UnivHypGeom3: Pappus' theorem and the cross ratio

Pappus' theorem is the first and foremost result in projective geometry. Another of his significant contributions was the notion of cross ratio of four points on a line, or of four lines through a point.

We discuss various important results: such as the Cross ratio theorem, asserting the invariance of the cross ratio under a projection, and Chasles theorem for four points on a conic. We show that the notion of cross ratio also works for four concurrent lines.

CONTENT SUMMARY: Pappus' theorem @00:52 cross ratio @02:46 cross ratio transformation theorem @11:08 cross ratio theorem @13:54 Chasles theorem @16:19 The cross ratio is the most important invariant in projective geometry 9:09

Category:

Education

Tags:

• UnivHypGeom3
• hyperbolic
• geometry
• Wildberger
• Pappus
• cross
• ratio
• projection
• Chasles
• Desargues
• conic
• transformation

[footstep002]

Another question.

In previous examples of cross ratios, It seems that it's all about straight lines. But suddenly in Chasles theorem the cross ratio is applied in conic including a circle.

I'm curious. Is cross ratio work on curved interval, like the arc interval between alpha and beta. Or Is it only work on direct straight interval between alpha and beta.

Why this cross ratio works on a circle, thought null points on a circle are not collinear on a straight line.

[njwildberger]

@footstep002 It is a good question. By projecting from a point on the circle to some tangent line of the circle, we get a correspondence between points on the circle and tangent line. This correspondence is quite remarkable, and it turns out that the notion of cross ratio can be transferred to a circle from a tangent line using it. This is a key point in projective geometry.

[joncui]

In your illustration of Chasles Theorem, your point η was OUTSIDE the arc αβγδ.

Does the theorem still work if η lies AMONGST the points α, β, γ, and δ ?

[njwildberger]

@joncui The answer is yes: in fact the notion of inside/outside an arc is not entirely obvious in projective geometry. We want theorems that hold in general.

[madier1000]

Thank you for such lecture.It helps me quit a lot in my understanding of concepts related to algebraic geometry.

[EmptySpaceEnterprise]

Pappus' theorem @00:52

cross ratio @02:46

cross ratio transformation theorem @11:08

cross ratio theorem @13:54

Chasles theorem @16:19

The cross ratio is the most important invariant in projective

geometry @19:09

[QuantumMaths]

I have the same question essentially as footstep002. I will rephrase it a touch: must a2 and b2 always lie between a1 and a3 or b1 and b3 respectively?

[footstep002]

In exercise 3.1 I don't understand what you mean by a_2b_3 and a_3b_2 are parallel. Normally those lines meet and form X shaped crossing. But if it is parallel, the orders of a_2, a_3 or b_2, b_3 have to be reversed. Is this the answer to the problem? Or is the problem asking about something else?

[MrFunatabi]

Thank you for another enlightening lecture :)