Quadratics

6.2 Light Cones

Have you ever played with a flashlight – shining the light onto the ground or onto a wall at different angles admiring the different shapes of light that arise?

The rays of light emanating from the flashlight make a cone:

Q6_2_pic1

If you point the light directly on to a wall, the shape of light produced is a circle.

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Angling the flashlight produces a shape of light more oval like.

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Placing the flashlight right up against the wall produces a U-shape of light.

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The Greek scholars of ancient times, naturally curious about the world, wondered if it was possible to describe the shapes that arise this way mathematically. They realized that these shapes are slices of a cone, and they gave the different forms of slices they saw the following names: ellipse, parabola, and hyperbola(Just to be as general as possible they looked at slices of double cones of light, noting that one could get double sections for the hyperbola.)

Q6_2_pic5

These curves are called the conic sections.

NOTE: The parabola is the “first” non-closed curve. It is produced by slicing the cone parallel to the edge of the cone, and so misses slicing the top section of the cone.

 

Question: If you hold a ball up towards the sunlight, the shadow cast on the ground is the shape of an ellipse. Do you see why?

Q6_2_pic6

 

ON THE USE OF THE WORD PARABOLA:

In these notes I have been careful to avoid the word “parabola” when describing the shapes of quadratic graphs. I do this because there is absolutely no reason to believe that the curve we are seeing by slicing a cone is the same shape as the curves given by quadratics. (After all, a hanging chain also looks like a parabola – but it is not!)

Many algebra books simply tell students that parabolas and quadratic curves are indeed the same – but they don’t explain why.

In the COMPANION GUIDE to this QUADRATICS course I give a proof that these curves are indeed the same.  But be warned: it is hard work! (This is why algebra books don’t do it. Fair enough, I suppose.)

 

ASIDE: SOME HISTORY OF CONIC SECTIONS

The Greeks of antiquity studied these curves with no practical applications in mind. They were naturally curious and pursued the topic solely for its beauty and its intellectual rewards.

APOLLONIUS OF PERGA (ca 225 B.C.E.) wrote a series of eight books, titled Conic sections, in which he thoroughly investigated these curves. He was the one to introduce the names parabola, ellipse and hyperbola. ARCHIMEDES OF SYRACUSE (ca. 287–212 B.C.E.) also wrote about these curves.

It turns out, as the Greeks discovered, that these curves have a number of remarkable properties that happen to lead to a number of significant practical applications.

Scientific observations some 2000 years later showed that these curves also appear in nature in a number of surprising places.

  • In 1604 Galileo discovered that objects thrown in the air follow parabolic paths (if air resistance can be neglected).
  • In 1609 astronomer JOHANNES KEPLER discovered that the orbit of Mars is an ellipse. He conjectured that all planetary bodies have elliptical orbits, which, 60 years later, ISAAC NEWTON was able to prove using his newly developed law of gravitation.
  • Recently scientists have discovered that the path of an alpha particle in the electrical field of an atomic nucleus is a hyperbola.

 

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