What is...a projective space?

[@solveig758]

I am doing a presentation on projective geometry and tried to find the Wolfram alpha demonstration, but it is not there. How did you find the one you used? Your video was great

[@VisualMath]

It should be this one: demonstrations.wolfram.com/OnePointPerspective/ I hope that helps 😃

[@karinasakurai6599]

Nice explanation!

[@VisualMath]

I am glad that you liked it! This explanation worked for me, but that doesn’t imply that it works for everyone, so thank you for the feedback.

[@cherma11]

So, from your video I got the intuition, that I can represenr a higher dimensional space by it's projective space. Meaning R^3 can be represented by RP^2 - why? Because I have a topology on RP^2 that preserves the information of R^3 via defined mapping in RP^2 for example Vector addition. Therefore the analogy of the painting fits so well because it maps RP^2 to R^3. Meaning if I would add(?) one more cell complex to RP^2 it would allow me to switch back to R^3. 🤔 Thank you for the informative video, greetings from Germany! 👋

[@VisualMath]

Yes, RP^2 is "essentially" S^2 topologically (RP^2=S^2/antipodal points), and S^2 is R^2-0 up to scaling. So a bit is lost when going from R^3 to RP^2, but ,depending on what one wants to do, maybe not that much! Wunderbar: Gruss zurück nach Deutschland ;-)

[@mattiapiras3331]

Nice video! if I may Sir , I'd have a question, and I apologize in advance if it is trivial or stupid. This concept can be, as you mentioned, generalized in a multivariable system. If this starting nD system of figures, or for better saying, of equations, is projected in a greater dimensional space (that could be, as analogy, an hypersphere centered in the origin with radius=1) I will end up with an (n+1)D system with a new variable, let's call that 'h', and an additional equation representing the previously mentioned hypersphere (x1² + ... + xn² + h² -1 = 0). Correct me if I am wrong,so, homogenizing the system with the variable h means multiplying each monomial of every equation by h as many time as needed to have the same degree for all terms in the same equation but every equation can have different degree from the other. If I found a solution at infinity and if I computed the Jacobian of the new homogenized system in that solution (so original system but homogenized plus the equation of the hypersphere) this would make it rank deficient, if I am not wrong. So, this solution at infinity seems to be singular but I can't understand the real motivation. Do you have any thoughts about it, Sir? Do you think that it is caused by the homogenization process or is because I'm thinking inside R^n? Sorry if this concept it is not expressed in a very formal way, but after having seen your video I tried to think about it more in detail but I'm a little stuck since I'm not an expert in mathematics :) . Thanks in advance

[@VisualMath]

I am glad that you liked the video! I am not sure whether I understand the question; let me try to answer what I think you are asking. Maybe let us do the circle as a low-dimensional example. I will write (x0,x1,x2) for projective coordinates, which is the more classical notation. In any case, x0 = your h. The points at infinity are (0,x1,x2). The only not allowed point is (0,0,0). The circle is defined by x^2+y^2-1=0; in homogeneous coordinates the equation becomes -x0^2+x1^2+x2^2=0. The solution at infinity can thus only exists over the complex numbers, e.g. (0,i,1). The Jacobian is the vector (2x0,2x1,2x2) which only vanishes for (0,0,0). P.S.: I go by they/them, so “Sir” is a bit of an unfortunate choice.

[@mattiapiras3331]

@@VisualMath First thing first, thanks very much for the answer. Second I apologize for the "Sir", I'm deeply sorry. Third, I'd have another doubt: so in the 2D case above mentioned, the only solution that could decrease the rank of the Jacobian is the trivial one (0,0,0) so this means that a solution at infinity doesn't always do that. But could it be any correlation with a REAL solution at infinity (0,x1,x2) and the rank reduction of the Jacobian Matrix? If not, don't worry, maybe I shouldn't have so many head trips :). Thanks again for your time.

[@VisualMath]

@@mattiapiras3331 Its all good, don't worry about it! Anyway, I am still not quite following. There is no real solution as (0,0,0) is the one point in homogeneous coordinates that is not allowed.

[@mattiapiras3331]

@@VisualMath Don't worry, I' m not very good at explaining things ;). In any case, my idea concerned an initial system that lets say, in 2D (2 equations) has no solution, while its projection on the 3D sphere -x0^2+x1^2+x2^2=0 has one solution at inf (2 initial equations homogenized + the sphere equation). That's why I was talking of rank of the Jacobian, cause the system would be squared. My doubts of the Real Solution (that means using Real Numbers for Homogeneous coordinates of a point) concerned the fact that I know that there is a solution at INF but I don't understand why the rank of the Jacobian of the final system decreases whenever I compute it at (0, x1*, x2*) -> making it singular: (x1*, x2* solution and Real Numbers ). In any case don't worry, I'll try to investigate further more 👍

[@VisualMath]

@Mattia Piras Well, its difficult to explain things here in the comments ;-) If I come up with something I let you know. Also keep us in the loop ;-)

[@samirelzein1978]

tried your best, you need to prepare more, write your script if you want. Good job, keep them coming.

[@VisualMath]

Thanks for the feedback. Great that you liked it.