Conic sections
Conic sections, commonly referred to as "conics," are a group of curves that arise from the intersection of a plane with a double cone. These curves include ellipses, hyperbolas, and parabolas, each defined by specific geometric properties. Conics were first systematically studied by Apollonius of Perga in ancient Greece, and they have since become foundational in various fields, including mathematics, physics, and engineering.
The classification of conic sections depends on the angle at which the intersecting plane meets the cone: an ellipse forms when the plane intersects one sheet of the cone without being perpendicular, a parabola when it is parallel to a generating line, and a hyperbola when it intersects both sheets. Each type of conic has unique characteristics and equations, which are derived based on their geometric definitions.
In modern applications, conic sections play crucial roles in fields such as astronomy—where celestial orbits are elliptical—and engineering, where they are utilized in designing optical devices, navigation systems, and even certain medical procedures. The study of conic sections continues to influence advancements in technology and science, illustrating their enduring significance across disciplines.
Conic sections
Summary: Conic sections have many interesting mathematical properties and real-world applications.
Conic sections, or simply “conics,” are the simplest plane curves other than straight lines. Students in the twenty-first century begin to study these curves in middle school. In coordinate geometry, they can be expressed as polynomials of degree 2 in two variables while straight lines are polynomials of degree 1 in two variables. Conic sections can further be divided into three types: ellipse, parabola, and hyperbola. Conics were named and systematically studied by Apollonius of Perga (262–190 b.c.e.). At that time, the study of conics was not merely to explore the intrinsic beauty of the curves but to develop useful tools necessary for applications to the solution of geometric problems. Today, the theory of conics has numerous applications in our daily lives including the designs of many machines, optical tools, telecommunication devices, and even the tracks of roller coasters.
Representations of Conics and Their Applications
One can generate a two-sheet circular cone by fixing a straight line as the axis of the cone in the space first. Choose a fixed point on it as the vertex of the cone. Rotating another straight line through the vertex that makes a fixed angle with the axis, we obtain the desired cone as the trace of the rotating line. Any straight line on the trace is called a “generating line” of the cone. Conic sections are obtained by intersecting the two-sheet cone with planes not passing through its vertex as shown in Figures 1A–C.
The three types of conic sections are generated according to the positions of the intersection.
Ellipse
When the intersecting plane cuts only one sheet of the cone and the intersection is a closed curve, an ellipse is created. A circle is obtained when the intersecting plane is perpendicular to the axis of the cone; an ellipse is obtained when the intersecting plane is not perpendicular to the axis of the cone. A circle, as such, can be considered as a particular case of an ellipse (Figure 1A). As illustrated in Figure 2, an ellipse is the collection of points in a plane that the sum of distances from two fixed points F1 andF2, the foci, to every point in the collection is constant.
On the coordinate plane, if the foci are located on the x-axis at the points (-c,0) and (c,0) and the constant distance between F1 andF2 is 2a, the equation of an ellipse can be derived as
Parabola
When the intersecting plane cuts only one sheet of the cone and is parallel to exactly one generating line of the cone, the intersection is a non-closed curve—a parabola (Figure 1B). A parabola is the collection of points in a plane that are equidistant from a fixed point f (called “focus”) and a fixed line (called “directrix”). The graph of the parabola is illustrated in Figure 3A. The graph is symmetric with respect to the line through the focus and perpendicular to the directrix. This line of symmetry is called the “axis” of the parabola. The intersection of the graph with the axis is called the “vertex” of the parabola. On the coordinate plane, if the vertex is located at the origin O, and the focus at the point (0,p), then its directrix will be on the line y=-p (Figure 3B), and the equation of the parabola can be derived as x2=4py.
Hyperbola
When the intersecting plane meets both sheets of the cone, the intersection is a hyperbola, which consists of two identical non-closed parts, each located in one of the two sheets of the cone (Figure 1C). A hyperbola is the collection of all points in a plane that the difference of distances from two fixed points F1 and F2, the foci, to every point in the collection is constant. The graph of a hyperbola is drawn as shown in Figure 4A.
On the coordinate plane, if the foci (-c,0) (c0) are located on the x-axis and the differences of distance is ¦±2a, then the equation of the hyperbola can be derived as
A Brief History of Conic Sections
Between 460 b.c.e. and 420 b.c.e., three famous geometry problems were posed by the ancient Greeks. These problems were (1) the trisection of an angle, (2) the squaring of the circle, and (3) the duplication of the cube. The last problem merely asks that given any cube of side length a, can one construct another cube with exactly twice the volume, 2a3. Hippocrates of Chios (circa 470–410 b.c.e.) had the idea of reducing that problem by finding two quantities x and y such that
Then, x2 = ay, y2 = ax, and xy = 2a2.
As such, x is the required solution for the problem. This solution is equivalent to solving simultaneously any two of the three equations (x2=ay , y2=2ax and xy=2a2) that represent parabolas in the first two and a hyperbola in the third. However, no explicit construction of the conic sections was given. Menaechmus (380–320 b.c.e.) is believed to be the first mathematician to work with conic sections systematically, which is theorized to have arisen because of curves traced out by sundials. At his time, the conic sections were formed by cutting a right circular cone with a plane perpendicular to a side.
The sections were named according to whether the vertex angle was acute, right, or obtuse (Figure 5). Menaechmus constructed conic sections that satisfied the required algebraic properties suggested by Hippocrates and thus obtained the points of intersection of these conic sections that would lead to the solution of the problem of the duplication of the cube.
The breakthrough in the study of conics by the ancient Greeks was attributed to Apollonius of Perga. His eight-volume masterpiece Conic Sections greatly extended the existing knowledge at the time (one of the eight books has been lost to history). Apollonius’ major contribution was to treat the conic sections as plane curves and use their intrinsic properties to characterize them. This method allowed conic sections to be analyzed in great detail by the ancient Greeks.
Abu Ali al-Hasan ibn al-Haytham studied optics using conic sections in the tenth and eleventh centuries. Omar Al-Khayyami (Omar Khayyam) authored Treatise on Demonstration of Problems of Algebra in the eleventh century. This work showed that all cubic equations could be classified using geometric solutions that involve conic sections. Later, in the seventeenth century, Gerard Desargues (1591–1662) and Blaise Pascal (1623–1662) connected the study of conic sections to developments from projective geometry. At the same time, René Descartes (1596–1650) and Pierre de Fermat (1601–1665) also connected it with the developments from coordinate geometry. Eventually, problems of conics in geometry could be reduced to problems in algebra.
Johan Kepler (1571–1630) revolutionized astronomy by introducing the notion of elliptical orbits. According to Isaac Newton’s later law of universal gravitation, the orbits of two massive objects that interact are conic sections.
If they are bound together, they will both trace out ellipses; if they move apart, they will both follow parabolic or hyperbolic trajectories.
The Applications of Conic Sections
Besides applications in astronomy, conics have many other applications.
In an ellipse, any light or radiation that begins at one focus will be reflected to the other focus (Figure 6 on following page). This property can be used in theater designs. In an elliptical theater, the speech from one focus can be heard clearly across the theater at the other focus by the audience. It can also be applied in lithotripsy, a medical procedure for treating kidney stones. The patient is placed in an elliptical tank of water, with the kidney stone fixed at one focus. High-energy shock waves emitted at the other focus can be directed to pulverize the stone. Also, elliptical gears can be used for many machine tools.
In a parabola, parallel light beams will converge to its focus (see Figure 7 on following page). Parabolic mirrors are used to converge light beams or heat radiations, and parabolic microphones are used to perform a similar function with sound waves.
In reverse, if a light source is placed at the focus of a parabolic mirror, the light will be reflected in rays parallel to said axis. This property is used in the design of car headlights and in spotlights because it aids in concentrating the parallel light beam. Hyperbolas are used in a navigation system known as Long Range Navigation (LORAN). Hyperbolic—as well as parabolic—mirrors and lenses are also used in systems of telescopes.
Bibliography
Akopyan, A. V., and A. A. Zaslavsky. Geometry of Conics. Providence, RI: American Mathematical Society, 2007.
Courant, R., and H. Robbins. What Is Mathematics? New York: Oxford University Press, 1996.
Downs, J. W. Practical Conic Sections: The Geometric Properties of Ellipses, Parabolas and Hyperbolas. Mineola, NY: Dover, 2003.
Kendig, K. Conics (Dolciani Mathematical Expositions). Washington, DC: The Mathematical Association of America, 2005.
Kline, M. Mathematical Thought From Ancient to Modern Times. New York: Oxford University Press, 1972.
Suzuki, Jeff. A History of Mathematics. Upper Saddle River, NJ: Prentice Hall, 2002.