Projective geometry | Math History | NJ Wildberger
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- Опубліковано 19 чер 2024
- Projective geometry began with the work of Pappus, but was developed primarily by Desargues, with an important contribution by Pascal. Projective geometry is the geometry of the straightedge, and it is the simplest and most fundamental geometry. We describe the important insights of the 19th century geometers that connected the subject to 3 dimensional space.
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In the 19th century it slowly became clearer that most other geometries (Euclidean, spherical, hyperbolic, inversive) can be built from projective geometry. It is also the main framework for modern algebraic geometry, which grew out of it.
It's amazing
i have been an architect, a programming instructor that was interested in computer graphics, at no point in my career and education did i receive clear understanding of these topics, nobody understood proj geom, homogeneous coord, matrix transform, even perspective, ending up with rote dissemination and applications of these laws, equations and programs etc. In this very brief lecture, you managed to thoroughly illuminate me, so rather belatedly, kudos from a retired student！
I taught myself descriptive geometry with only an innate understanding of projective geometry.
@@4DMovie You are a very clever boy then . You can have a badge.
A student is never retired 😂. Just Joking
Thank you sir for such useful advice when captured by aliens
One of the most profound lectures of all time. The understanding of art, maths and perspective, extremely humbling.
I was reading a short story by Haruki Murakami where a character puzzled about "the circle with many centers and no circumference." I later thought it could be thought of as the line at infinity. Indeed, when I did a google search, I see results return about "the infinite sphere with center everywhere and circumference nowhere" which was a phrase attributed to Pascal. I'm more or less convinced that Pascal was talking about the line at infinity when he used that phrase. The non-orientability of projective plane puzzled me at first when I read it but the way you explained it makes it clear to me how the points at infinity loop around each other.
at 37:37 the fuse is carefully lit and it blows my mind by the end of Professor Wildberger's lecture...just a hyperbola, so to speak😮❤
Great video. Used the begining of it as an introduction to perpective drawing in a high school class going on a trip to Rome.
A reference for those of you who are interested in digging a bit more in this matter: N.J. Hitchin "Linear Geometry", Oxford 1987. Hitchin explains how the projective geometry can be considered using linear algebra (matrices and stuff). I used the paper for my Bachelors project back in 1992 :-)
@EmanT777 Yes projective geometry is indeed a unifying framework for other geometries. This is not properly appreciated these days, one of the reasons students often miss out on this important geometry. Projective geometry and Mobius geometry are also closely related. I will discuss such topics in my Universal Hyperbolic Geometry series.
@madier1000 You might like to know that in my WildTrig series there is a 8-10 part segment on projective geometry, if you are particularly interested in that topic.
I'm reading Multiple View Geometry in Computer Vision, and this was very helpful. Thanks!
I feel like I gained a new perspective in my life. I can't thank you enough for this clear explanation of the topic Professor
Thank you very much for posting these online, Professor! I'm watching them all and I have to say this particular lecture simply blew my mind. I wish I had been introduced to these concepts earlier.
Thank you, thank you, thank you Prof. Wildberger for thinking to put these videos up. I am revising after many years out for a CS PhD studying impossible objects. I need a deep understanding of topology and projective geometry and your lecture series is a fantastic start.
The study of imposable objects must start with a read of "Fundamentals of Three-Dimensional Descriptive Geometry" and "Four-Dimensional Descriptive Geometry" by Steve M. Slaby and C. Ernesto S. Lindgren.
You sir saved the day, i am currently studying computer vision and your examples made these ideas clearer to me. Have a nice day/(or night)
I learned something from this lecture, thank you. I especially like the way you narrate the theorem that may sound very abstractive to the laymen of this field, great.
We have gone down the rabbit hole, Dr Wildberger
N. J Wildberger. Me acaba de cambiar la vida. Increíble sus videos. Explicaciones geniales.
Thanks for these great lectures! I've seen homogenous coordinates in Computer Graphics (makes translations a lot easier by making it possible to work with Matrices for chaining them and so on) but I didn't know that their roots go back to the 18th century and they have something to do with homogenous functions. Probably Homogenous ODEs are also named homogenous for the very same reason (right hand side is homogenous in all variables).
Epic! This video should be the first ingredients for persons like me, who have never come across projective planes before. Nice work!
Great explanations- I have always wanted to, but never understood this til now so thank you very much!
This is AMAZING! Thank you so much for these, Sir!
I enjoyed this lecture very much and look forward to the whole serie.
is it weird that I'm just captivated by the coolness of this guy? the dude is just confident and well dressed and smart, and like a cocky cool guy.
The alien metaphor is so great, it makes me very happy! Thank you.
Good use of coloured chalk. Makes things a lot clearer than teachers who stick to one colour. Thank you.
This lecture really infuriates me.I suggested to a young friend of mine who just did a Doctorate in physics and neurology that he should take a look at Projective Geometry to expand his researches.He told me tonight that he couldn't get a handel on it.I couldn't understand why he couldn't get it until I came across your mutilation of Geometrical Beauty.
Excectly how stupid are you that you are completely oblivious that you are butchering the beauty of Projective Geometry.
Fascinating stuff! Keep up the good work Professor Wildberger! Really enjoy your videos.
Thanks!
Finished this one. Some questions asked previously remain open, hopefully can be answered by later lectures. Thanks professor.
Thanks to you I'm learning something interesting while improving my English listening.
I’m half into this and, as always with your other lectures in this series, found it very educative and interesting. One feeling however makes me inclined to believe the legitimacy of infinity, out of instinct, whereas before this point I used to be joining you in doubting this because of its seemingly logical flaw - with a fictional line of infinity in the projective plane, the system of ideas in projective geometry (with a perfect symmetry between lines and points) seems complete and intuitively sound, it just looks beautiful without a proof.
Just holding my thought and impression here but will wait and see what happens down the journey of mathematics along with you.
Thank you professor.
Btw, my way of thinking might not sound very logical but I’m a strong believer that, because the beauty of mathematics somehow describes and reflects the beauty of the nature, discoveries of its secrets are more likely to be made following instincts.
this guy is an amazing teacher!!
Thank you very much! This was quite easy to understand, and the thought experiment with the parabola was very helpful.
I particularly like your advice on the best way of convincing an ET you are an intelligent person. Brilliant!
Excellent lecture. I especially admire your ability to accurately diagram on black board.
This has been very helpful... I'm watching this in 2022 still very fascinating thank you for this information keep up the good work.
I am so enjoying this. 💕. Thankyou.
Slight correction at 12:20. It should read "triangle [a1,b1,c1] perspective with triangle [a2,b2,c2]".
I am gaining so many new insights about Math and its history from your lecture series! Thankyou for posting them.
Excellent lecture, thank you so much
Thank you very much ! This video is very useful !
Awesome intro. Loved it! ❤
Professor Wildberger, this is a lovely talk. Fascinating! And very helpful for one like me to whom it gives a perspective he does not get from his textbook. I do not see why you say the projective plane is non-orientable at time t=55:50, but I will try and figure that out.
Thank you for sharing this amazing lecture; very useful.
lovely video and quite helpful for our teaching staff
minutes 37 to 39 will take some effort. but once u understand, u get a grt feeling ..... thks Professor, u r the BEST
This is excellent!
Thank you for this lesson! It's really helpful and you presented it so well! I do really appreciate it! I found CG community use 4x4 matrix to do affine transformation, but few teachers do explain the reason this specifically.
I watched this video whole day and take my note with some pictures I made in Rhino.
Here's the note: drive.google.com/file/d/1svSKEk4jApfo_x35fO5H6CffYzVeRUDE/view?usp=sharing
Thank you to make this quality lecture!! THANKS
Thanks for the nice comment. You put unite a lot of work into the Notes you made, and they look great! Well done. If you don’t mind, perhaps I could link to your notes in the video description? That way other viewers can also benefit. If OK, please also give me your (English) name so I can attribute.
@@njwildberger Thanks!! My name is Jim Yuan.
@@user-hn1hf3rw9n Thanks Jim, I have now posted the link to your notes in the video description.
Awesome. Thanks!
Thank you so much for this! :)
Thanks!
very helpful!
There is a beautiful perspective mural in Pompei. The Renaissance artists rediscovered what was known to the Romans.
simple but very nice illustration of projective and perspectives. in your lecture you spoke of "line at infinity", but you sir don't agree with "infinity"? just kidding. very helpful lecture. thank you!
Great explanation
Around 30 minutes, if every two points are connected by a line, then what line connects the points on the line at infinity? And if you took a parallel line to said line, where would they meet? Sorry if this question gets answered later in the video! On more question, does your WildTrig videos cover 19th century work or does it stick to the 1600s?
Brilliant channel by the way.
Wonderful. Fun fact, Desargues and Descartes were friends! They wrote to each other. As the prof says, Desargues' ideas were not entirely lost. In my addled imagination they are a kind of subversive undercurrent to the main development of practical mathematics via Newton, Euler et al. In the nineteenth century, curved spaces came in by abandoning Euclid's 5th postulate (parallel lines don't meet). But if you abandon 3 and 4 (effectively disowning distances and angles) and change 5 to "any two distinct lines meet in a single point (with no exceptions)" you get projective geometry.
The notation at around 12 min in Desargues thm: should it be the two triangels a1b1c1 and a2b2c2?
I think so.
thanks for the videos, very helpful. where did this lecture take place?
Two points at infinity are connected by the line at infinity. This is the one line we need to add to the existing line to go from the affine to the projective plane.
As for the WildTrig series of videos on Rational Trigonometry, that is 21st century mathematics all the way! But still with its origins in the work and thinking of the ancient Greeks.
Does this surface behave similarly to being on a sphere and the poles are the horizon?
The word Homogeneous which we call it همگن(ham-goon) is a Persian word for sure although Google wrongly report it as a Greek word. It has to parts. The prefix 'ham-' which means the same and the noun 'Goon' which means kind
Thank You You're Amazing!!! I'm a high schooler and I have a presentation tomorrow and you definitely saved me
Thank you!
I had 1 month of projective geometry in my linear algebra class
49:00 Since we see in 2D , 3th D can be looked as function of Time (passing).
Desargues' "points of infinity" is perhaps before the concept of non-euclidian space.
If there are assumptions, they should be defined.
Thank you very much
Excellent, Wildberger. At 45', it might be slightly better to assert P = R union, not R plus, infinity.
Very interesting video.
I don't understand at 45.30 : why we meet the same point at infinity in the 2 directions ?
why projective line is a circle ? Because for me a circle is not infinite it has values between -R and R.
Also : what does it mean : perspective via a line L ?
Thank u for the video
Hi!!! can you explain the development of Fourier series, transform and Laplace transform using just geometry please ? Thank you.
Here I'm a ug student of physics , wasn't able to understand physics deeply so started differential geometry now that this lecture was suggested ...so I can literally feel those 19th century's mathematicians. Without deeper knowledge of mathematics, I dunno how people do physics !
Thanks a lot for that great tour of the projective realm!
Had a question at about 1:04. Will the projection of the parabola in Z = 1 on the sphere be a circle instead of an ellipse? Seems it will always be a circle on the sphere for any conic in Z = 1?
It seems so from the picture. However, it is not true. The parabola can be arbitrarily thin, and then its projection onto the sphere will also be thin (and a circle cannot be thin, of course).
For your example at 51.15 : why the projective line is y=1 in particulary and not y=5 ? It is the same ?
41:05 How about if we draw the lines y=x and y=-x one the view of perspective, would they still appear straight lines? Seems not, so this contradicts with the previous assumption that a straight line will still be a straight line?
Very good.
One trivial point as there are streams of student coming in what happen to them after a decade I wonder.
if we drew two parallel lines W and Y which are 0.5 (1/2) apart and a line Z cuts perpendicular to the two of them at points a and b, does point c at infinity where projective geometry purports W meets Y complete a triangle with two right angles and angle ∆=0 at c?
This is pretty good.
Congratulation Teacher. The lecture very good.
I'm glad you like it.
Who's the composer for the music at the beginning?
Do 2 parallel lines meet twice, at two different and opposed infinities?
you look like Steven Martin (cheaper by the dozen) , anyway I learnt a lot , thak you , this is an oustanding courses you made, clear and straight to the point (line, plane ... )
Nice pun.
thank you sir
Is it necessary to know how to plot irrational numbers on a number line.
We also write AB ∩ CD =E, when line AB intersect line CD at point E.
I love his style. Very easy to follow. I subscribed and will follow lectures as I'm finishing up my BFA. Thank you Sir.
Thankyou
48:28 Shouldn't the projective plane be a 2-dim subspace, not 1-dim?
38:50 Fascinating
31:03 is like if that "flat plane" is in a sphere? And the line at infinity is the visible border of the sphere?
😅ok now i saw the hole video
Anyone out there happen to know of any established mathematical ideas wherein infinity coincides with the infinitesimal? or zero?
In so many ways, infinity and zero are very much alike, and the way that negative and positive infinity coincide in Projective Geometry is rather similar to this idea.
Would infinity and zero coincide if we were to do the same thing with only the non-negative (strictly positive) side of a number line? i.e. from zero to infinity?
52:40 Why does the line need to be at y=1 and not another value?
thank you
I suppose any nonzero value would work, for example z=2. Taking z=1 probably just makes computations (like homogeneous coordinates) easier. You could also think at 1 as being your "unit" distance beteween your eye and the plane where you project figures.
Hi 172Break This year there will be 12. But next year I hope to add some more.
I have a few questions about this, but I would really like to understand it better, let me know if you offer tutoring or teaching private rates. I would really love to understand the whole concept better.
@Kevin Matson, I am too busy unfortunately to offer private lessons at this point: but I have quite a few other videos that explain aspects of projective geometry further. Start with the Wild Trig series with ua-cam.com/video/fX7IOxuIjKY/v-deo.html and following. Then you can have a look at the Universal Hyperbolic Geometry playlist with more advanced material: see for example ua-cam.com/video/EvP8VtyhzXs/v-deo.html
Great lecture. I have one question:
At around 29 min, the c2 c1 line was not drawn parallel to the a1 b2 and a2 b1, but then you explain that all parallel lines meet at only one point in infinity, so if the rule works both ways (if only the parallel lines meet at the some particular point at inf.), it seems that all 3 lines meet at c3 and they should be parallel?
True. This drawing is not perfect :p
Why is it that each family of parallel lines meet at infinity yet not at minus infinity? Or to put it another way, why does the family of lines not meet at the antipolar point of the declared intersection?
Thank you for your time.
In projective geometry, there is only one point ``at infinity'' on a line. In other words, the two points infinity and minus infinity coincide.
njwildberger An example for your OA: great circles on an arbitrarily-large sphere meet twice but both meets are arbitrarily far away; the pair of meets correspond morphologically to the single meet of non-parallel lines on Euclid’s plane. Then again, I don’t remember a demonstration of Pappus’ theorem on the surface of a sphere. Must do homework!
1h 5m - Intersection of a non-circular cone with a sphere, can't be an actual ellipse, because a (non-circular) ellipse can't lie on a sphere and be planar.
It is, however, ellipse-like. Of course, if the cone is circular, the intersection is an ellipse, but one that is a circle.
Wondering about that. The 3d points (0,1,0) and (0,0,1) are on that ellipse/circle, regardless if the original parabola is y=x^2 or y=m x^2 for some positive m. Changing m should change the ellipse but cannot change the circle. I guess it is an "ellipse-like" think on the sphere. Norman?!
1:07:50
I think the absence of Projective Geometry from the curriculum would be a minor detail for this hypothetical time traveler.
If a Uni student from the 1800s saw Unis today he'd be depressed to see that other than marginal improvements in the quality of the chalk, higher education is ridiculously anachronistic and has gone backwards in more respects than it has improved.
It really boggles the mind.
Unis today resemble medieval institutions from the mid 1600s. in the 1800s and early 1900s education was central - as opposed to publishing - there were fewer students per class and the approach was a lot more interactive and hands-on. There was a constant dialogue student-teacher.
Reading about the Uni experiences of Planck and Einstein, it's just amazing how much better they were than "modern"* Uni (*in reality we are in the postmodern era, they were in the modern era). It's like Universities were on a race to self-destruct. Not even going into the completely wasted, superficial, merely skeuomorphic use of widely available technology. The 1800s person would be dumbfounded about how could they possibly devalue higher ed this much with so much more and better resources.
35:21 is that the meet between psicology and math?
Are they teaching this way at the University? In the USA? God, this class could have been given 80 years ago.
Elliptic curve cryptography makes a lot more sense now.