Introduction to Projective Geometry via Tic-Tac-Toe Grids 

Very nice! The cross-ratio is one of my favorite less well-known topics. With the Mobius transformation view, we can see that the cross-ratio of four complex numbers is real if and only if they lie on the same line or circle.

when i was younger, every time i tried to get some bit of projective geometry, it was just a mysterious mess to me. well, maybe, i could remember something, do formulas, make some constructions, but the essence was still unaccessible. now, when some maths intuition settled in my head, i am starting to rediscover a lot of mathematical concepts, so videos like this are very useful not for getting some rigid knowledge (you may always read a textbook or a Wiki page for this purpose) but learn some motivation of a concept, get the intuition, highlight the connections with other math stuff and well, really understand what's going on. so i'd like to thank a lot the author for the video, it really uncovered some bits of information needed to order my previous knowledge. thank you and good luck in developing the channel!

The mathematics, explanations, and animations are beautiful, but the background music is too loud and drowns out your voice for me :(

Really nice video. I've just made my final bachelors degree project about projective geometry and used de cross-ratio (which i knew as double ratio) to make an euclidean clasification of the conics.

Great Video! The way of looking at the harmonic construction as a projective parallelogram was really nice, I wish we would have been shown that in our geometry lecture!

This was absolutely fascinating - great job!

I'm really happy you decided to share this video as it was very clearly presented, well animated and well paced. Love the subject and how you made it easy to follow your train of thought by using (wonderful) animations. Great job!!

This was a pretty fun watch, thank you

Shocked me when this channel only had 1k subs. This is a gold mine. Keep it up!

AMAZING video!!! thank you so much!!!

Amazing video!! Love it!!

BEAUTIFUL! I can finally get a proper introduction to projection geometry without the need to look up all different books on the internet! Nice!

This is a cool topic, thank you!

Both explanation and visualization are just perfect. Thank you!

Very good exposition. Really really clear, I thoroughly enjoyed it!

The channel looks brand new. Best of luck. It’s gonna grow! Mark my words

Really, really nice! And what a great starter for any lecture about Projective Geometry. If all maths and maths ed students just knew what you showed in the video, the world would be a better one!

This video came as an suggestion. It's a good video for understanding, as I find it's your first video.. I am hoping you do more of this🥳 love from Nepal 🇳🇵

Nice video. It reminded me of some topics we talked about in the geometry class I was in while working on my master's degree.

Great video! 🔥 Good luck in peer-review!

I was considering doing a video entry on projective geometry after listening to Prof. Richter Geberts great lecture, I but scrapped it due to time issues. What a coincidendce now to see you make this wonderful entry :)

3:13 This moment is brilliant and enlightening

I was following along with the progress on this video on the SoME2 discord! I'm so glad you were able to get it out before the deadline. Awesome stuff :).

Fantastic Video!

Please do more videos, your explanations are very clear, great job

I recall from a drafting & architecture class in high school that the formula for measuring along the axis in perspective was something simple involving the cosine function. I suppose that was an approximation. When I took computer graphics in college, I recall the extra dimension allowing perspective transformations, and that moving a parameter changed the perspective; but I don't recall any formula for how that relates to the vanishing point.

When I did a perspective drawing in high school, I tried to make a street with many (a dozen or so) buildings of equal width. I used the Fibonacci sequence to measure out their respective widths and it looked good enough. I now wonder how it aligned to this reality of math.

thanks ; very interesting (sehr angenehm :) )

Amazing

This video helped me figure out some constructions in Pythagorea

I genuinely thought this was one of those professional videos made by big math-stuff content creators, you did a great job! This was a very friendly-speaking informative video and I think you narrated everything pretty well! I'm not a native english speaker but I didn't really think you had an accent hahaha, your english is great! :)

Where are your subscribers? This video was very well made.

Cool animating style

I so wish I had found this video 2 or 3 years ago when I kept googling for this exact problem 😅

I like your style

Bellissimo video! Grazie. Io sto scrivendo un libro che riguarda il birapporto armonico con svariate costruzioni grafiche. Mi fa piacere vedere che nel mondo ci sono altre persone che studiano e divulgano la Geometria Proiettiva! Quando finirò il libro avrò piacere di condividerlo.

Cool!

Your accent is beautiful

Oh man! What a treat! How I wish you had been around in 1985 when I was doing 3-point perspective in technical drawing class. This is such a clear and thoughtful exposition. How long did it take you to put together??

Very good video! 15:38 should the right side yellow equation be (1, inf | 2, 0)?

Yea, I use all of this when drafting and/or illustrating all the time. This is core artisan measurement and partitioning. Thing is, I was forced to figure it out by myself. I ended up creating a chart of 'natural' divisions with this method (from corner to centerline) - with a really weird holdout of 3/11 which is more likely an exotic number stuck in between 1/3 and 1/5. There are some other exotic ones out there ( like 4/7 ), but not until you get to the point this method is no longer useful. You will also stumble over the link between music theory, resonance, and design theory just sitting in the middle of all of this - as these natural divisions are how the old artisans basically did everything - and it will also make things like the "imperial" measurement system make a WHOLE lot more sense and base 12 numerical systems in general. Copy all of that on top of Babylonian 60-divison clocks and how that translated into its own unique measurement system and - boom - you got the background for so much of our technological progress. I love this stuff, and its nice to see other people talking about it. I have a whole "Artisan Cheat Sheet" that I post regularly for other artists/craftsmen to use as it can be really hard to find this information collected or easily digested - as computers pretty much killed this whole way of thinking.

Everything is clearly explained @sumandproduct, great video, congratulations! I just wonder which software did you use for the figures and animations? Thank you

Awesome video! What tools have you used to create it? It's of very high quality.

Great Video. Would love to see more like this from you. Btw what was that geogebra while editing the ruler's pic? can u gimme ur version please.

great video, but we're a bit confused about the start of the "school method" section... how do you equally divide the auxiliary line in a way that doesn't also work directly on the original line?

3:05 This is correct if one assumes that there is no atmosphere or its refraction is zero. If you include the atmospheric refraction, the (horizontal) vanishing points should be a bit further above the horizon, right?

I have a 2-Point Perspective Drawing that I'm working on for my Art Class and I've determined the position of the Cube's Bottom-Left Corner to be at 1937 units away from the Closest Edge of the Cube from the Viewer. When using a Ruler to measure 0 to 1937, I get Roughly 5.4cm. I want to find the Position of 4527 units from 0. How would I do this?

Very nice. 09:25 doesn't how well this work with a photo depend on how much distortion is added by the camera? wiki: Distortion_(optics)

Nice tricks I finally understood some things that were exposed to me in art class, but not in an enough mathematical way for me. Btw, i got some difficulties understanding your speach. As a non nativue english speaker, I needed all my attention to graps the sounds you were saying. I don't have this problem with most english videos, and I think it is due to 2 things. 1. The music The music is fuzzy (not sure what that mean, but dense and chaotical) and loud. For you, used to all kind of parasite noise in english, no problem But I need all the details of the speech to understand what's going on. With a little of noise, your brain can easily recover the original sounds emitted, byt my untrained brain can used those redundancy rules. 2. The micro I can't know, but it seems that your sound is a little bith saturated. Maybe it's just that I learn english with a different accent then yours Well Thank you for the vid, I hope I could help

Reading the description, I think you've exaggerated the audio quality/accent "problems" quite a lot. I've heard a lot worse in both of these regards, and I didn't really find the audio or accent disturbing while watching. The only audio nitpick I have is the abrupt ending, I think you could've let the music ring out or fade out a bit at the end without it affecting the video negatively in any way. In either case, it was a nice video, good job!

This is great. Can you tell what you use to create these nice animations and the text morphing animations?

Loved it 9.5, not a perfect 10 due to loud music

Ho guardato con maggiore attenzione. Al minuto 12:51 dici che, dati tre punti A, B e C, non è possibile costruire il quarto punto D del doppio rapporto (ABCD). Invece è possibile! Se vuoi ti mando la costruzione.

instead of using -1 and +1, it would probably be better to use a fictitious number ω and its negative, to avoid confusion with an "actual" 1.

First.

reversibility does not imply surjectivity.

voice over your lecture is somewhat distracting, otherwise thanks.

You really need to work on your didactic skills, talking to an empty screen and such :-)