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Introduction Parabolas Ellipses Projectivity

Geometric Approaches to Conics

David Altizio

CMU Undergraduate Summer Lecture Series

22 June 2018

Introduction Parabolas Ellipses Projectivity

Geometry Review: Cyclic Quadrilaterals

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

AB

CD

Some properties of cyclic quadrilaterals:

∠ABC + ∠ADC = 180◦,∠BCD + ∠BAD = 180◦

∠ABD = ∠ACD, etc.

(Ptolemy) AB · CD + AD ·BC = AC ·BD

The key is that these properties also go the other way - in otherwords, 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 passingthrough A, B, C , and D.

AB

CD

Some properties of cyclic quadrilaterals:

∠ABC + ∠ADC = 180◦,∠BCD + ∠BAD = 180◦

∠ABD = ∠ACD, etc.

(Ptolemy) AB · CD + AD ·BC = AC ·BD

The key is that these properties also go the other way - in otherwords, 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 passingthrough A, B, C , and D.

AB

CD

Some properties of cyclic quadrilaterals:

∠ABC + ∠ADC = 180◦,∠BCD + ∠BAD = 180◦

∠ABD = ∠ACD, etc.

(Ptolemy) AB · CD + AD ·BC = AC ·BD

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

Introduction Parabolas Ellipses Projectivity

A Sample Applications

Example

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

A

B C

D

E

Introduction Parabolas Ellipses Projectivity

Two Sample Applications

Example 1

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

A

B C

D

E

Introduction Parabolas Ellipses Projectivity

What is a parabola?

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

FX

P`

Introduction Parabolas Ellipses Projectivity

Reflection Property for Parabolas

Theorem

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

FX

P`

m

Introduction Parabolas Ellipses Projectivity

Reflection Property for Parabolas

Theorem

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

FX

P`

m

Q

Y

Introduction Parabolas Ellipses Projectivity

Reflection Property for Parabolas

Corollary

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

FX

P`

m

Introduction Parabolas Ellipses Projectivity

Simson Lines

Theorem (Simson)

Let ABC be a triangle with circumcircle ω, and let P be a point inthe plane. Then the projections of P onto the sides of 4ABC arecollinear iff P lies on ω.

A

B C

P

X

Y

Z

ω

Introduction Parabolas Ellipses Projectivity

Simson Lines

Theorem

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

A

B C

P

H

X

Y

Z

ω

Introduction Parabolas Ellipses Projectivity

Applications of Simson Lines

Consider the following scenario, where the pairwise intersectionpoints of three distinct tangents to a parabola P form a triangleABC . We wish to find interesting relationships between P and4ABC .

P

Q

R

F

A

B

C

Introduction Parabolas Ellipses Projectivity

Applications of Simson Lines

P

Q

R

F

A

B

C`

Introduction Parabolas Ellipses Projectivity

Applications of Simson Lines

Theorem

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

P

Q

R

F

A

B

C

Introduction Parabolas Ellipses Projectivity

Applications of Simson Lines

Theorem

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

P

Q

R

F

H

A

B

C`

Introduction Parabolas Ellipses Projectivity

What is an ellipse?

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

F1F2

P

E

Introduction Parabolas Ellipses Projectivity

Reflection Property for Ellipses

Theorem

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

F1F2

P

E

`

Introduction Parabolas Ellipses Projectivity

Reflection Property for Ellipses

F1F2

P

E

`

F ′1

Introduction Parabolas Ellipses Projectivity

Isogonal Property of Ellipses

Theorem

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

F1F2

X

P

Q

E

Introduction Parabolas Ellipses Projectivity

Isogonal Property for Ellipses

F1F2

F ′1

F ′2

X

P

Q

E

Introduction Parabolas Ellipses Projectivity

Distances from Foci to Tangent

Theorem

Let E be an ellipse with foci F1 and F2. Let ` be an arbitrarytangent to E at a point P. Then

dist(F1, `) · dist(F2, `) =(PF1+PF2

2

)2−(F1F22

)2;

in particular, this quantity does not depend on P.

F1F2

P

E

`

Introduction Parabolas Ellipses Projectivity

Distances from Foci to Tangent

F1F2

Q

X

P

E

P1

P2

Introduction Parabolas Ellipses Projectivity

An Application (aka shameless advertising)

CMIMC 2018 G9, Gunmay Handa

Suppose E1 6= E2 are two intersecting ellipses with a common focusX ; let the common external tangents of E1 and E2 intersect at apoint Y . Further suppose that X1 and X2 are the other foci of E1and E2, respectively, such that X1 ∈ E2 and X2 ∈ E1. IfX1X2 = 8,XX2 = 7, and XX1 = 9, what is XY 2?

X1 X2

X

Y

Introduction Parabolas Ellipses Projectivity

An Application (aka shameless advertising)

X1 X2

X

Y

Introduction Parabolas Ellipses Projectivity

Projective Transformations

A projective transformation Φ is a transformation of the planewhich preserves lines, i.e. A, B, and C are collinear iff Φ(A), Φ(B),Φ(C ) are.

If we work with the usual R2 plane, however, thenparallel lines remain parallel because transformations are injective.

Definition

The line at infinity is a “line” which has points corresponding tointersections of parallel lines (one for each direction). The normalplane joined with the line at infinity is called the projective plane.

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

Introduction Parabolas Ellipses Projectivity

Projective Transformations

A projective transformation Φ is a transformation of the planewhich preserves lines, i.e. A, B, and C are collinear iff Φ(A), Φ(B),Φ(C ) are. If we work with the usual R2 plane, however, thenparallel lines remain parallel because transformations are injective.

Definition

The line at infinity is a “line” which has points corresponding tointersections of parallel lines (one for each direction). The normalplane joined with the line at infinity is called the projective plane.

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

Introduction Parabolas Ellipses Projectivity

Projective Transformations

A projective transformation Φ is a transformation of the planewhich preserves lines, i.e. A, B, and C are collinear iff Φ(A), Φ(B),Φ(C ) are. If we work with the usual R2 plane, however, thenparallel lines remain parallel because transformations are injective.

Definition

The line at infinity is a “line” which has points corresponding tointersections of parallel lines (one for each direction). The normalplane joined with the line at infinity is called the projective plane.

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

Introduction Parabolas Ellipses Projectivity

Projective Transformations

A projective transformation Φ is a transformation of the planewhich preserves lines, i.e. A, B, and C are collinear iff Φ(A), Φ(B),Φ(C ) are. If we work with the usual R2 plane, however, thenparallel lines remain parallel because transformations are injective.

Definition

The line at infinity is a “line” which has points corresponding tointersections of parallel lines (one for each direction). The normalplane joined with the line at infinity is called the projective plane.

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

Introduction Parabolas Ellipses Projectivity

Projective Transformations

One can consider projective transformations as central projectionsbetween planes. It turns out that all projective transformationsmust be of this form.

Introduction Parabolas Ellipses Projectivity

Facts about Projective Transformations

For any quadruples of points A, B, C , D and A′, B ′, C ′, D ′ ingeneral position, there exists a projective transformationsending A→ A′, B → B ′, C → C ′, and D → D ′.

Given a circle ω and a point P, there exists a projectivetransformation sending ω to a circle ω′ and sending P to thecenter of ω′.

Given a circle ω and a line `, there exists a projectivetransformation sending ω to a circle and sending ` to the lineat infinity.

Introduction Parabolas Ellipses Projectivity

Facts about Projective Transformations

For any quadruples of points A, B, C , D and A′, B ′, C ′, D ′ ingeneral position, there exists a projective transformationsending A→ A′, B → B ′, C → C ′, and D → D ′.

Given a circle ω and a point P, there exists a projectivetransformation sending ω to a circle ω′ and sending P to thecenter of ω′.

Given a circle ω and a line `, there exists a projectivetransformation sending ω to a circle and sending ` to the lineat infinity.

Introduction Parabolas Ellipses Projectivity

Facts about Projective Transformations

For any quadruples of points A, B, C , D and A′, B ′, C ′, D ′ ingeneral position, there exists a projective transformationsending A→ A′, B → B ′, C → C ′, and D → D ′.

Given a circle ω and a point P, there exists a projectivetransformation sending ω to a circle ω′ and sending P to thecenter of ω′.

Given a circle ω and a line `, there exists a projectivetransformation sending ω to a circle and sending ` to the lineat infinity.

Introduction Parabolas Ellipses Projectivity

Example: Pappus

Theorem (Pappus)

Let ABC and A′B ′C ′ be two distinct lines in the plane. ThenA′B ∩ AB ′, B ′C ∩ BC ′, and C ′A ∩ CA′ are collinear.

AC

C′

A′

B

B′

XY

Z

Introduction Parabolas Ellipses Projectivity

Example: Pappus

Theorem (Pappus’)

Let ABC and A′B ′C ′ be two distinct lines in the plane. IfAB ′ ‖ A′B and BC ′ ‖ B ′C , then AC ′ ‖ A′C .

A

CB

B′C ′

A′

Introduction Parabolas Ellipses Projectivity

Projective Transformations and Conic Sections

Remember this?

Introduction Parabolas Ellipses Projectivity

Projective Transformations and Conic Sections

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

Theorem

Let C1 and C2 be any two conics. Then there exists a projectivetransformation sending C1 to C2.

Note that under this framework, parabolas intersect the line atinfinity at one point (in the direction perpendicular to thedirectrix) 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 theidea of central projections. Thus, the image above suggests that

Theorem

Let C1 and C2 be any two conics. Then there exists a projectivetransformation sending C1 to C2.

Note that under this framework, parabolas intersect the line atinfinity at one point (in the direction perpendicular to thedirectrix) 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 , andCD ∩ FA are collinear.

A

BC

D

E

F

XY

Z

Introduction Parabolas Ellipses Projectivity

A Very Useful Example

Theorem (Pascal’)

Let ABCDEF be a cyclic hexagon. If AB ‖ DE and BC ‖ EF , thenCD ‖ FA.

A

B

C

D

EF

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.

A

BC

D

E

F

X Y Z