Math & Beyond

Conics in Real Life

27th Oct '20384 views8 mins read

27 October 2020                

Read time: 8 minutes

Introduction

  • Conics or conic sections were studied by Greek mathematicians, with Apollonius of Pergo’s work on their properties around 200 B.C.

    Apollonius of Pergo’s
    Source: i.gr-assets.com

  • Conics sections are planes, cut at varied angles from a cone. The shapes vary according to the angle at which it is cut from the cone.

  • As they are cut from cones, they are called Conies. Further, they have some common properties as they all belong to cones. These curved sections are related to.

    curved sections
    Source: upload.wikimedia.org

  • Also read 

  • Why Is Math So Hard? 


What is Conic Section?

  • Conic section is a curve obtained by the intersection of the surface of a cone with a plane.

  • In Analytical Geometry, a conic is defined as a plane algebraic curve of degree 2. That is, it consists of a set of points which satisfy a quadratic equation in two variables. This quadratic equation may be written in matrix form. By this, some geometric properties can be studied as algebraic conditions.

  • Thus, by cutting and taking different slices(planes) at different angles to the edge of a cone, we can create a circle, an ellipse, a parabola, or a hyperbola, as given below

    Conic section is a curve
    Source: ellipsesconicsections.weebly.com

  • The circle is a type of ellipse, the other sections are non-circular. So, the circle is of fourth type.

Focus, Directrix and Eccentricity

The curve is also defined by using a point(focus) and a straight line (Directrix).

If we measure and let

a – the perpendicular distance from the focus to a point P on the curve,

and b – the distance from the directrix to the point P,

then a: b will always be constant.

\(a: b < 1\) for ellipse

\(a: b= 1\) for parabola as \(a= b\)

and \(a: b> 1\) for hyperbola.

Focus, Directrix and Eccentricity
Source: .wikimedia.org

Eccentricity: The above ratio a: b is the eccentricity.

Thus, any conic section has all the points on it such that the distance between the points to the focus is equal to the eccentricity times that of the directrix.

Thus, if eccentricity \(<1\), it is an ellipse.

if eccentricity \(=1\), it is a parabola.

and if eccentricity \(=1\), it is a hyperbola.

Eccentricity
Source: https://3.bp.blogspot.com

For a circle, eccentricity is zero. With higher eccentricity, the conic is less curved.

Latus Rectum

The line parallel to the directrix and passing through the focus is Latus Rectum.

LATUS RECTUM
Source: i2.wp.com/c1.staticflickr.com

Length of Latus Rectum = 4 times the focal length

Length \(=\frac{2b^2}{a}\) where \(a =\frac{1}{2}\) the major diameter

and \(b =\frac{1}{2}\) the minor diameter.

= the diameter of the circle.

Ellipse has a focus and directrix on each side i.e., a pair of them.

Equations

General equation for all conics is with cartesian coordinates x and y and has \(x^2\) and \(y^2\) as

the section is curved. Further, x, y, x y and factors for these and a constant is involved.

Thus, the general equation for a conic is

\[Ax^2 + B x y + C y^2+ D x + E y + F = 0\]

Using this equation, following equations are obtained:

For ellipse, \(x^2a^2+y^2b^2=1\)

For hyperbola, \(x^2a^2-y^2b^2=1\)

For circle, \(x^2a^2+y^2a^2=1\) (as radius is a)


Parabola in Real Life

  • Parabola is obtained by slicing a cone parallel to the edge of the cone. It is of U – shape as a stretched geometric plane. This formula is \(y =x^2\) on the x – y axis.

  • Mathematician Menaechmus derived this formula.

    Mathematician Menaechmus
    Source:
    alchetron.com

  • Parabola is found in nature and in works of man.

  • Water from a fountain takes a path of parabola to fall on the earth.

    fountain takes a path of parabola
    Source: s3-us-west-2.amazonaws.com

  • A ball thrown high, follows a parabolic path.

  • A roller coaster takes the path of rise and fall of a parabolic track of the sea.

  • An architectural structure built and named The Parabola in London in 1962 has a copper roof with parabolic and hyperbolic linings.

    architectural structure built
    Source: phaidon.com

  • The Golden Gate Bridge in San Francisco in California is famous with parabolic spans on both sides.

    Golden Gate Bridge in San Francisco in California
    Source: askideas.com

  • In light houses, parabolic bulbs are provided to have a good focus of beam to be seen from distance by mariners.

    light house

  • Automobile headlights are also with parabola type.

    Automobile headlights with parabola type
    Source: pict1.reezocar.com

  • The stretched arc of a rocket launch is parabolic.

    rocket launch in parabolic
    Source:
    pixy.org

  • The satellite dish is a parabolic structure facilitating focus and reflection of radio waves.

    satellite dish
    Source: i.pinimg.com


Ellipses in Real Life

  • Electrons in the atom move around the nucleus in an elliptical path of orbit.

    Electrons in the atom move around
    source: lh3.googleusercontent.com

  • Property of Ellipse to reflect sound and light is used in pulverizing kidney stones. The patient is laid in an elliptical tank of water. Kidney stones being at the other focus are concentrated and pulverized.

    Ellipse to reflect sound and light kidney stone
    Source: img.brainkart.com

  • Paul’s Cathedral is an elliptical shaped structure to facilitate talking at one end is heard at the other end using the property of ellipse.

  • There is an ellipse shaped park in front of White House in Washington.

    ellipse shaped park in front of White House in Washington
    Source:
    lh3.googleusercontent.com

  • When a tumbler of water is tilted, an elliptical surface of water is seen.

    tumbler of water an elliptical surface
    Source:
    encrypted-tbn0.gstatic.com

  • Food items carrot, cucumber cut at an angle to its main axis results in elliptical shape and elegant look.

    Food items in elliptical shape
    Source:
    cdn.xl.thumbs.canstockphoto.com

  • Whispering galleries at US Statutory capital and St. Paul’s Cathedral, London demonstrates the property of the ellipse that one’s whisper from one focus can be heard at the other focus by only a person to whom it is sent
     

    St. Paul’s Cathedral, London  elliptical shape
    Source: 
    strangesounds.org

  • Elliptical training machines enable running or walking without straining the heart.

    Elliptical training machines
    Source: cdn.sweatband.com


Hyperbolas in Real Life

  • A guitar is an example of hyperbola as its sides form hyperbola.

     guitar is an example of hyperbola
    Source:
    keisan.casio.com

  • Dulles Airport has a design of hyperbolic parabolic. It has one cross-section of a hyperbola and the other a parabola.

    Dulles Airport has a design of hyperbolic parabolic
    Source: insider.com

  • Gear Transmission having pair of hyperbolic gears. It is with skewed axles and hourglass shape giving hyperbola shape. The hyperbolic gears transmit motion to the skewed axle.

    Gear Transmission of hyperbolic gears
    Source: Wolfram.com

  • The Kobe Port Tower has hourglass shape, that means it has two hyperbolas. Things seen from a point on one side will be the same when seen from the same point on the other side.

    Kobe Port Tower has hourglass shape it has two hyperbolas
    Source:
    upload.wikimedia.org

  • Satellite systems, Radio systems use hyperbolic functions.

  • Inverse relationship is related to hyperbola. Pressure and Volume of gas are in inverse relationships. This can be described by a hyperbola.

    Inverse relationship is related to hyperbola
    Source:
    edu.glogster.com

  • Lens, monitors, and optical glasses are of hyperbola shape.


Summary

Conic or conical shapes are planes cut through a cone. Based on the angle of intersection, different conics are obtained. Parabola, Ellipse, and Hyperbola are conics. Circle is a special conic. Conical shapes are two dimensional, shown on the x, y axis. Conic shapes are widely seen in nature and in man-made works and structures. They are beneficially used in electronics, architecture, food and bakery and automobile and medical fields.

Conic or conical shapes are planes
Source: google content


Frequently Asked Questions (FAQs)

What are the 4 types of conic sections?

According to the angle of intersection between a plane and a cone, four different conic sections are obtained. They are Parabola, Ellipse, Hyperbola, and Circle. They are two dimensional on the x-y axis.

4 types of conic sections
Source: oxfordlearnersdictionaries.com

How do we obtain Conic Sections?

Conic section involves a cutting plane, surface of a double cone in hourglass form and the intersection of the cone by the plane. According to the angle of cutting, that is, light angle, parallel to the edge and deep angle, ellipse, parabola and hyperbola respectively are obtained. Circle is also conic, and it is cut parallel to the circular bottom face of the cone.

Conic Sections
Source:
assignmentpoint.com

How do you find the area of an ellipse?

Area of an ellipse is \((a \times b \times π)\) sq. units.

Area of ellipse
Source:
i.pinimg.com

where a = length of major axis of ellipse

b = length of minor axis of ellipse.

What are some real-life applications of conics?

Planets travel around the Sun in elliptical routes at one focus.

Mirrors used to direct light beams at the focus of the parabola are parabolic.

Parabolic mirrors in solar ovens focus light beams for heating.

Sound waves are focused by parabolic microphones.

Car headlights and spotlights are designed based on parabola’s principles.

The path travelled by objects thrown into air is parabolic.

basket ball thrown in air is parabolic
Source: scarface tutorials

Hyperbolas are used in long range navigation systems called LORAN.

Hyperbolas are used in Loran
Source: wikimedia.org

Telescopes use parabolic mirrors.


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