geometric approaches to conics - university of illinoisintroduction parabolas ellipses projectivity...

Introduction Parabolas Ellipses Projectivity

Geometric Approaches to Conics

David Altizio

CMU Undergraduate Summer Lecture Series

22 June 2018


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!


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


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


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`


Reflection Property for Parabolas

Theorem

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

FX

P`

m



Theorem

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

FX

P`

m

Q

Y



Corollary

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

FX

P`

m


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


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


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



P

Q

R

F

A

B

C`



Theorem

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

P

Q

R

F

A

B

C



Theorem

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

P

Q

R

F

H

A

B

C`


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


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

`


Reflection Property for Ellipses

F1F2

P

E

`

F ′1


Isogonal Property of Ellipses

Theorem

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

F1F2

X

P

Q

E


Isogonal Property for Ellipses

F1F2

F ′1

F ′2

X

P

Q

E


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

`


Distances from Foci to Tangent

F1F2

Q

X

P

E

P1

P2


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


An Application (aka shameless advertising)

X1 X2

X

Y


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.



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.



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


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.


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


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′


Projective Transformations and Conic Sections

Remember this?


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).


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



Theorem (Pascal’)

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

A

B

C

D

EF



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

geometric approaches to conics - university of illinoisintroduction parabolas ellipses projectivity...

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