Device Constructions with Hyperbolas Alfonso Croeze 1 William Kelly 1 - - PowerPoint PPT Presentation

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Device Constructions with Hyperbolas Alfonso Croeze 1 William Kelly 1 William Smith 2 1 Department of Mathematics Louisiana State University Baton Rouge, LA 2 Department of Mathematics University of Mississippi Oxford, MS July 8, 2011 Croeze,

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Device Constructions with Hyperbolas

Alfonso Croeze1 William Kelly1 William Smith2

1Department of Mathematics

Louisiana State University Baton Rouge, LA

2Department of Mathematics

University of Mississippi Oxford, MS

July 8, 2011

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Hyperbola Definition

Conic Section

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Hyperbola Definition

Conic Section Two Foci Focus and Directrix

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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The Project

Basic constructions Constructing a Hyperbola Advanced constructions

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Rusty Compass

Theorem Given a circle centered at a point A with radius r and any point C different from A, it is possible to construct a circle centered at C that is congruent to the circle centered at A with a compass and straightedge.

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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C A B X Y

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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C A B X Y D

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Angle Duplication

A X

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Angle Duplication

A X A B X Y

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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A B C X Y Z

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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A B C X Y Z A B C X Y Z

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Constructing a Perpendicular

C A B C

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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A B C X Y A B C X Y O

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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We Need to Draw a Hyperbola!

Trisection of an angle and doubling the cube cannot be accomplished with a straightedge and compass.

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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We Need to Draw a Hyperbola!

Trisection of an angle and doubling the cube cannot be accomplished with a straightedge and compass. We needed a way to draw a hyperbola.

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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We Need to Draw a Hyperbola!

Trisection of an angle and doubling the cube cannot be accomplished with a straightedge and compass. We needed a way to draw a hyperbola. Items we needed:

  • ne cork board
  • ne poster board
  • ne pair of scissors
  • ne roll of string

a box of push pins some paper if you do not already have some a writing utensil some straws, which we picked up at McDonald’s

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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The Device

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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F2 F1 P R = length of tube C = length of string

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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F2 F1 P C = PF1 + (R – PF2) + R PF1 – PF2 = C – 2R

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Hyperbolas and Triangles

Lemma Let △ABP be a triangle with the following property: point P lies along the hyperbola with eccentricity 2, B as its focus, and the perpendicular bisector of AB as its directrix. Then ∠B = 2∠A. A B P c c a h g b

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Proof. h2 = a2 − (c − b)2 h2 = g2 − (c + b)2 ... a b = 2 ... 2 h g b + c g

  • =

h a 2 sin(∠A) cos(∠A) = sin(∠B) sin(2∠A) = sin(∠B) 2∠A = ∠B

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Lemma Let △ABP be a triangle such that ∠B = 2∠A. Then point P lies along the hyperbola with eccentricity 2, B as its focus, and the perpendicular bisector of AB as its directrix. A B P c c a h g b

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Proof. 2∠A = ∠B sin(2∠A) = sin(∠B) 2 sin(∠A) cos(∠A) = sin(∠B) 2 h g b + c g

  • =

h a ... (a − 2b)(2c − a) = a b = 2

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Result

Theorem Let AB be a fixed line segment. Then the locus of points P such that ∠PBA = 2∠PAB is a hyperbola with eccentricity 2, with focus B, and the perpendicular bisector of AB as its directrix. A B P c c a h g b

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Trisecting the Angle - The Classical Construction

Let O denote the vertex of the angle. Use a compass to draw a circle centered at O, and obtain the points A and B on the angle. Construct the hyperbola with eccentricity ǫ = 2, focus B, and directrix the perpendicular bisector of AB. Let this hyperbola intersect the circle at P. Then OP trisects the angle.

A B O P

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Trisecting the Angle

Given an angle ∠O, mark a point A on on the the given rays. A O

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Trisecting the Angle

Draw a circle, centered at O with radius OA. Mark the intersection on the second ray B, and draw the segment AB. A B O

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Trisecting the Angle

Draw a circle, centered at O with radius OA. Mark the intersection on the second ray B, and draw the segment AB. A B O

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Divide the segment AB into 6 equal parts: to do this, we pick a point G1, not on AB, and draw the ray AG1. Mark points G2, G3, G4, G5, and G6 on the ray such that: AG1 = G1G2 = G2G3 = G3G4 = G4G5 = G5G6. A B G1 G2 G3 G4 G5 G6

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Draw G6B. Draw lines through G1, G2, G3, G4 and G5 parallel to

  • G6B. Each intersection produces equal length line segments on
  • AB. Mark each intersection as shown, and treat each segment as a

unit length of one. A B G1 G2 G3 G4 G5 G6 C D1 V

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Extend AB past A a length of 2 units as shown below. Mark this point F2. Construct a line perpendicular to AB through the point

  • D1. Using F2 and B as the foci and V as the vertex, use the device

to construct a hyperbola, called h. Since the distance from the the center, C, to F1 is 4 units and the distance C to the vertex, V , is 2 units, the hyperbola has eccentricity of 2 as required. F2 A B C D1 V

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Mark the intersection point between the hyperbola, h, and the circle OA as P. Draw the segment OP. The angle ∠POB trisects ∠AOB. A P B C D1 V O

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Constructing

3

√ 2

Start with a given unit length of AB. A B

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Constructing

3

√ 2

Start with a given unit length of AB. A B Construct a square with side AB and mark the point shown. A B E

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Draw a line l through the points A and E. Extend line AB past B a unit length of AB. Draw a circle, centered at A, with radius AC and mark the intersection on l as V . Draw a circle centered at E with radius AE and mark the intersection on l as F1. Draw the circle centered at A with radius AF1 and mark that intersection on l as F2. Bisect the segment EB and mark the point O. A B C E O V F1 F2

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Draw a circle centered at O with a radius of OA. Using the device, draw a hyperbola with foci F1 and F2 and vertex V . A B C E O V F1 F2

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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The circle intersects the hyperbola twice. Mark the leftmost intersection X and draw a perpendicular line from AC to X. This segment has length

3

√ 2. A C D X

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Construction Proof

We can easily prove that the above construction is valid if we translate the above into Cartesian coordinates. If we allow the point A to be treated as the origin of the x - y plane and B be the point (1, 0), we can write the equations of the circle and hyperbola. The circle is centered at 1 unit to the right and 1

2 units up,

giving it a center of (1, 1

2) and a radius of

  • 5
  • 4. This gives the

circle the equation: (x − 1)2 +

  • y − 1

2 2 = 5 4.

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The hyperbola, being rectangular with vertex ( √ 2, √ 2), has the equation xy = 2, so y = 2/x. Substituting this expression into the circle’s equation and solving for x yields the following: (x − 1)2 + 2 x − 1 2 2 = 5 4 x2 − 2x + 4 x2 − 2 x = 0 x4 − 2x − 2x3 + 4 = 0 (x3 − 2)(x − 2) = 0 From here we can see that both x =

3

√ 2 and x = 2 are solutions. This proves that the horizontal distance from the y-axis to the point X is

3

√ 2.

Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas

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Bibliography

Heath, T., A History of Greek Mathematics, Dover Publications, New York, 1981. Apostol, T.M. and Mnatsakanian, M.N., Ellipse to Hyperbola: With This String I Thee Wed, Mathematics Magazine 84 (2011) 83-97. http://en.wikipedia.org/wiki/Compass equivalence theorem

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Acknowledgements

  • Dr. Mark Davidson
  • Dr. Larry Smolinsky

Irina Holmes

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Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas