Conic Sections and Dandelin Spheres l v4 9ly s1

8/11/2019 Conic Sections and Dandelin Spheres l v4 9ly s1

http://slidepdf.com/reader/full/conic-sections-and-dandelin-spheres-l-v4-9ly-s1 1/10

Conic Sections and Dandelin

Spheres

Raja Almukahhal

Larame Spence

Mara Landers

Nick Fiori

Art Fortgang

Melissa Vigil

Say Thanks to the Authors

Click http://www.ck12.org/saythanks(No sign in required)

http://www.ck12.org/saythanks




To access a customizable version of this book, as well as other

interactive content, visit www.ck12.org

CK-12 Foundation is a non-profit organization with a mission to

reduce the cost of textbook materials for the K-12 market both

in the U.S. and worldwide. Using an open-content, web-based

collaborative model termed the FlexBook®, CK-12 intends to

pioneer the generation and distribution of high-quality educational

content that will serve both as core text as well as provide an

adaptive environment for learning, powered through the FlexBook

Platform®.

Copyright © 2012 CK-12 Foundation, www.ck12.org

The names “CK-12” and “CK12” and associated logos and theterms “FlexBook®” and “FlexBook Platform®” (collectively

“CK-12 Marks”) are trademarks and service marks of CK-12

Foundation and are protected by federal, state, and international

laws.

Any form of reproduction of this book in any format or medium,

in whole or in sections must include the referral attribution link

http://www.ck12.org/saythanks (placed in a visible location) in

addition to the following terms.

Except as otherwise noted, all CK-12 Content (includingCK-12 Curriculum Material) is made available to Users

in accordance with the Creative Commons Attribution/Non-

Commercial/Share Alike 3.0 Unported (CC BY-NC-SA) License

(http://creativecommons.org/licenses/by-nc-sa/3.0/ ), as amended

and updated by Creative Commons from time to time (the “CC

License”), which is incorporated herein by this reference.

Complete terms can be found at http://www.ck12.org/terms.

Printed: January 1, 2013

AUTHORS

Raja Almukahhal

Larame Spence

Mara Landers

Nick Fiori

Art Fortgang

Melissa Vigil

http://www.ck12.org/


http://creativecommons.org/licenses/by-nc-sa/3.0/

http://www.ck12.org/terms

http://www.ck12.org/terms

http://creativecommons.org/licenses/by-nc-sa/3.0/





www.ck12.org Concept 1. Conic Sections and Dandelin Spheres

CONCEPT 1Conic Sections and

Dandelin Spheres

Here you will explore the fascinating proofs of conic section properties conceived in 1822 by the French mathemati-

cian Germinal Dandelin. The proofs use what are now called "Dandelin Spheres."

One of the major activities of mathematicians and scientists of all kinds is learning, and attempting to prove, how

and why things work.

Since Ancient Greece at the beginning of the millennium, and likely well before, the properties of conic sections

have been studied. One that has been explored extensively is the "Focal Property", which we have referred to in

nearly every lesson in this section.

There are a number of mathematical proofs of the focal property, but trying to explain such proofs to someone who

does not dedicate his or her life to mathematics is difficult. The Dandelin Spheres are one solution.

What are "Dandelin Spheres?"

Watch This

Embedded Video:

MEDIA

Click image to the left for more content.

- James Sousa: Graphing Quadratics

Guidance

Dandelin Spheres and the equation of an ellipse

It wasn’t until 1822 that the French mathematician Germinal Dandelin thought of this very clever construction.

Dandelin found a way to find the foci and prove the focal property in one fell swoop.

Take the conic section in question. Then choose a sphere that is just the right size so that when it’s dropped into

the conic, it touches the intersecting plane, as well as being snug against the cone on all sides. If you prefer, you

can think of the sphere as a perfectly round balloon that is blown up until it “just fits” inside the cone, still touching

the plane. Then do the same on the other side of the plane. After we’ve drawn both of these spheres we have this

picture:

1





http://youtu.be/oH6eZ6oDRBI











http://www.ck12.org/flx/show/video/user%3Ack12editor/http%3A//www.youtube.com/embed/oH6eZ6oDRBI%3Fwmode%3Dtransparent%26rel%3D0%26hash%3Df1d9dda3fd80f06a754debb33fc41511




www.ck12.org

or

These spheres are often called “Dandelin spheres”, named after their discoverer. It turns out that not only is our

shape an ellipse (which, like all ellipses satisfies the focal property), but these spheres touch the ellipse exactly at

the two foci. To see this, consider this geometric argument.

The first thing to notice is that the circles C 1 and C 2 shown on the diagram below, where each sphere lies snug against

the cone, lie in parallel planes to one another. In particular, each line passing through these circles and the vertex of

the cone, such as the line l drawn below, cuts off equal segments between the two circles. Let’s call d the shortest

distance along the cone between circles C 1 and C 2. This can also be thought of as the shortest distance between C 1and C 2 that passes through the vertex of the cone.

2









The next thing to remember is a property of tangents to spheres that you may have learned in geometry. If two

segments are drawn between a point and a sphere, and if the line containing each segment is tangent to the sphere,

then the two segments are equal. In the diagram below, AB = AC . (This follows from the fact that tangents are

perpendicular to the radii of a sphere and that two congruent triangles are formed in this configuration. See this

description for more about this property.)

Now consider the point P on the ellipse drawn below. Let QR be the segment of length d between C 1 and C 2 that

passes through P. The distances between the two foci are marked d 1 and d 2. But d 1 = RP and d 2 = PQ by the

property of tangents to spheres discussed above. So d 1 + d 2 = RP + PQ = QR = d . And this sum will always equal

d , no matter what point P on the ellipse is chosen. So this proves the focal property of ellipses: that the sum of the

distances between any point on the ellipse and the two foci is constant.

3








www.ck12.org

Dandelin and the Parabola

Like the ellipse, the parabola has a focal property. And, also like the ellipse, a construction similar to Dandelin’s

with the spheres can show us what it is. Dandelin himself didn’t prove the focal property for parabolas that we are

about to discuss, but Pierce Morton used a sphere construction similar to Dandelin’s to prove the focal property of

parabolas in 1829. We’ll look at Morton’s argument here.

In contrast with the argument we made for the ellipse, for a parabola we can only fit one tangent sphere inside thecone. That is, only one sphere can be tangent to both the cone and the cutting plane. In the diagram below, the

sphere fits underneath the cutting plane, but there is no room for a sphere to lie on top of the cutting plane and still

be tangent to the cone.

As with the ellipse, the point where the sphere intersects the plane is called a focus. But because there is only one

sphere in this construction, and this is related to the fact that a parabola has only one focus. The other geometric

object of interest is called the directrix. This is the line that results from the intersection between the cutting plane

and the plane that contains the circle of contact between the sphere and the cone. In the diagram below, the directrix

is labeled l and is found by intersecting the plane defined by circle C and the cutting plane (the planes are shown in

dashed lines for clarity). Finally, we will call the angle between the planes θ.

4









In the above diagram, we have labeled the point where the sphere contacts the cutting plane with F , and we’ll call

that point the focus of the parabola. Suppose P is an arbitrarily chosen point on the parabola. Then, let Q be the point

on circle c such that PQ is tangent to the sphere. In other words Q is chosen so that PQ lies on the cone itself. Let

L be the point on the directrix l such that PL is perpendicular to l . Then PF = PQ since both segments are tangents

to the sphere from the same point P. We can also show that PQ = PL. This follows from the fact that the cutting

plane is parallel to one side of the cone. Consider the point P that is the projection of P onto the plane containing

circle C . Then PPQ and PP L are both right angles by the definition of a projection. PQP and PLP are

both equal to the angle 90−θ , where θ is the angle defined above, because the cutting plane and the cone both have

an angle of θ

with the horizon. Since they also share a side, triangles

PQP

and

PLP

are congruent by AAS .So the corresponding sides PQ and PL are congruent. By the transitive property we have PF = PL, so the distance

between the point P on the parabola to the focus is the same as the distance between P and the directrix l . We have

just proven the focus-directrix property of parabolas.

Dandelin spheres and hyperbolas

To prove the focal property of hyperbolas, we examine Dandelin’s sphere construction. Unlike the construction for

ellipses, which used two spheres on one side of the cone, and the sphere construction for parabolas, which used one

sphere on one side of the cone, this construction uses two spheres, one on each side of the cone. As with the ellipse

construction, each sphere touches the plane at one of the foci of the hyperbola. And as with the argument for the

elliptical focal property, the argument uses the fact that tangents from a common point to a sphere are equal.

5








www.ck12.org

In the above diagram, suppose P is an arbitrary point on the hyperbola. We would like to examine the difference

PF 2 −PF 1. Let C 1 be the point on the upper sphere that lies on the line between P and the vertex of the cone.

Let C 2 be the point on the upper sphere when this line is extended (so P, C 1, and C 2 are all on the same line and

PC 1 + C 1C 2 = PC 2 and the cone By the common tangent property, PF 1 = PC 1 and PF 2 = PC 2 for some points C 1and C 2 on the circles where the spheres meet the cone. So PF 2−PF 1 = PC 2−PC 1 = (PC 1 +C 1C 2)−PC 1 = C 1C 2.

But C 1C 2 is the distance along the cone between the two circles of tangency and is constant regardless of the choice

of C 1 and C 2. So the difference PF 2−PF 1 is constant.

Example A

What would the Dandelin Spheres look like if used to explore a circle?

Solution

Since the spheres touch the figure within just at the foci, and since a circle is an ellipse with both foci at the same

point, the spheres would sit directly above each other, and would touch the circle at the center point on both sides.

Example B

Conceptually speaking, why is there only one Dandelin Sphere used in the proof of a parabola?

6









Solution

As we learned in the proof for an ellipse, the Dandelin sphere must be tangent to both the cone, e.g., the sphere

touches the cone all the way around in a circle, and the plane, where the tangency is a single point. In the case of a

parabola, the sphere below the plane "fits" just fine, but a sphere above the plane would "sit on" the plane, and not

touch the cone all the way around. The figure only allows for one sphere to be tangent.

Example C

Explain why for any positive number b and a, there exists a c such that b2 = c2−a2.Let c =

a2 + b2.

Solution

Since a2 + b2 is always positive for positive a and b, this number is always defined. Geometrically, let c be the

hypotenuse of a right triangle with side lengths a and b.

Vocabulary

The Dandelin Spheres are balls used to geometrically define the conic sections.

A tangency is a location or set of locations which "just touches" a second figure.

Practice

1. Who was the first mathematician who conceptualized dandelin spheres? When?

2. What is proved by dandelin spheres?

3. How do you identify the foci of an ellipse using dandelin spheres?

4. If two tangents are drawn from a single point to a sphere, what can you say about the line segments formed?

5. How do the tangents relate to a radius of a sphere?

6. Describe the focal property of ellipses.

7. How and when did Germinal Dandelin prove the focal property for parabolas?

8. What is the line that results from the intersection between the cutting plane and the plane that contains thecircle of contact between the sphere and cone?

9. What is defined by the point where the sphere intersects the cutting plane?

10. Which construction uses two spheres in a single cone?

11. Which construction uses one sphere in a single cone?

12. Which construction uses two spheres and two cones?

Identify the parts listed on the diagram as specified below:

7








www.ck12.org

13. Directrix Line - Small Sphere

14. Directrix Line - Large Sphere

15. Focus - Small Sphere

16. Focus - Large Sphere17. Vertex - Small Sphere

18. Vertex - Large Sphere

19. Directrix Plane - Small Sphere

20. Directrix Plane - Large Sphere

21. Cutting Plane

22. What conic section is illustrated here?

8






Documents

Conic Sections and Dandelin Spheres l v4 9ly s1