Introduction Parabolas Ellipses Projectivity

Geometric Approaches to Conics

David Altizio

CMU Undergraduate Summer Lecture Series

22 June 2018 B A Some properties of cyclic quadrilaterals: ◦ ∠ABC + ∠ADC = 180 , ◦ ∠BCD + ∠BAD = 180 ∠ABD = ∠ACD, etc. (Ptolemy) AB CD + AD BC = AC BD CD · · ·

The key is that these properties also go the other way - in other words, e.g. ∠ABD = ∠ACD implies ABCD is cyclic!

Introduction Parabolas Ellipses Projectivity Geometry Review: Cyclic Quadrilaterals

A quadrilateral ABCD is cyclic if there exists a circle passing through A, B, C, and D. The key is that these properties also go the other way - in other words, e.g. ∠ABD = ∠ACD implies ABCD is cyclic!

Introduction Parabolas Ellipses Projectivity Geometry Review: Cyclic Quadrilaterals

A quadrilateral ABCD is cyclic if there exists a circle passing through A, B, C, and D.

B A Some properties of cyclic quadrilaterals: ◦ ∠ABC + ∠ADC = 180 , ◦ ∠BCD + ∠BAD = 180 ∠ABD = ∠ACD, etc. (Ptolemy) AB CD + AD BC = AC BD CD · · · Introduction Parabolas Ellipses Projectivity Geometry Review: Cyclic Quadrilaterals

A quadrilateral ABCD is cyclic if there exists a circle passing through A, B, C, and D.

B A Some properties of cyclic quadrilaterals: ◦ ∠ABC + ∠ADC = 180 , ◦ ∠BCD + ∠BAD = 180 ∠ABD = ∠ACD, etc. (Ptolemy) AB CD + AD BC = AC BD CD · · ·

The key is that these properties also go the other way - in other words, e.g. ∠ABD = ∠ACD implies ABCD is cyclic! Introduction Parabolas Ellipses Projectivity A Sample Applications

Example Let ABC be a triangle with ABC = 90◦. Point D lies on AB, 4 ∠ and E is the foot of D onto AC. Then ∠ABE = ∠ACD.

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E

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BC Introduction Parabolas Ellipses Projectivity Two Sample Applications

Example 1 Let ABC be a triangle with ABC = 90◦. Point D lies on AB, 4 ∠ and E is the foot of D onto AC. Then ∠ABE = ∠ACD.

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E

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BC Introduction Parabolas Ellipses Projectivity What is a parabola?

A parabola is the locus of points X which have equal distances P to some fixed point F (called the focus of ) and some fixed line ` (called the directrix of ). P P

X F

` P Introduction Parabolas Ellipses Projectivity Reflection Property for Parabolas

Theorem Let m denote the tangent to at the point X . Then m bisects P ∠FXP.

X F

m

` P Introduction Parabolas Ellipses Projectivity Reflection Property for Parabolas

Theorem Let m denote the tangent to at the point X . Then m bisects P ∠FXP.

X F

Y

m

` Q P Introduction Parabolas Ellipses Projectivity Reflection Property for Parabolas

Corollary The reflection of the focus F over the tangent line m lies on the directrix of . P

X F

m

` P Introduction Parabolas Ellipses Projectivity Simson Lines

Theorem (Simson) Let ABC be a triangle with circumcircle ω, and let P be a point in the plane. Then the projections of P onto the sides of ABC are 4 collinear iff P lies on ω. Z A P

Y

ω

BC X Introduction Parabolas Ellipses Projectivity Simson Lines

Theorem Let H be the orthocenter of ABC. Then for any P on (ABC), the Simson line of P wrt ABC4 bisects PH. 4

Z A P

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H ω

BC X Introduction Parabolas Ellipses Projectivity Applications of Simson Lines

Consider the following scenario, where the pairwise intersection points of three distinct tangents to a parabola form a triangle ABC. We wish to find interesting relationshipsP between and ABC. P 4

P R

F Q B C

A Introduction Parabolas Ellipses Projectivity Applications of Simson Lines

P

R

F

B Q

C `

A Introduction Parabolas Ellipses Projectivity Applications of Simson Lines

Theorem The focus F of lies on the circumcircle of ABC. P 4

P R

F Q B C

A Introduction Parabolas Ellipses Projectivity Applications of Simson Lines

Theorem The orthocenter H of ABC lies on the directrix of . 4 P

P

R

F

Q B

H C `

A Introduction Parabolas Ellipses Projectivity What is an ellipse?

An ellipse is the locus of points P in the plane such that the sum E of the distances from P to two fixed points F1 and F2 (called the foci of ) is a constant. E

P

F2 F1

E Introduction Parabolas Ellipses Projectivity Reflection Property for Ellipses

Theorem Let ` denote the tangent to an ellipse at a point P. Then the E acute angles made by F1P and F2P with respect to ` are equal.

P `

F2 F1

E Introduction Parabolas Ellipses Projectivity Reflection Property for Ellipses

F10

P `

F2 F1

E Introduction Parabolas Ellipses Projectivity Isogonal Property of Ellipses

Theorem Let XP and XQ be tangents to the ellipse . Then E ∠F1XP = ∠F2XQ.

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Q

F2 F1

E Introduction Parabolas Ellipses Projectivity Isogonal Property for Ellipses

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F10 P

F20 Q

F2 F1

E Introduction Parabolas Ellipses Projectivity Distances from Foci to Tangent

Theorem

Let be an ellipse with foci F1 and F2. Let ` be an arbitrary tangentE to at a point P. Then E  2  2 PF1+PF2 F1F2 dist(F1, `) dist(F2, `) = ; · 2 − 2 in particular, this quantity does not depend on P.

P `

F2 F1

E Introduction Parabolas Ellipses Projectivity Distances from Foci to Tangent

P2

P P1 X

Q F2 F1

E Introduction Parabolas Ellipses Projectivity An Application (aka shameless advertising)

CMIMC 2018 G9, Gunmay Handa

Suppose 1 = 2 are two intersecting ellipses with a common focus E 6 E X ; let the common external tangents of 1 and 2 intersect at a E E point Y . Further suppose that X1 and X2 are the other foci of 1 E and 2, respectively, such that X1 2 and X2 1. If E ∈ E 2∈ E X1X2 = 8, XX2 = 7, and XX1 = 9, what is XY ?

X

Y X1 X2 Introduction Parabolas Ellipses Projectivity An Application (aka shameless advertising)

X

Y X1 X2 2 If we work with the usual R plane, however, then parallel lines remain parallel because transformations are injective. Definition The line at infinity is a “line” which has points corresponding to intersections of parallel lines (one for each direction). The normal plane joined with the line at infinity is called the projective plane.

For the remainder of this talk, we will be working in the projective plane.

Introduction Parabolas Ellipses Projectivity Projective Transformations

A projective transformation Φ is a transformation of the plane which preserves lines, i.e. A, B, and C are collinear iff Φ(A), Φ(B), Φ(C) are. Definition The line at infinity is a “line” which has points corresponding to intersections of parallel lines (one for each direction). The normal plane joined with the line at infinity is called the projective plane.

For the remainder of this talk, we will be working in the projective plane.

Introduction Parabolas Ellipses Projectivity Projective Transformations

A projective transformation Φ is a transformation of the plane which preserves lines, i.e. A, B, and C are collinear iff Φ(A), Φ(B), 2 Φ(C) are. If we work with the usual R plane, however, then parallel lines remain parallel because transformations are injective. For the remainder of this talk, we will be working in the projective plane.

Introduction Parabolas Ellipses Projectivity Projective Transformations

A projective transformation Φ is a transformation of the plane which preserves lines, i.e. A, B, and C are collinear iff Φ(A), Φ(B), 2 Φ(C) are. If we work with the usual R plane, however, then parallel lines remain parallel because transformations are injective. Definition The line at infinity is a “line” which has points corresponding to intersections of parallel lines (one for each direction). The normal plane joined with the line at infinity is called the projective plane. Introduction Parabolas Ellipses Projectivity Projective Transformations

A projective transformation Φ is a transformation of the plane which preserves lines, i.e. A, B, and C are collinear iff Φ(A), Φ(B), 2 Φ(C) are. If we work with the usual R plane, however, then parallel lines remain parallel because transformations are injective. Definition The line at infinity is a “line” which has points corresponding to intersections of parallel lines (one for each direction). The normal plane joined with the line at infinity is called the projective plane.

For the remainder of this talk, we will be working in the projective plane. Introduction Parabolas Ellipses Projectivity Projective Transformations

One can consider projective transformations as central projections between planes. It turns out that all projective transformations must be of this form. Given a circle ω and a point P, there exists a projective transformation sending ω to a circle ω0 and sending P to the center of ω0. Given a circle ω and a line `, there exists a projective transformation sending ω to a circle and sending ` to the line at infinity.

Introduction Parabolas Ellipses Projectivity Facts about Projective Transformations

For any quadruples of points A, B, C, D and A0, B0, C 0, D0 in general position, there exists a projective transformation sending A A0, B B0, C C 0, and D D0. → → → → Given a circle ω and a line `, there exists a projective transformation sending ω to a circle and sending ` to the line at infinity.

Introduction Parabolas Ellipses Projectivity Facts about Projective Transformations

For any quadruples of points A, B, C, D and A0, B0, C 0, D0 in general position, there exists a projective transformation sending A A0, B B0, C C 0, and D D0. → → → → Given a circle ω and a point P, there exists a projective transformation sending ω to a circle ω0 and sending P to the center of ω0. Introduction Parabolas Ellipses Projectivity Facts about Projective Transformations

For any quadruples of points A, B, C, D and A0, B0, C 0, D0 in general position, there exists a projective transformation sending A A0, B B0, C C 0, and D D0. → → → → Given a circle ω and a point P, there exists a projective transformation sending ω to a circle ω0 and sending P to the center of ω0. Given a circle ω and a line `, there exists a projective transformation sending ω to a circle and sending ` to the line at infinity. Introduction Parabolas Ellipses Projectivity Example: Pappus

Theorem (Pappus) Let ABC and A0B0C 0 be two distinct lines in the plane. Then A0B AB0, B0C BC 0, and C 0A CA0 are collinear. ∩ ∩ ∩

Z Y X

B C A

C0 B0 A0 Introduction Parabolas Ellipses Projectivity Example: Pappus

Theorem (Pappus’) Let ABC and A0B0C 0 be two distinct lines in the plane. If AB0 A0B and BC 0 B0C, then AC 0 A0C. k k k

C B A

C0 B0 A0 Introduction Parabolas Ellipses Projectivity Projective Transformations and Conic Sections

Remember this? Note that under this framework, parabolas intersect the line at infinity at one point (in the direction perpendicular to the directrix) and hyperbolas intersect the line at infinity at two points (in the direction of their asymptotes).

Introduction Parabolas Ellipses Projectivity Projective Transformations and Conic Sections

Notice that the cone shape has very strong connections with the idea of central projections. Thus, the image above suggests that Theorem

Let 1 and 2 be any two conics. Then there exists a projective C C transformation sending 1 to 2. C C Introduction Parabolas Ellipses Projectivity Projective Transformations and Conic Sections

Notice that the cone shape has very strong connections with the idea of central projections. Thus, the image above suggests that Theorem

Let 1 and 2 be any two conics. Then there exists a projective C C transformation sending 1 to 2. C C Note that under this framework, parabolas intersect the line at infinity at one point (in the direction perpendicular to the directrix) and hyperbolas intersect the line at infinity at two points (in the direction of their asymptotes). Introduction Parabolas Ellipses Projectivity A Very Useful Example

Theorem (Pascal) Let ABCDEF be a cyclic hexagon. Then AB DE, BC EF , and CD FA are collinear. ∩ ∩ ∩ X Y Z

C B D

A E

F Introduction Parabolas Ellipses Projectivity A Very Useful Example

Theorem (Pascal’) Let ABCDEF be a cyclic hexagon. If AB DE and BC EF , then CD FA. k k k C D

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A

E F Introduction Parabolas Ellipses Projectivity A Very Useful Example

Theorem (actually Pascal) Let ABCDEF be six points on a conic. Then AB DE, BC EF , and CD FA are collinear. ∩ ∩ ∩

X Y Z

C B D

E A

F