Projective geometry and homogeneous coordinates | WildTrig: Intro to Rational Trigonometry

[@njwildberger]

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

[@muhammadadnan0075]

Holy moly the only video on KZbin that explains it clearly. KZbin needs to correct it's video recommendation algorithm. I came to this channel after very deep linking videos. I wish i had found this early. Thanks Professor.

[@WildEggmathematicscourses]

Hi Mak, Glad to have you here. There are lots more videos on this channel: I am attempting to not only renovate mathematics education, but also pure mathematics research too.

[@fhydan]

Mr Wildberger. You are my new hero. Sometimes reading math books can be mind bending to a hobbyist like myself and I really need to look for a more visual explanation like what you present.

[@oneplusonetwo]

Thank you very much! I watched a lot of videos on youtube, and this is the only one that makes me understand homogeneous coordinates.

[@elultimopujilense]

exactly my thoughts! i have made many CV projects with homogeneous coordinates, read many books and articles, but non of them showed me how these coordinates work and why they are important. They assume you know this stuff, I thought i was dumb for not getting it, but now i know that most people in the field of CV dont have an idea of how these coordinates work and they dont have enough time or curiosity to understand this.

[@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.

[@troooooper100]

Only source on internet that explains it!!!!!

[@elultimopujilense]

Exactly! I have searched everywhere, I made projects, and nowhere i could find how these work. Incredible.

[@sundaranarasimhan58]

Prof.Njwildberger .Thank you for your selfless help to students around the globe by explaining these beautiful ideas in a very nice manner.

[@bboysil]

This is the best explanation of homogeneous coordinates I have ever seen.

[@ChristopherBreeden85]

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

[@Urza26]

Hey... I'd like to personally thank you very much for such an excellent video. Perhaps one of the only, if not the only one on the net that explains it well.

[@chrischoir3594]

Brilliant explanation, but more importantly clear and concise, thank you !!!!

[@LAnonHubbard]

The penny's starting to drop now. Thanks for making these videos. You currently pretty much have the market covered on KZbin for homogeneous coordinates which is surprising given their utility.

[@Metricspaces]

This is explained in a very structured and easy to follow way, great video!

[@martinzen]

Very clear explanation. I'm in the middle of implementing NURBS curves so I totally needed this. Thanks a lot!

[@wilismatrix9847]

Thank you for the lesson !! This laws are used in 3d application to represent 3d models and project textures on them ! this is why there is always a bounding box for each mesh in 3d applications and a world position.

[@owen7185]

The most interesting math videos are on your channel. Awesome

[@carolinaarnosti866]

Thank you very much! I'm from Brazil and you helped me a lot!

[@elultimopujilense]

HOLY SHIT! I am so happy! I felt dumb when doing CV projects because I didnt got this concept, but its so simple, yet so elegant and genius. Thank you sooo much!!!

[@Jianju69]

Ok, I just spent about two hours reading about abstract algebra, group theory, homomorphisms, and so forth, finally getting to a point where I thought, "Ok, I think I understand projective geometry." Now, after spending seven minutes here, I'm wondering why I didn't come here first. Great lecture! Heading off to the next one...

[@ChrisDjangoConcerts]

I think this projective stuff is very beautiful and I would have loved to have learned this in high school and college. Unfortunately I didn't, but fortunately I found it here

[@ashishjain871]

Thinking in terms of polar coordinates really elucidates the idea of homogeneous coordinates. Furthermore, similar triangles can be used to determine why the cartesian coordinates in the projective plane are (x/z, y/z).

[@njwildberger]

Yes they are different. A linear subspace does not have points at infinity as does a projective space.

[@senuu5932]

Thank you! Great video

[@ChrisDjangoConcerts]

The choice of z =1 determines how we collapse (project) all parallel planes. The line at infinity is then the horizon in the previous lecture. Since for each direction of parallel lines in the plane, we have 1 point at infinity (and minus infinity) the line at infinity (horizon) is more like a circle, since all directions though a point can be viewed a parametrisation of a circle.

[@weetabixharry]

I wish my undergraduate lecturer 12 years ago had explained projective geometry so clearly as this. I didn't really understand, but studied enough to get an A on the exam. Now I need to design and implement distortion rectification in a camera system, so I can't get away with being ignorant any more.

[@njwildberger]

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

[@rijulsingh9803]

I hate to comment on KZbin videos but this video here, belongs to the set of videos which are the best explanations of their respective topics/fields.

[@Андреич-с4н]

Great! First ever explanation from a commonsense view point. Other explanations are one mathematician mind talks to another mathematician mind. Definition "The point represented by a given set of homogeneous coordinates is unchanged if the coordinates are multiplied by a common factor" drives common sense crazy. We who live in 3D world cannot imagine a 4D world and therefore see no sense in it while mathematician mind feels comfortable in any N-D space. This video attempts to visualise the concept and thus making it digestible for a layman (I guess the target audience of this lesson is a layman). After this lesson I understand that homogeneous coordinates are directly linked to computer vision which tries to emulate human eye. Therefore the "orign" is my eye. If I look at the street through the window, the window glass is projection plane and if I firmly fix my had, I can draw any the object on the street on the glass with a feltpen. The curbs of a long streight street will try to meet at some point. To this point will direct all other lines parallel to the street, like, say electrical cables. And if I look at the Moon, any object between me and the Moon will look the same. And the Moon might seem of a coin size, or of a plate size or of a weel size. In this light "if the coordinates are multiplied by a common factor, the vector still represent the same point" begins making sense.

[@fhydan]

Incidentally I was trying to stare at lines between tiles down the hallway a couple of weeks ago and I tried to understand why they always look like they 'will meet but couldn't. Now it is clear.

[@oier2995]

God bless you. Really.

[@fatimaalhoti7804]

Thanks very much

[@LowLifeGraphicsProgrammer]

That intro music is INTENSE man

[@Mahmood42978]

This video is 11 years old now. Zany to think about.

[@danielordonez3246]

Amazing explanation. Educational videos must rely first on the geometrical intuition instead of a bunch of definitions of mathematical objects without an apparent real utility for the student.