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, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
Hyperbola Definition Conic Section Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
Hyperbola Definition Conic Section Two Foci Focus and Directrix Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
The Project Basic constructions Constructing a Hyperbola Advanced constructions Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
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
X B C A Y Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
X D B C A Y Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
Angle Duplication A X Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
Angle Duplication A X A B X Y Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
C Z A B X Y Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
C Z A B X Y C Z A B X Y Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
Constructing a Perpendicular C C A B Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
C C X X O A B A B Y Y Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
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
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
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: one cork board one poster board one pair of scissors one 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
The Device Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
R = length of tube C = length of string P F 1 F 2 Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
C = PF 1 + (R – PF 2 ) + R PF 1 – PF 2 = C – 2R P F 1 F 2 Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
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. P g a h b c c A B Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
Proof. a 2 − ( c − b ) 2 h 2 = g 2 − ( c + b ) 2 h 2 = ... a = 2 b ... � h � � b + c � h 2 = g g 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
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. P g a h b c c A B Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
Proof. 2 ∠ A = ∠ B sin(2 ∠ A ) = sin( ∠ B ) 2 sin( ∠ A ) cos( ∠ A ) = sin( ∠ B ) � h � � b + c � h 2 = g g a ... ( a − 2 b )(2 c − a ) = 0 a = 2 b Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
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. P g a h b c c A B Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
Trisecting the Angle - The Classical Construction Let O denote the vertex of the A angle. P Use a compass to draw a circle B centered at O , and obtain the points A and B on the angle. Construct the hyperbola with eccentricity ǫ = 2, focus B , O and directrix the perpendicular bisector of AB . Let this hyperbola intersect the circle at P . Then OP trisects the angle. Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
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
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
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
Divide the segment AB into 6 equal parts: to do this, we pick a point G 1 , not on AB , and draw the ray AG 1 . Mark points G 2 , G 3 , G 4 , G 5 , and G 6 on the ray such that: AG 1 = G 1 G 2 = G 2 G 3 = G 3 G 4 = G 4 G 5 = G 5 G 6 . G 6 G 5 G 4 G 3 G 2 G 1 A B Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
Draw G 6 B . Draw lines through G 1 , G 2 , G 3 , G 4 and G 5 parallel to G 6 B . Each intersection produces equal length line segments on AB . Mark each intersection as shown, and treat each segment as a unit length of one. G 6 G 5 G 4 G 3 G 2 G 1 A B C D 1 V Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
Extend AB past A a length of 2 units as shown below. Mark this point F 2 . Construct a line perpendicular to AB through the point D 1 . Using F 2 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 F 1 is 4 units and the distance C to the vertex, V , is 2 units, the hyperbola has eccentricity of 2 as required. F 2 A C D 1 V B Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
Mark the intersection point between the hyperbola, h , and the circle � OA as P . Draw the segment OP . The angle ∠ POB trisects ∠ AOB . P A C D 1 V B O Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
√ 3 Constructing 2 Start with a given unit length of AB . A B Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
√ 3 Constructing 2 Start with a given unit length of AB . A B Construct a square with side AB and mark the point shown. E A B Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
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 F 1 . Draw the circle centered at A with radius AF 1 and mark that intersection on l as F 2 . Bisect the segment EB and mark the point O . F 1 V E O A B C F 2 Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
Draw a circle centered at O with a radius of OA . Using the device, draw a hyperbola with foci F 1 and F 2 and vertex V . F 1 V E O A B C F 2 Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
The circle intersects the hyperbola twice. Mark the leftmost intersection X and draw a perpendicular line from AC to X . This √ 3 segment has length 2. X D A C Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
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 5 2 ) and a radius of 4 . This gives the circle the equation: � � 2 y − 1 = 5 ( x − 1) 2 + 4 . 2 Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas
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