Convert points from a 2d space with a real dimension x and an imaginary dimension y

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MrY
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Convert points from a 2d space with a real dimension x and an imaginary dimension y

Postby MrY » Tue Feb 21, 2017 5:33 pm UTC

Hi,

I have points pi in a 2d space with a real dimension x and an imaginary dimension y: pi(xi;i*yi).
I want to represent them in an euclidean edit:Minkowski? space so I search: p'i(x'i;y'i).

For example, p'1 and p'2 must have a distance of sqrt(abs((x1-x2)²-(y1-y2)²)) in this euclidean space (I guess).

I am not sure where to start.
Can you help me?
Last edited by MrY on Wed Feb 22, 2017 9:13 am UTC, edited 1 time in total.

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Re: Convert points from a 2d space with a real dimension x and an imaginary dimension y

Postby Xanthir » Tue Feb 21, 2017 6:06 pm UTC

MrY wrote:Hi,

I have points pi in a 2d space with a real dimension x and an imaginary dimension y: pi(xi;i*yi).
I want to represent them in an euclidean space so I search: p'i(x'i;y'i).

That's the standard embedding, yes - map the real part to X and the imaginary part to Y.

For example, p'1 and p'2 must have a distance of sqrt(abs((x1-x2)²-(y1-y2)²)) in this euclidean space (I guess).

That's definitely not the Euclidean distance metric! Euclidean distance is just sqrt((x1-x2)² + (y1-y2)²); note the addition and lack of abs().

What exactly are you trying to do? It would help us understand your needs if we had details about what you're using this for.
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Re: Convert points from a 2d space with a real dimension x and an imaginary dimension y

Postby Flumble » Tue Feb 21, 2017 6:08 pm UTC

Do you mean this?

Code: Select all

x' = Re(p) = 1/2 (p+p*)
y' = Im(p) = 1/(2i) (p-p*)

(with p a complex number, p* its conjugate and x' and y' real numbers)

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Re: Convert points from a 2d space with a real dimension x and an imaginary dimension y

Postby MrY » Tue Feb 21, 2017 8:08 pm UTC

I have read this:
http://www.gregegan.net/DICHRONAUTS/00/DPDM.html

The story of this book take place in a universe with 2 dimensions of space and 2 dimensions of time.


I want a 2d space (no spacetime) but with 1 space-like dimension (real) and 1 time-like dimension (imaginary), like this:

Image

Here, the u dimension is a time-like dimension.

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Re: Convert points from a 2d space with a real dimension x and an imaginary dimension y

Postby Xanthir » Tue Feb 21, 2017 8:42 pm UTC

I'm very confused; you seem to be contradicting yourself - you want a 2d *space* (not spacetime), which suggests two spatial dimensions, but you also want 1 space dimension and 1 time dimension, forming a 2d spacetime. Which do you want?

If you want a spacetime that works like ours does, just with only a single spatial dimension, that's fine; just use ordinary Euclidean distance, but make sure you preserve the imaginary unit when doing the math, so squaring the time difference results in a negative number. That's equivalent to doing sqrt(Δspace² - Δtime²). Note: no abs(); the end result is allowed to go imaginary. (In fact, all actual paths need to be imaginary, if the time dimension is imaginary; if the distance between two spacetime points is real, it means you need FTL travel to get between them. This is why the convention of treating space distance as imaginary/negative is better than the one that treats time distance as imaginary/negative.)

If you're using all real numbers, then the term for what you're looking for is *Minkowskian* geometry, not Euclidean - Minkowskian captures the "subtract one of the terms, rather than adding them all" concept.
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Re: Convert points from a 2d space with a real dimension x and an imaginary dimension y

Postby cyanyoshi » Tue Feb 21, 2017 8:55 pm UTC

There is a thing called split-complex numbers that act like complex numbers but with an imaginary unit j that is a square root of +1. Maybe this is what you are looking for.

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Re: Convert points from a 2d space with a real dimension x and an imaginary dimension y

Postby MrY » Tue Feb 21, 2017 9:15 pm UTC

Xanthir wrote:I'm very confused; you seem to be contradicting yourself - you want a 2d *space* (not spacetime), which suggests two spatial dimensions, but you also want 1 space dimension and 1 time dimension, forming a 2d spacetime. Which do you want?


I want a space "time-like" dimension :D a space dimension which combined with a common space dimension behave like Minkowski diagram but in 2d space!

So yes you are right, it is not an euclidean space but a Minkowski space.

Thinks cyanyoshi, I did not know that.

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Re: Convert points from a 2d space with a real dimension x and an imaginary dimension y

Postby Soupspoon » Tue Feb 21, 2017 9:27 pm UTC

cyanyoshi wrote:There is a thing called split-complex numbers that act like complex numbers but with an imaginary unit j that is a square root of +1. Maybe this is what you are looking for.

Despite reading as far down that page as it seems to still promise the proper explanation, I'm still somewhat unsure as to what j is, if it isn't ±1... (To be precise, (+1)²=+1, (-1)²=+1, and ±1, though not being a single value (in fact I'd call that a "split" or "double" number in a data definition that had ±ness as well as simple +ve and -ve available to its values) when used as (±1)²=+1 is as good a value of j as any.)

And it looks nothing like the j (or k) in the quaternion notation, unless I've been misreading something... Which is possible.

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Re: Convert points from a 2d space with a real dimension x and an imaginary dimension y

Postby gmalivuk » Tue Feb 21, 2017 9:38 pm UTC

j isn't ±1 because it's not defined to be.

I think the useful thing to note on that page is the geometry section, which goes over how it can describe a Minkowski space rather than a Euclidean one (as the complex numbers do), which is precisely the feature MrY seems to want.
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Re: Convert points from a 2d space with a real dimension x and an imaginary dimension y

Postby MrY » Tue Feb 21, 2017 9:53 pm UTC

I will rephrase my request:

It is possible to have:
- a first 2d diagram with points pi(xi;yi), for example a segment p1p2
- a second 2d diagram representing corresponding p'i which behave like Minkowski space if a rotate or move my segment in the first 2d diagram? So for the exemple if a rotate my segment in the first diagram it will grow in the second diagram, like the gif?

If it is possible, what are the equations of x'i and y'i?

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Re: Convert points from a 2d space with a real dimension x and an imaginary dimension y

Postby MrY » Wed Feb 22, 2017 9:28 am UTC

Basically I would like to redo the following application but in 2d:
Http://www.gregegan.net/DICHRONAUTS/02/Interactive.html

I imagine that the parallelepipeds have coordinates, and according to their position in space their representation is deformed, but not the object itself.


I'm wrong?

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Re: Convert points from a 2d space with a real dimension x and an imaginary dimension y

Postby Derek » Wed Feb 22, 2017 12:33 pm UTC

Soupspoon wrote:
cyanyoshi wrote:There is a thing called split-complex numbers that act like complex numbers but with an imaginary unit j that is a square root of +1. Maybe this is what you are looking for.

Despite reading as far down that page as it seems to still promise the proper explanation, I'm still somewhat unsure as to what j is, if it isn't ±1... (To be precise, (+1)²=+1, (-1)²=+1, and ±1, though not being a single value (in fact I'd call that a "split" or "double" number in a data definition that had ±ness as well as simple +ve and -ve available to its values) when used as (±1)²=+1 is as good a value of j as any.)

And it looks nothing like the j (or k) in the quaternion notation, unless I've been misreading something... Which is possible.

j is a non-real number whose square is +1. That's all there is to it. Don't try to relate it to anything you are already familiar with.

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Re: Convert points from a 2d space with a real dimension x and an imaginary dimension y

Postby Soupspoon » Wed Feb 22, 2017 1:01 pm UTC

Derek wrote:j is a non-real number whose square is +1. That's all there is to it. Don't try to relate it to anything you are already familiar with.

Yep (to you and gmalivuk, both), I'm not going to go and do a full Treatid on this issue, but that wiki page just really didn't (at first glance) give me the insight that I needed to understand the actual basis of the definition.

And that's with having been a vociferous arguer that there were such things as "negative numbers", to my disbelieving peers, in primary school, and picking up on imaginary numbers well before I should. I've a feeling I need to go back to my Physical Systems notes on non-Euclidean representations and try to match those against the Minkowski stuff that we evidently didn't study at that area and level of undergrad studies.

(It might also affect my little project that simulates a stable finite unbounded (i.e. hyperspherical, inclusive of time) universe system in a sort of Galilean geometry way. But that's a more playing about than serious, so it's only ever going to be as much a toy as the Asteroids universe geometry is. ;) )


Sorry to OP for butting in, though, I just was momentarily a bit too much bothered by it to let it pass without comment.

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Re: Convert points from a 2d space with a real dimension x and an imaginary dimension y

Postby Demki » Wed Feb 22, 2017 1:35 pm UTC

Soupspoon wrote: I'm not going to go and do a full Treatid on this issue

[offtopic]
This became a thing, now it's time to spread it in 4chan so everyone would use it without knowing what the hell it is.
[/offtopic]

MrY, are you looking for a transformation T from R^(2) to R^(2) such that for any pair of points x and y, the euclidean distance between x and y is equal to the minkowski distance between T(x) and T(y)?

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Re: Convert points from a 2d space with a real dimension x and an imaginary dimension y

Postby MrY » Wed Feb 22, 2017 2:02 pm UTC

I don't know exactly what I'm looking for, but yes I search this kind of transformation I guess, but not only.

Did you play with the java application?

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Re: Convert points from a 2d space with a real dimension x and an imaginary dimension y

Postby gmalivuk » Wed Feb 22, 2017 5:35 pm UTC

I still think you're looking for the split-complex numbers.

The unique thing about dichronauts seems to be that there are two time-like dimensions and one (or more) space-like, but if you're reducing it to 2 total dimensions, you've just got "standard" R1,1 Minkowski space.

It's not a regular metric space with a simple always-positive "distance", because the analogue of the square of the distance can be negative, and the points at zero "distance" from a given point are the past and future light cone from that point.
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Re: Convert points from a 2d space with a real dimension x and an imaginary dimension y

Postby MrY » Wed Feb 22, 2017 6:06 pm UTC

Yes, I agree, but I don't know how to use them.

Is my example with 2 diagrams correct?
It is possible to have x'=f(x,y) and y'=g(x,y) ?

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Re: Convert points from a 2d space with a real dimension x and an imaginary dimension y

Postby gmalivuk » Wed Feb 22, 2017 6:31 pm UTC

I guess I still don't know what you're trying to do, and why you can't just map x+j*y to (x,y), like how we map x+i*y to (x,y) to show complex numbers.

There are differences in how stuff moves and rotates and what "circles" look like, but that's all for curves and transformations, not individual points.
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MrY
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Re: Convert points from a 2d space with a real dimension x and an imaginary dimension y

Postby MrY » Wed Feb 22, 2017 7:50 pm UTC

In the java application, I take two identical blocks and rotate one of them:
dichronauts.png


The red side of the second block is no longer a rectangle, and CD distance seems to be greater than AB.
But it is just a representation, the second block is not deformed, and we still have AB=CD, right?

Is there a diagram in which these two blocks are rectangles (one of which is rotated) and which, by a transformation, becomes the red faces that are seen here?

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Re: Convert points from a 2d space with a real dimension x and an imaginary dimension y

Postby gmalivuk » Wed Feb 22, 2017 8:46 pm UTC

Isn't that what your 2d animation was doing above?

The wiki page about split-complex numbers talks about rotations.
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