Homogeneous points & coordinates

[Severian596]

Hello all! I'm taking a look at Meserve's "Fundamental Concepts of Geometry" for an introduction to the world of geometry. Section 1-7, A geometry of number triples completely confunded me. It introduced homogeneous points, and in my opinion it either a) did a horrible job, or b) did a horrible job explaining the significance of of these things. Furthermore after reading section 1-7 at least 5 times and still not really understanding what the point of these strange points are, they have not been mentioned in the subsequent sections yet.

Mathworld's page on the topic here didn't help me out, either.

In short I'm curious what the heck these points are all about. The pages seem to say that for a given triplet $$(x_{1}, x_{2}, x_{3})$$ any other point which is a multiple of it, for example $$(kx_{1}, kx_{2}, kx_{3})$$, is actually the same point. Furthermore it says that there is no triplet (0,0,0), and that usually $$k = 1/x_{3}$$. This looks like smashing a 3D coordinate system in (x,y,z) into a plane at (x,y,1)...

Why? What for? What's the point of starting with an (x,y,z) and "smashing" to z=1? How do you start with the triplet? What does this triplet mean?

Any help is very much appreciated, I feel like I'm close but still far enough away that I'm completely confounded about homogeneous points' purpose or use.

Thanks!

-sev

 

[matt grime]

Try looking for other resources on projective space, which is what I'd usually call this topic.

Projective space is very important in mathematics, and mathematical physics, though it's often disguised by looking at varieties and quasi projective varieties (or schemes, coherent schemes and quasi-coherent schemes if you want to really use a sledgehammer to crack a nut).

The idea is that rather than look at the space C^n (C for complex numbers), of n'tuples, you can look at the coordinate ring C[x_1,...,x_n] of complex polynomials in n commuting variables., which you can think of as defining functions from C^n to C. A set of points can then be described as the set of zeroes of some set of polynomials, and a single point (z_1,...z_n) corresponds to the polynomal (x_1-z_1)...(x_n-z_n), which defines a maximal ideal in the coordinate ring., and so points in C^n are the same as maximal ideals in a commutative ring.

There is an important variant of this, and that is the space defined by the homogeneous ring of functions in n variables.

How we doing so far?

There is a useful model of this if we take C^n and remove the point at the origin and define an equivalence relation on all the other remaining points: two points are equivalent iff they are a complex multiple of each other. So imagine actually we're doign this for R^2, then omit the origin, draw a line through the origin, all points on that 'ray' become identified, that's the squashing down part.

For the 3-d thing you've got in the post, we can't actual properly identify this space with (x,y,1) because there are points such as (1,2,0) and (1,1,0) which aren't in that set, and those points aren't equivalent either.

However, if you fix the third coordinate at 1 then that subspace is the same as R^2, which is fairly interesting, and you see that fixing another coordinate at 1 is another copy of R^2 and that real projective 3 space is a few of copies of R^2 glued together in a funny way.

 

[Severian596]

Hi Matt! Thank you very much for your reply. Strangely enough I am completely lost in your post up until the remark "How are we doing so far?" At that point I begin understanding part of what you said. After the body of my post here I'm going to include a verbatim transcription of the section in the book I'm reading. I'm still not sure if I'm completely over my head, or the language involved is the barrier. I'm not a math major so much of the terminology you use is archaic and heavy, but not impossible for me to look up. The language in the book I'm reading is not heavy at all...it just doesn't explain anything! Well, without further adeu I'll post the verbatim, with some comments on what you wrote following the transcription:

"Fundamental Concepts of Geometry" by Bruce E. Meserve,
ISBN 0-486-63415-9
Copyright 1955 and 1983

(this is the FIRST chapter, and the sections leading up to this one discuss reasoning and postulates...section 1-7 seems independent and completely off topic to me)

[Begin transcription]
1-7 A geometry of number triples. We are accustomed to representing points on a plane by pairs of real numbers (x,y) and lines by equations of the for ax + by + c = 0. These representations will be obtained after coordinates are introduced in Section 3-7:Nonhomogeneous Coordinates. For the present, we may dismiss visual geometric concepts and consider points abstractly as number triples $$(x_{1}, x_{2}, x_{3})$$, subject to the following two conditions:

(i) $$(kx_{1}, kx_{2}, kx_{3})= (x_{1}, x_{2}, x_{3})$$ when k != 0
(ii) there is no point corresponding to (0,0,0)

When $$x_3$$ not equal to 0, we may choose $$k=\frac{1}{x_{3}}$$ and associate with each point a number triple (x,y,1) where $$x = \frac{x_{1}}{x_{3}}$$ and $$y = \frac{x_{2}}{x_{3}}$$. Then for real values of the coordinates x, y we have, essentially, the points (x,y) on an ordinary plane. The two representations (x,y) and $$(x_{1}, x_{2}, x_{3})$$ of the points on an ordinary plane are called, respectively, nonhomogeneous and homogeneous coordinates of the points. When $$x_3=0$$, there exist number triples (points) that cannot be represented in the form (x,y,1) and therefore cannot be represented using nonhomogeneous coordinates. For real values of the coordinates, the totality of points $$(x_{1}, x_{2}, x_{3})$$ will be called the real projective plane (Chapter 4).
[End transcription]

He goes on to define the equations of a line in homogeneous points and the symbolism for them, but because I'm not following the concenpt I'll not print them here. I'm confused about whether I'm "too confined" to the cartesian system to properly visualize the point of these "points." How does representing a point as a triplet "dismiss visual geometric concepts and consider points abstractly," as Meserve says.

I'm sorry and I hope you don't feel like your time is wasted, Matt. I want to start off on the right footing but I'm simply swimming in this concept and can't touch bottom, and I'm afraid your approach of trying to explain to me that "[the polynomial] defines a maximal ideal in the coordinate ring, and so points in C^n are the same as maximal ideals in a commutative ring," does absolutely nothing for me! I have not learned the terminology and therefore am thinking that

1) I should not be learning homogeneous coordinate systems right now, and
2) Placing a discussion of the homogeneous coordante system in the FIRST CHAPTER of this geometry book was a complete mistake. He does not explain it well...

What are your thoughts? Matt? Anyone?

Thanks again,
-sev

 

[matt grime]

The initial bit you didn't follow was supposed to be the 'correct setting' in which to think about these things. Or at least, part of an answer as to the "what is this used for?" bit.


What he's saying is that R^2 is a subspace of real projective plane.

He's attempting to get you to think about geometry by equations, not visualizing pictures of planes and so on.

It is possible to visualize the real projective line because it is a circle, but not the higher ones.

I think the problem is that you've got hold of a book on geometry thinking geometry is like drawing triangles and using trig. It isn't; geometry is one of the most misleading titles for the uninitiated. Geometry is these days algebraic geometry where it is the coordinate ring that is important not the picture you can draw on the page. Geometry now is not really the ideas of Euclid's elements.


I don't think you want to use the word archaic. What you appear to want geometry to be is the archaic and slightly quaint thing mathematically.

I think that as you're not a math major then you can safely try different books. If you want things like the locus of points equidistant from a point and directrix is a parabola then you need to avoid all books that deal with (algebraic) geometry. Otherwise you need to start with a good grounding in commutative algebra and in particular ideals, maximal and prime, polynomial rings especially.

 

[Severian596]

The initial bit you didn't follow was supposed to be the 'correct setting' in which to think about these things. Or at least, part of an answer as to the "what is this used for?" bit.


What he's saying is that R^2 is a subspace of real projective plane.

He's attempting to get you to think about geometry by equations, not visualizing pictures of planes and so on.

It is possible to visualize the real projective line because it is a circle, but not the higher ones.

I think the problem is that you've got hold of a book on geometry thinking geometry is like drawing triangles and using trig. It isn't; geometry is one of the most misleading titles for the uninitiated. Geometry is these days algebraic geometry where it is the coordinate ring that is important not the picture you can draw on the page. Geometry now is not really the ideas of Euclid's elements.


I don't think you want to use the word archaic. What you appear to want geometry to be is the archaic and slightly quaint thing mathematically.

I think that as you're not a math major then you can safely try different books. If you want things like the locus of points equidistant from a point and directrix is a parabola then you need to avoid all books that deal with (algebraic) geometry. Otherwise you need to start with a good grounding in commutative algebra and in particular ideals, maximal and prime, polynomial rings especially.

Thanks again, Matt. I picked up this (and other) books on the topic of Geometry because I discovered just that; that the topic is only barely approached with drawing triangles on a page. The idea that 3 space is one kind of space and that there are others, almost as many as you can define (as long as you set up rules for the space and the components of the space are consistent) is, put simply, pretty damn awesome. Thus I began with a book entitled the fundamentals of geometry, fully expecting it to introduce me to a the topic in full, in all its glory and/or terror.

At this point I don't feel overwhelmed, just improperly conditioned. I'm conditioned with Cartesian coordinates and calculus that works with either degrees or radians. I feel unprepared for this surprise section 1-7 only because it jumps quite far from the preceeding topics and does not assist enough in nailing down the fundamentals of points systems in general. Heck even saying, "There are other ways of defining points in space than (x,y,z) and here's why you would do that," would've helped.

I'm interested in becoming a math major in the future. For now I wanted to see if geometry was as interesting of a concept as I anticipate, because studying relativity and the nature of space, and then finally reading about Reimann and Gauss and their noneuclidian spaces, REALLY picked my interest. I will take a look at your first post again and pursue the terminology as far as the definitions take me. Just because I haven't seen them yet doesn't mean I can't learn them.

I was hoping for something less involved, but if involvement is required (as you've stated) and I need to be familiar with the terms you used then so be it. I'll do that. I just think they're terribly misplaced in his book unless there's some prerequisits that aren't spelled out in his publication. Because the book doesn't supply the phrases such as "ring of complex polynomials in n commuting variables." ;-)

I'll post a follow-up once I research what you were kind enough to write for me. Hopefully it will clear up this topic and something will "click"

-sev

 

[Severian596]

Finding this site was almost fate:
http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node6.html

It has an illustration (Figure 1) that reflects exactly what I had imagined when reading about homogeneous coordinates. It was quite a godsend to see a picture of the concept I had only imagined up to that point.