Conic Sections: Types, Applications, Examples

The conic sections are the curves obtained by intercepting a plane with a cone. There are several ways to do this; for example, if the plane is made to pass perpendicular to the axial axis of the cone, a circumference is obtained.

By tilting the plane a little with respect to the axial axis of the cone, an ellipse is obtained, a curve that is closed, but if we incline it even more, a parabola or a hyperbola is obtained, as can be seen in the animation in figure 1.

Conic sections are part of nature and the world around us. Engineering, architecture, and astronomy are important branches of knowledge that make use of conics.

Conditions for conic sections

Conic sections are defined as loci that satisfy the following conditions:

Parable

It is the locus of all the points that lie in a plane equidistant from a fixed point called the focus F and a fixed straight line, called the directrix .

Ellipse

A point on the plane belongs to an ellipse if the sum of the distances between that point and two other fixed points, called foci and located on the ellipse’s major axis , remains constant.

Circumference

It is the locus of all the points that maintain the same distance to another point called the center. This distance is the radius of the circumference.

Hyperbola

Set of points in the plane such that the difference between their distance to two fixed points called foci is constant.

Applications

Let’s see some of the applications of conic sections:

Parables  

-When an object is thrown, the trajectory that follows is shaped like a parabola.

-The parabolas have notable applications in Engineering, for example in suspended bridges the cables hang in the form of parabolas.

-The parabolas are also good for making reflectors and telescopes. This is thanks to an interesting property: when placing a luminaire in the focus of a parabolic cross-sectional surface, the light will travel in rays parallel to the axis of the parabola.

-If the light rays parallel to the axis of symmetry approach the parabolic surface, it concentrates them in the focus, a circumstance used to make reflecting telescopes, such as the Hale telescope on Monte Palomar.

Ellipses

-The planets of the solar system move following elliptical trajectories, quite close to the circumference in the case of the major planets, the Earth included. The Sun is not in the center, but in one of the foci.

-The ellipse is widely used in architecture as a decorative and design element.

-When placing a reflector in one of the sources of an ellipse, the light is reflected towards the other source. The same happens with sound. For this reason, in ellipse-shaped rooms, those who speak in a low voice while located in one focus are clearly heard by listeners located in the other focus.

-This same property has a surprising application in the field of medicine. Kidney stones can be destroyed by sound. High intensity ultrasound waves are generated in one of the foci of an elliptical tub filled with water, and the patient is located in the other foci. Sound waves strike and reflect on the stone, breaking it up into small pieces with their energy, which the person then easily expels during urination.

Hyperbolas

-Some comets in the Solar System follow hyperbolic trajectories, always with the Sun in one of the foci.

-The foci of hyperbolas are also very interesting to study the phenomena of wave reflection. For example, when directing a beam of light to the focus of a parabolic mirror, it is reflected in the other focus, a very useful property to build telescopes, since the light can be focused on a parabolic mirror and be redirected to another more appropriate place according to design.

-The cooling towers of nuclear power plants have a silhouette in the shape of hyperbolas.

-Before the advent of GPS, hyperbolas were used in navigation to locate boats. The ships carried on board receivers of signals emitted simultaneously by radio stations A and B and a computer was in charge of recording the differences in the arrival times of the signals, to transform them into differences in distances. In this way the ship is located on the branch of a hyperbola.

The procedure is repeated with two other radio stations C and D, which places the ship on the branch of another hyperbola . The final position of the boat is the intersection of both hyperbolas.

Circumferences

-The arrival of the wheel changed the course of history.

-Circular motion is very common, many parts rotate to produce various effects, from mills to fans.

-Although the trajectories of the major planets are elliptical, circular trajectories are good approximations in many cases.

-The circumferences are frequent elements in architecture, design, engineering and construction. The list of pieces with a circular or disk shape is endless: coins, CD’s, watches and more.

Examples

Below are two conics in the plane, a circumference and an ellipse.

Each one has an analytical equation:

Circumference

(xh) 2 + (yk) 2 = R 2

Where h and k are the coordinates of the center and R is the radius. For the circumference shown in the figure the equation is:

(x + 2) 2 + (y-2) 2 = 4

Ellipse

The equation of the ellipse whose center is the coordinate point (h, k):

[(xh) 2 / a 2  ] + [(yk) 2 / b 2  ] = 1

Where a and b are the semi-axes of the ellipse. For the ellipse shown, the center is at the point 0,0, the semi-major axis is equal to 5 and the semi-minor axis is 4. Therefore, its equation is:

(x 2 /25) + (y 2 /16) = 1

References

  1. Hoffman, J. Selection of Mathematics Topics. Volume 2.
  2. Jiménez, R. 2008. Algebra. Prentice Hall.
  3. Stewart, J. 2006. Precalculus: Mathematics for Calculus. 5th. Edition. Cengage Learning.
  4. Wikipedia. Conic section. Recovered from: es.wikipedia.org.
  5. Zill, D. 1984. Algebra and Trigonometry. McGraw Hill.

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