WT33: Projective geometry and homogeneous coordinates

[Nameless3567]

Thanx a million Sir!

[auspicious99]

Thanks again for an excellent video! BTW, the document math.ucdavis.edu/~temple/MAT21­­D/SUPPLEMENTARY-ARTICLES/Crow­e­_History-of-Vectors.pdf that you suggested was not found on the server. Might there be a spelling error in the URL?

[njwildberger]

@auspicious99 If you Google Crowe History of Vectors you will find it.

[auspicious99]

@njwildberger thanks, found it

[breedencm]

Great! I really need a quick explanation on homogenous coordinates, and that hit the spot. Thanks!

[beardymonger]

Oh, forgot to mention i like your videos very much :-)

What i meant is we need new technology. A system that will allow revisions (a la wiki), allow commenting or selecting text from a specific location, interactive (3D) graphics (with some open format and animations), multiple heading levels (like Word's) which will allow the student to chose the level of exposition. All while keeping the level teaching as high as here on this video :-).

I don't know why, but Google seems the one to pull it

[beardymonger]

Here's an idea for you: Try to get Google to hire you to change the Math/Sci educational scene. Make it free AND high quality by using hi-tech OPEN technologies.

[njwildberger]

@beardymonger I am already trying to do that. My videos are free and hopefully of reasonable quality, and as time goes by I hope they present educationalists with an alternative to the status quo. And I already have a very pleasant job teaching at UNSW.

[ic3sk83r]

Wow... I'm just amazed at how well you describe everything! Your videos are really helping me out a lot with my project :) hehe. Thanks for all the videos! Keep it up!

[MrFunatabi]

I wonder if this idea extends to 4-D. Like every plane through the origin corresponds to a line in the [x_1, x_2, x_3, 1] space...?

[njwildberger]

Hi MrFunatabi

That is correct. And it is a very natural and important way of describing lines in 3D space, which are otherwise a bit hard to specify uniquely.

[lisp21]

quotient space

[qin0712]

Excellent! Thanks for the instruction.

[campusGraphics]

What do we mean by locating the plane at infinity? We already know it is ( 0, 0 ,0, 1) then why again we want to locate that? cold you please explain.

[noisygrass]

I saw the video several times and googled a lot, but I still don't get why and how were they invented? all explanations that I found are about transformation matrices.

why did Möbius invented them? was he trying to construct the transformation matrices? how did he found them? did he found them by accident or was he searching for them?

and who, why and how actually discovered that they are so cool for transformation matrices?

I know I ask questions like a baby but it drives me crazy ;]

[njwildberger]

Hi noisygrass,

Homogeneous coordinates are very natural for describing lines through the origin. It should not be a surprise that Moebius invented them, and I can only guess that he was just trying to understand the mathematics as best as he could. He would not have been thinking about transformation matrices., I think.

Crowe has written a History of Vector Analysis, that might have some more information on the story of matrices etc.

[samruby82]

@njwildberger fyi, a synopsis can be found here: math.ucdavis.edu/~temple/MAT21­D/SUPPLEMENTARY-ARTICLES/Crowe­_History-of-Vectors.pdf

Though the synopsis makes no mention of matrices (haven't found the book yet).

I'm sure a lot of people stumble upon these videos looking for an understanding of the math behind computer graphics. Any chance you could cover this connection between projective geometry and projection/transformation matrices in a video down the road?

[njwildberger]

@samruby82 Yes that is a good suggestion that others have made too. I will try to do that one of these days: perhaps in my Linear Algebra series (WildLinAlg)...

[samruby82]

@njwildberger excellent. looking forward to it.

[albertsCan]

This helped immensely. thank you!

[786HKHM]

many thanks.

[tibetefendi]

I've read about the homogeneous coordinates in three different books. But in this video, in 8 minutes, I've understood much moore. Thank you very much!

[mmorrell1]

I'm new to projective geometry. In fact, I have no geometry knowledge whatsoever. Could you suggest a text book that I could buy that would explain projective geometry from an elementary level? I'm also interested in videos and I'm greatly appreciative of the ones you've posted thus far on projective geometry.

[njwildberger]

Projective geometry is an old subject, and probably the best texts have been around for a while. Have a look for books on the subject by Young, Faulkner and Coxeter. Hartshorne also has a book on Foundations of Projective Geometry. All these require some geometric maturity, and of course patience.

[wetcoloredarch]

... the "jump" to argumentation that horizontal lines are points of infinity confused me. I assume, but am not sure, this is because x/0 and y/0 is indeterminate/infinity?.

More confusing is that there is a horizontal line for every family of parallel lines in a Cartesion plane.Your discussion explained points in the cartesian plane therefore I do not understand where the idea of family of parallel in cartesion plane. Are these simply the same parallel lines from your 2D introduction?

[njwildberger]

Those central lines (lines in 3D through the origin) that are not horizontal intersect the Euclidean plane z=1 in a unique point, so these correspond to `ordinary points'.

Those central lines that are horizontal correspond to `points at infinity', namely to families of parallel lines in the Euclidean (z=1) plane, where parallel means the usual thing. And the correspondence is very simple, since every such family of parallel lines is parallel to exactly one central line.

[njwildberger]

Yes, that is it exactly! As with most really important mathematical ideas, it is surprisingly simple once you get it.

[RationArtiste]

OK, I get it. The projective plane was viewed as R2 + the points at infinity. With this new construction every point of the PP is represented by a line. Horizontal lines represent points at infinity and the others represent the ordinary point that are situated where they meet the plane

z=1