the Society for the Study of Myth & Tradition, concerned with exploring the inner being through myth and its manifestations. Published quarterly. Address: 656 Broadway, New York, NY 10012. Website: http://www.parabola.org/.

Sources:

Parabola Online. http://www.parabola.org/. March 8, 2000.

parabola Mathematical curve, a conic section traced by a point that moves so that its distance from a fixed point, the focus, is equal to its distance from a fixed straight line, the directrix. It may be formed by cutting a cone parallel to one side. The general equation of a parabola is y = ax² + bx + c, where a, b, and c are constants.