https://en.m.wikipedia.org/wiki/Real_projective_line
Wouldn't this imply that inf = -inf? Is this way of picturing the number line actually valid, and if so why?
Hello ! Does someone know how to prove continuity of the map given in the question https://math.stackexchange.com/q/3609525/727001 ? Thanks for your help and stay healthy !
Suppose I have two non-overlapping, non-parallel line segments. Given a point not on these two line segments. Is it possible to construct a line through this point such that all three lines would converge on some other [vanishing] point? The trick here is that we assume the vanishing point is so far off the page so that we cannot draw it (hence cannot use it).
Application: this problem comes from perspective drawing. Often we have a few βparallelβ lines who converge to a vanishing point that is very far off the page, but we want to add another parallel line from a given point.
EDIT: I think I found a solution. It solves the problem with a finite number of steps, but I wonder if there's another simpler solution.
Given two lines l_1 and l_2, I transform one of them using a homography
l_3 = H^{-T} l_1
I want to know if l_3 corresponds to l_2. if it was mapped over it, If it were a point I could dehomogenize it and use a l2 norm to check the distance, but with a line I am not sure how to proceed.
My goal is to integrate the first Chern class of the tautological line bundle. To this end I calculated the connection matrix for the connection obtained by projecting the canonical flat connection of C^2 onto the line bundle and using the frame e([1:z]) = e_1 + z e_2, where {e_1, e_2} is the canonical basis of C^2 . This yields for the only element of the connection matrix the 1-form w = z*/(1+|z|^2 ) dz, with z* the complex conjugate of z. This is still in accordance with exercise 18, page 24 of http://www.staff.science.uu.nl/~crain101/DG-2015/main10.pdf
Now to compute the curvature matrix I wanted to use k = dw + w ^ w, using ^ to indicate the wedge product. The second summand is 0 (right?) so we get k = dw = d/dz* (z*/(1+zz*)) dz* ^ dz = -1/(1+zz*)^2 dz* ^ dz. However, this disagrees with both example 9.1 page 3 of http://www.maths.manchester.ac.uk/~tv/Teaching/Differential%20Geometry/2008-2009/lecture9.pdf and https://mathoverflow.net/questions/227233/first-chern-class-of-the-tautological-line-bundle-over-mathbbcpn
What am I doing wrong? Also, if anyone could show how one actually goes about calculating the integral I would appreciate it
I'm working on a project that is involving perspective projection/rectification. I'm having trouble grasping the concept of the 'line at inifinity', or the 'ideal line' and why it is represented in a homography matrix as the last row =[0,0,1].
Can someone share with me how they visualize homotopy equivalent between two spaces.
I have always visualize homeomorphism as being able to deform your space in such a way without stretching to morph into one another.
It seems to me homotopy equivalence is something else, I can understand the definition but I'm not getting a good handle of homotopy equivalent. It's almost like homeomorphism but not quite, so it has been really hard for me to get a feel for what I need to do when I'm being asked to prove that two spaces are homotopy equivalent.
Furthermore, I am finding it increasingly hard to visualize many things like, projective plane, connected sum of real projective plane, earlier I was asked to prove things like
the connected sum of projective planes are homotopy equivalent to bouquet of circles.
or that the union of 3 circles laid out on the x-axis horizontally, touching a point on each end is homotopy equivalent to bouquet of 3 circles (I am done with the homework, or rather, the deadline is due so this isn't a homework-seeking help post). (Professor in case you are reading this XD)
I thought I understood homotopy, homotopy equivalent, but I really don't, and I can really use some help.
thanks!
Reading up on Fundamental Polygons on Wikipedia I came upon this section which described the Sphere, the Torus, the Klein Bottle and the Real Projective Plane. I've heard of the first three, but the fourth is new to me. To try to get my head round it, I tried to imagine what it would would like if the Universe was in the shape of each of these.
So if the Universe was a Hypersphere, you might fly off from Earth in a straight line and eventually get back where you started. Whichever direction you go, it would take roughly the same amount of time. This is the four-dimensional equivalent of travelling all the way around a sphere.
If the Universe was a Hypertorus, and imagining it happened to be oriented exactly to the Earth's axis to make describing this easier, You could fly off the Earth from the North Pole in a straight line and eventually reach reach Earth again, at the South Pole. If you were to leave from the equator and travel in a straight line, you would arrive back at Earth but it would take much longer. This is the equivalent of going the long way around a ring doughnut, instead of just going down the hole and back up round.
Stop me if I'm getting this wrong, but if the Universe was a Klein bottle, just as with the Hypertorus above you could go the short distance from the Pole and end up back at Earth as normal, but if you chose to go the long way you would end up back at Earth but everything would look backwards - like a mirror image, and would have opposite spin making it react like antimatter. Or maybe you and your spaceship is the Antimatter, I don't know. But this would only happen if you went the long way, not the short way. This is the four-dimensional equivalent of travelling along a Mobius strip and finding yourself on the reverse side of the point of the strip where you started.
I don't for one second imagine that the Universe is a Real Projective Plane, this is just a Thought Experiment, but if it was what would that mean from within? Would it mean that whichever direction you went in a straight line you'd end up at the reverse-spin Earth?
Does anyone have suggestions on removing appendages from self enclosed two dimensional manifolds?
An order n projective plane has n^(2) + n + 1 points. Since 4^(2) + 4 + 1 = 21 and there are 21 consonants in English, this suggests assigning letters to each of these 21 points so that each line corresponds to a word (or short phrase); there are 5 points on each line, so to build a word, we allow ourselves to use 5 consonants perhaps with repeats, and to throw in vowels at will. The result is 21 words (okay, phrases) with the property that any two of them share exactly one consonant.
- phleboid
- qualitatively
- squabbing
- wrinkle
- magicalize
- overjudge
- oryzopsis
- jonathanization
- mikvahs
- woodcuts
- jackboy
- whiggify
- myxinoid
- tubeform
- fivepence
- exchequer
- quaked fizz
- jump quaw
- vibex weeze
- jollifies ox
- gatekeep tax
There are various games that one can imagine playing with such cards. Can redditors come up with more (and better) lists of such words?
Alignment is one the most highly debated topics in Dungeons and Dragons. Few people can agree on what what a particular alignment means or how heavily it should affect a character. I think that I have been able to boil it down into two very broad categories for how to use it, though.
The first one is projective. This is the traditional style where you your alignment at character creation, and it shapes your role play. It might change over the course of a campaign, but it is mostly an established trait.
The second field is reflective. This is the style where you make your character, and decide based on your backstory and in game choices what your alignment is. It is more open to change, and is a less concrete part of your character.
How do you use alignment in D&D? Do you think those categories are a fair description of its main uses? I am very interested in hearing what you all have to say.
Yours in D&D
TheAres1999
Now i know that I do have different kinds of synesthesia and ideaesthesia but I was always wondering if it is projective or associative. Maybe some folks of you who know for sure that they have projective synesthesia can enlighten me.
I will just talk about my sound->visual synesthesia since it is probobly my strongest form of synesthesia.
When i listen to sounds/music I do kinda see the colors, shapes and spatial arrangements of the sounds BUT I do not see them exactly like I see true visual stimuli... So when I see music and look at my soundspeakers where the music is coming from, I can still see the speakers very clearly; my normal visual sense is NOT interrupted or overlayed by the synesthesia. It is more like involuntarily imagining something visual... but not really in your head but more so outside of the head, probably kinda where the music/sound is coming from (but it also depends on the type of sound... if I'm using in-ear headphones or hear specific psychoacoustic sounds it can also be spatially located in my head).
So the forms of projective and associative synesthesia have always confused me... When there is someone who has for example projective number->color synesthesia, do they REALLY the colors ON the numbers? If that was the case, wouldn't they not be able to see the true color of the numbers anymore (in this case for example the numbers being printed in regular black)?
Now I also wonder if every synethesia can be projective. What about letters->personality syesthesia? How could they project the personality onto the letters like normal sensory stimuli? Is projective synesthesia even like the regular sensory stimuli? Or is it more like seeing something with your mind's eye but not really in your head, but still in another way so that it doesn't interrupt the normal senses?
I tried to google it, but it almost always is described like projective synesthets LITERALLY perceive the synesthesia EXACTLY like any other sense (but I have also read varying descriptions, maybe even by people who don't even have synesthesia but like to write about neuroscience)... But that wouldn't really make sense... It would make projective synesthets practically disabled since it would basically be hallucinations that are interrupting the regular senses... Are all the bloggers writing about synesthesia just bad at describing it? Is projective synethsesia more on another dimensional plane that the regular senses or would that be associative synesthesi
... keep reading on reddit β‘My apologies if this sounds stupid, I'm just wondering.
For example, in sound-to-colour synesthesia, can you have both associative and projective synesthesia? Because when I listen to certain songs, they look like they're one colour in my mind, but they feel like a completely different colour. (Both involuntary)
I find myself doubting my own synesthesia when this happens. Sorry if this is confusing. Thanks :)
Unfortunately I am not a mathematician by training, but there is a question that I would like to know an answer to concerning geometry.
Is it possible to have a standard 3-dimensional euclidean space (about 3-D scenes for instance) and without additional information turn it into a Projective Space with 3 (+ 1) dimensions? I know getting a projected plane out of euclidean 3 space is possible. But is it possible mathematically to artificially add another dimension (e.g. by means of points at infinity being interpreted as the vantage point from which the euclidean 3-Space is viewed) and view the 3D objects from different perspectives in space, depending on the subsequent transformations?
I hope to find an answer on this or a hint at something where I can read about it!
Guys, I've been out well over a year. I've been swamped by the emotional turmoil. I've pondered killing myself. I thought I was fucking crazy and needed help. Pyscho. A piece of shit. A crazy stalker.
But why?? None of that is me. I'm a lover, not a fighter. I'm kind, empathetic, thoughtful, and considerate of the people around me. Yeah I have a temper but I've worked hard to get it under control and not let it control me. So why?? Why the fuck did I feel like the problem?
It's something called Projective Identity. It's an extension of regular projection and even more insidious.
Numerous times on here I've explained to people asking "am I the crazy one" that in the beginning they mirror us, showing us how awesome we are as partners. Then as the mirror cracks they start to project their issues on us. And since the person we love, this perfect person is saying these things about us they must be right, right? So we internalize it. We believe it. And guess what, we eventually identify with that, and become that!!!
That is Projective Identification. I found this post that explains it in different terms. But either way, we can get better. I promise you guys we can. It takes time and intention, but we sure as hell can. No matter how bad it is. You got this.
Here's your take away. Do you remember how awesome the honeymoon phase was? When they mirrored you?? That's how awesome you are!! Not them, you!!!
Projective Identification is the process whereby a pwBPD, in an unconscious attempt to 'get rid of' overpowering internal states or feeling, attempts to externalise them by unconsciously projecting them on to, and eliciting them in, an 'other'. In this way they can externalise the self-destructive or self-persecuting feelings, and through a process of projection and transference, 'transfer' them on to the 'other'.
This explains, to me, in large part the source of the abuse that I experienced with my pwBPD. Why she made everything impossible - because, on an unconscious level, she didn't even want things to work. She wanted to make it impossible, so that she could then be in a position of feeling abused/abandoned/heartbroken - she deliberately sabotaged our relationship and tried to make herself heartbroken', so that she could have someone to blame for it. This was an unconscious reliving, playing out of her early childhood experiences of abandonment.
To her, I was nothing other than the object on to which she projected all of her own feelings - which explains why it was always about her. She was not prepared to have a relationship with another autonomous human being with their own thoughts and feelings. Just a crash test dummy, a mail-order, identikit male, who was there to meet her own various needs.
She made me into someone I wasn't, tried to provoke me into becoming someone that she could blame as her 'abuser' - which explains why her ex was also an 'abuser'. I honestly feel sorry for him too. To be put through what she put me through.
I'm not a mathematician, so I apologize for using any terminology incorrectly. Please correct me I love to learn!
I came across these after being amazed at the properties of my nephew's card game "Spot It". In Spot It, there are 31 cards and 31 animals. Each card has six animals on it, each animal exists on six cards, and each pair of cards has exactly one animal that exists on both cards (the object of the game is to look at a pair of cards and spot the animal shared between them). I was curious if arbitrarily large Spot It could be constructed. I learned that it could and how to do it (I made a Spot It with 30 Pokemon on each card).
My question is: can I enumerate the animals on each card 0 through n-1, and construct spot it such that for each card each animal exists on it has a unique "position" (0, 1, ..., n-1)?
For my Pokemon spot it I was able to brute force find a solution where each Pokemon was guaranteed to have a card where it existed at "position" 0 (I made the position 0 Pokemon large and centered on the cards - the remaining 29 were placed randomly). But this search space was enormous and it takes 30 seconds to find a solution.
