Conic Sections are the Shapes that Shape Our World

Super fact 80 : A conic section is a shape formed by slicing a cone with a plane. There are four such shapes, circle, ellipse, parabola, and hyperbola. The conic sections universally describe motion under gravity. The orbits of planets around their stars are circles or ellipses, comets fly around space in elliptical orbits, or parabolic or hyperbolic paths. Objects thrown up in the air follow parabolic paths. They are the basis for a huge amount of engineering applications.

Esther’s writing prompt: January 21 : Shapes

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Four cones each shown with a plane section forming a specific conic section. | Conic Sections are the Shapes that Shape Our World
Types of conic sections : circle , ellipse , parabola , hyperbola Shutterstock Asset id: 2377159367 by ProfDesigner

The four conic sections, circle, ellipse, parabola and hyperbola are fundamental and very useful shapes in mathematics, physics and engineering. Well, a circle is a special case of an ellipse, so it is really only three conic sections. The motion of the planets and other stellar objects are described by the conic shapes. Isaac Newton derived his law of gravitation from Kepler’s laws, which describe planetary orbits as ellipses.

The conic sections are all described by second degree equations (quadratic equations) and are in that sense the simplest shapes aside from points and lines. It is important to understand that there is an infinite amount of shapes that are almost conic sections and look like conic sections, but it is the exact mathematical properties of the four conic sections that make them so common in physics, mathematics, nature and engineering.

The picture shows a cone with four planes slicing the cone in four ways. The resulting shapes are circle (red), ellipse (green), parabola (blue), hyperbola (orange).
The black boundaries of the colored regions are conic sections. Not shown is the other half of the hyperbola, which is on the unshown other half of the double cone. by Magister Mathematicae, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=18556148

It may not come as a surprise that the circle is a fundamental and important shape, but I believe that the fact that the other conic sections are also fundamental in mathematics, physics and engineering come as a surprise to people outside of the STEM fields. It is a true and an important fact regarding how our world works.

Conic Sections

As mentioned, the conic sections are fundamental shapes that appear in a lot of places in STEM. Below are a few examples.

Parabola

Math function parabola graphics illustration with a dark background. | Conic Sections are the Shapes that Shape Our World
Math function parabola. Shutterstock Asset id: 1628916337 by EleonoraDesigner

A parabola is formed when a plane cuts a cone, so the plane is parallel to a side of the cone. Parabolas are shapes that are roughly U-shaped and described by the equation y = x^2 or more generally by y = ax^2 + bx + c. Parabolas have a so called focus point. See the picture below. If you throw a ball, or any object, up in the air its trajectory will be a parabola (ignoring distortions caused by friction and wind). I should say the parabola you get in this case is upside down. The parabola is important when you design any kind of projectile.

U-shaped parabola with the focus shown. The pciture has an x-axis and a y-axis.
Part of a parabola (blue), with various features (other colours). The complete parabola has no endpoints. In this orientation, it extends infinitely to the left, right, and upward. Picture is from Wikipedia Melikamp, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons.

Antennas shaped like parabolas (in 3D) will direct incoming radiation and waves towards their focus point. If the surface is reflective a light located at the focus point will reflect to create a straight beam. Parabolas are used for radio telescopes, satellite dishes, car headlights, flashlights, solar cookers, solar power plants, water fountains, suspension bridges, business modelling and thousands of engineering applications. Parabolas like circles and the other conic sections shape our modern world (pun intended).

A parabola dish with equipment located at the focus.
Würzburg-Riese radar built by Germany in WW2 had a 7.4 meter (24 foot) dish. From this page. Alan Wilson from Stilton, Peterborough, Cambs, UK, CC BY-SA 2.0 https://creativecommons.org/licenses/by-sa/2.0, via Wikimedia Commons

Ellipse and circle

As mentioned, a circle and an ellipse are conic sections formed by intersecting a plane with a cone. You get a circle when the cuts perpendicular to the cone’s axis (see pictures above) and an ellipse form when the plane intersects the cone at a slant but not slanted so much that it becomes a parabola or a hyperbola. An alternative for an ellipse is that the sum of the distances from any point on the curve to two fixed points (called the foci) is a constant. See the picture below. The two definitions are identical. For a circle the two foci are merged into one point at the center.

The picture shows an ellipse and its two foci points. From the foci points there are lines going to a point P on the ellipse. The length of the two lines are added together and is the sum “2a” no matter where on the ellipse the point P is located. | Conic Sections are the Shapes that Shape Our World
Ellipse: definition by sum of distances to foci. Ag2gaeh, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

There are a lot of real world examples of ellipses. Planets orbit the Sun in elliptical paths. The sun is in one of the foci points. The orbits of other stellar objects and satellites are also elliptical. Charged particles follow elliptical paths within magnetic fields.  Elliptical patterns are observed in the rotation of ocean currents, elliptical models and algorithms are used in medical imaging, computer science and encryption. Also whispering galleries.

Hyperbola

Comets and spacecraft that are not orbiting another body, in other words, they have enough speed to escape the gravitational pull and continue into deep space, will travel along a hyperbola. The boundary of a shockwave from a supersonic jet (a sonic boom) creates a hyperbolic curve on the ground as it moves. The intersection of two sets of concentric ripples in water makes a hyperbola. The light beam from a lamp or flashlight makes an ellipse or an hyperbola on a plane depending on the angle.

Newton’s Law of Gravitation

Johannes was an early 17th century German mathematician who derived three laws that describe how planetary bodies orbit the Sun using the observational data collected by the Danish astronomer Tycho Brahe. The three laws are the following:

  • Planets move in elliptical orbits with the Sun as a focus.
  • A planet covers the same area of space in the same amount of time no matter where it is in its orbit.
  • A planet’s orbital period is proportional to the size of its orbit (its semi-major axis).
Kepler’s three laws are illustrated in a diagram for two planets.
Illustration of Kepler’s laws with two planetary orbits.
The orbits are ellipses, with foci F1 and F2 for Planet 1, and F1 and F3 for Planet 2. The Sun is at F1.
The shaded areas A1 and A2 are equal and are swept out in equal times by Planet 1’s orbit.
The ratio of Planet 1’s orbit time to Planet 2’s is (a1/a2)^3/2
Hankwang, CC BY-SA 3.0 <http://creativecommons.org/licenses/by-sa/3.0/&gt;, via Wikimedia Commons

Later Isaac Netwon would use Kepler’s three laws to derive his law of gravity. Newton showed that an inverse-square force (gravity) directed toward the sun was necessary to explain the orbits.

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Author: thomasstigwikman

My name is Thomas Wikman. I am a software/robotics engineer with a background in physics. I am currently retired. I took early retirement. I am a dog lover, and especially a Leonberger lover, a home brewer, craft beer enthusiast, I’m learning French, and I am an avid reader. I live in Dallas, Texas, but I am originally from Sweden. I am married to Claudia, and we have three children. I have two blogs. The first feature the crazy adventures of our Leonberger Le Bronco von der Löwenhöhle as well as information on Leonbergers. The second blog, superfactful, feature information and facts I think are very interesting. With this blog I would like to create a list of facts that are accepted as true among the experts of the field and yet disputed amongst the public or highly surprising. These facts are special and in lieu of a better word I call them super-facts.

48 thoughts on “Conic Sections are the Shapes that Shape Our World”

            1. I was afraid my last comment wouldn’t post but it did. I had to login even though I was already logged in. It has happened to me on other people’s blogs but it was the first time it happened on my own. I loved my topology class for being interesting and fascinating but I hated for being so abstract and difficult. It was the hardest math class I’ve ever taken. It is very useful for General Relativy / Gravity.

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                1. Wow! I believe that originated from algebraic topology and abstract algebra. I believe that is very abstract and difficult. I love math but that would be over my head. I wish him all the best with his challenging masters.

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  5. I was in junior high school when I first saw the movie Star Trek: The Motion Picture and there’s a line where Captain Kirk orders Mr. Sulu to plot a “conic section flight path to the object at the center of the cloud.” It stood out because it sounded very impressive. Later I would realize he was just asking for an orbital flight path, which of course a space ship would make when approaching some large object! Thanks for the great post about conic sections.

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      1. You are welcome, Thomas! Just in case, would like to let you know that I mostly post on my newer blog currently, at starrysteps.wordpress.com (I think WordPress still links my avatar to my older bilingual blog, at doarnicol.wordpress.com) 🙂

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      1. Bean failed to mention the part where it gave me flashbacks to Differential Equations back in college, or “Diff-EQ”, or “Diffy-Screw”, as we mostly called it. I had forgotten how many types of curves you get out of slicing cones …

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        2. BTW if you look a few comments above, Robbie’s son is doing a masters in homological algebra, which originated from algebraic topology and abstract algebra. I think that beats differential equations with respect to difficulty and being abstract. Well at least if you compare with a regular class in differential equations and not research.

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  12. What a fascinating post! Sorry I’ve been out of the loop for a while; my wordpress account has been playing up and I was under the weather for a bit and unable to find a chance to fix it. This is such a cool coincidence, as my son was trying to explain a parabola to be a few days ago (I must confess, I didn’t quite understand what he meant). He’d love this article!

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