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# Projective geometry

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1. I don't understand projective geometry and I've tried to find decent resources like HSM Coexter's book but it has weird notation that I don't understand. Could anyone suggest decent books or pdfs for olympiad prep.
2. (Original post by 11234)
I don't understand projective geometry and I've tried to find decent resources like HSM Coexter's book but it has weird notation that I don't understand. Could anyone suggest decent books or pdfs for olympiad prep.
I suppose it might help to know how you are (or how you wish to) approach the subject. You can either begin with axioms and build a theory introducing co-ordinates, or start with a projectivized co-ordinate space and then prove "axioms" such as "two lines in a plane meet in a unique point".

There is a short Oxford course which takes the second approach, with online notes at

http://www0.maths.ox.ac.uk/courses/course/28699

but you'd need to start being comfortable with vector spaces and dual spaces.

Alternatively you might also start trying to understand Mobius transformations and the Riemann sphere as a way in (the Riemann sphere being the complex projective line).

If you want an axiomatic approach then I like

https://www.amazon.co.uk/Projective-...ctive+geometry

3. (Original post by 11234)
X
More online notes

https://people.maths.ox.ac.uk/hitchi...e_geometry.pdf

Do you have any specific questions yourself?
4. (Original post by RichE)
More online notes

https://people.maths.ox.ac.uk/hitchi...e_geometry.pdf

Do you have any specific questions yourself?
Thanks, I'm just a bit stuck on understanding the difference between projectivity and perspectivity and the meaning of the ^ sign. I've just started learning it yesterday and have so far done la hire's theorem, harmonic divisions and a few lemmas on it and polars.
5. (Original post by 11234)
Thanks, I'm just a bit stuck on understanding the difference between projectivity and perspectivity and the meaning of the ^ sign. I've just started learning it yesterday and have so far done la hire's theorem, harmonic divisions and a few lemmas on it and polars.
That all sounds very much the axiomatic approach. Not knowing Coxeter's book I don't know what the ^ symbol denotes - can you describe it?

Perspectivities are types of projectivity (a.k.a. projective transformation). They don't themselves make a group but they do generate the group of projectivities. (Somewhat akin to how reflections generate the group of isometries).

PS La Hire's Theorem (just looked it up) would be, I guess, quite far beyond the basic definitions and on to poles, polars, polarities, quadrics.
6. (Original post by RichE)
That all sounds very much the axiomatic approach. Not knowing Coxeter's book I don't know what the ^ symbol denotes - can you describe it?

Perspectivities are types of projectivity (a.k.a. projective transformation). They don't themselves make a group but they do generate the group of projectivities. (Somewhat akin to how reflections generate the group of isometries).

PS La Hire's Theorem (just looked it up) would be, I guess, quite far beyond the basic definitions and on to poles, polars, polarities, quadrics.
Yea I just worked through some olympiad pdf which touched on projective geometry which looks quite interesting. I haven't come across vectors and duality yet but its one of the chapters in that book but I'm struggling to understand the notation tbh.
It says
"Thusthe range is a section of the pencil (namely, the section by the line o) and thepencil projects the range (from the point 0). As a notation for this elementarycorrespondence we may write eitherX ^ x,where X is a variable point of the range and x is the corresponding line of thepencil"

I know what a pencil is but not sure as to what the ^ sign means geometrically, whether its a pencil through O and say X due to x or something else. And if thats a projectivity, I don't know the difference between that and a perspectivity.
7. Yeah I've basically taken an axiomatic approach not sure about the vector approach though. However I haven't done pascals or pappus theorem yet but I've done desargues, la hires and brokard(don't know how to use it though)
8. (Original post by 11234)
Yea I just worked through some olympiad pdf which touched on projective geometry which looks quite interesting. I haven't come across vectors and duality yet but its one of the chapters in that book but I'm struggling to understand the notation tbh.
It says
"Thusthe range is a section of the pencil (namely, the section by the line o) and thepencil projects the range (from the point 0). As a notation for this elementarycorrespondence we may write eitherX ^ x,where X is a variable point of the range and x is the corresponding line of thepencil"

I know what a pencil is but not sure as to what the ^ sign means geometrically, whether its a pencil through O and say X due to x or something else. And if thats a projectivity, I don't know the difference between that and a perspectivity.
I'm afraid out of context I can't quite make sense of what is being said. It seems that the ^ thing relates to a correspondence between points and lines but I can't be sure.

But the ideas of perspectivity and projectivity are much more basic than any of this. A perspectivity is a fairly natural thing, a transformation between two lines (or equally between a line and itself) when viewed from a certain point. A projectivity is any map that can be written as a composition of perspectivities.
9. (Original post by RichE)
I'm afraid out of context I can't quite make sense of what is being said. It seems that the ^ thing relates to a correspondence between points and lines but I can't be sure.

But the ideas of perspectivity and projectivity are much more basic than any of this. A perspectivity is a fairly natural thing, a transformation between two lines (or equally between a line and itself) when viewed from a certain point. A projectivity is any map that can be written as a composition of perspectivities.
Oh ok makes sense . So surely any two points can be defined as a projectivitty cos I can go from one to the other in a finite number of perspectives?
10. (Original post by 11234)
Oh ok makes sense . So surely any two points can be defined as a projectivitty cos I can go from one to the other in a finite number of perspectives?
That phrase doesn't really make sense. A projectivity is a map between two lines (possibly the same line). So I can't quite make sense of that phrase.
11. (Original post by RichE)
That phrase doesn't really make sense. A projectivity is a map between two lines (possibly the same line). So I can't quite make sense of that phrase.
Yea sorry my fault. Maybe we can generalise to a range of points so thus making a line (correct me if my logic is wrong)
12. (Original post by 11234)

Yea sorry my fault. Maybe we can generalise to a range of points so thus making a line (correct me if my logic is wrong)
So for the diagram you've drawn

A-> A', B -> B'', C -> C'' is a perspectivity from line L to L'' (from P)

A' -> A', B'' -> B', C'' -> C' is a perspectivity from line L'' to L' (from P')

The composition of these two is a projectivity from L to L'

Does this help? Not sure what you're asking.
13. (Original post by RichE)
So for the diagram you've drawn

A-> A', B -> B'', C -> C'' is a perspectivity from line L to L'' (from P)

A' -> A', B'' -> B', C'' -> C' is a perspectivity from line L'' to L' (from P'

The composition of these two is a projectivity from L to L'

Does this help? Not sure what you're asking.

Hey so I found this worksheet and was wondering how theorem 7 proof worked. I don't understand how (C,D;A,A*)=(C,D;A*,A) I understand how they';re harmonic but not sure of this equality. Any help appreciated thnx
14. (Original post by 11234)

Hey so I found this worksheet and was wondering how theorem 7 proof worked. I don't understand how (C,D;A,A*)=(C,D;A*,A) I understand how they';re harmonic but not sure of this equality. Any help appreciated thnx
Sorry that seems a bit abstruse. Not necessarily hard, but I'm afraid I don't have the time to work out quite what it all means. It's certainly not standard stuff I can just quickly reply to - sorry.
15. (Original post by 11234)

Hey so I found this worksheet and was wondering how theorem 7 proof worked. I don't understand how (C,D;A,A*)=(C,D;A*,A) I understand how they';re harmonic but not sure of this equality. Any help appreciated thnx
Had a further look at the details. That equality just follows from an identity of the cross ratio.

If you swap the last two points you get its reciprocal. But as the points are harmonic then the cross ratio is -1. **
16. (Original post by RichE)
Had a further look at the details. That equality just follows from an identity of the cross ratio.

If you swap the last two points you get its reciprocal. But as the points are harmonic then the cross ratio is -1. **
But we don't know the points are harmonic isn't that the consequence of the identity. I'm confused on how they got the identity because it would give the reciprocal but how do we know they are equal
17. (Original post by 11234)
But we don't know the points are harmonic isn't that the consequence of the identity. I'm confused on how they got the identity because it would give the reciprocal but how do we know they are equal
Well it depends which four points you're talking about. The conclusion is that ABCD are harmonically separated. But it's an early claim of the proof that C_1 D_1 A A* are harmonically separated.

That follows from the (C1,D1;A,A*) = (C1,D1;A*A) = 1/(C1,D1;A,A*) and the fact that the cross ratio cannot equal 1, so it equals -1.

Were those the points you were referring to?
18. (Original post by RichE)
Well it depends which four points you're talking about. The conclusion is that ABCD are harmonically separated. But it's an early claim of the proof that C_1 D_1 A A* are harmonically separated.

That follows from the (C1,D1;A,A*) = (C1,D1;A*A) = 1/(C1,D1;A,A*) and the fact that the cross ratio cannot equal 1, so it equals -1.

Were those the points you were referring to?
yea thanks

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