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From: HenHanna <HenHanna@devnull.tb>
Newsgroups: sci.lang,sci.math
Subject: Re: f(x) = (x^2 + 1) --------- strange (curved Surface) Graph
Date: Mon, 29 Jul 2024 16:41:33 -0700
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On 7/29/2024 2:30 PM, guido wugi wrote:
> Op 29-7-2024 om 21:28 schreef HenHanna:
>>
>> When   this function      y =  f(x)  =  (x^2  +   1)      is first 
>> introduced, we learn its Graph to be a  simple  parabola.
>>
>> THEN  when we learn  that  x can be a complex number, so that
>> the Graph  is    2 (orthogonally) linked   Parabolas.
>> ---------- like this:
>>
>> https://phantomgraphs.weebly.com/uploads/5/4/5/4/5454288/4_4_orig.jpg
>>
>> https://www.geogebra.org/resource/czbugz9h/fofRh3ZjmwwISd2v/material-czbugz9h-thumb@l.png
>>
>>
>>
>> This graph   is   showing a smooth ,  curved  surface   -->
>>
>>                 https://i.sstatic.net/soSJ8.png
>>
>> What is this graph showing???
>>
>>                it purports to show    f(x)  =  (x^2  +   1)
> 
> Some 3D graphs include the surfaces of Re(f(z)), Im(f(z)), Abs(f(z)), 
> where w=f(z), z=x+iy and w=u+iv. The graphs you mentioned are (part of) 
> one of these.*
> 
> The 'true' graph of the function is a fourdimensional surface in 
> (x,y,u,v) space. No mainstream math grapher whatsoever has even come to 
> think about trying to visualise complex functions as 4D surfaces. But I 
> have, since college. I've been using such tools as mm-paper with a 
> programmable HP calculator, Amiga and Quick Basic, until I came across 
> the unpretentious Graphing Calculator 4.0 of Pacific Tech that came with 
> 4D included in its standard package! And now I've tricksed Desmos3D and 
> Geogebra as well into graphing 4D surfaces. All to be discovered in my 
> webpages and YT channel.
> 
> https://www.wugi.be/qbComplex.html
> https://www.wugi.be/qbinterac.html (Desmos and Geogebra examples, 
> ongoing and not up to date)*
> https://www.youtube.com/@wugionyoutube/playlists (look for "4D" and 
> "Complex Function" playlists)
> 
> So, for your parabola, ie, w=z^2:
> https://www.wugi.be/animgif/Parab.gif (QBasic)
> https://www.youtube.com/watch?v=wuviGuMTrTM&list=PL5xDSSE1qfb6Uh98_9vS4BEMEGJB2MZjs&index=2
> https://www.youtube.com/watch?v=oIyGTf1ZKCI&list=PL5xDSSE1qfb6FIk0Pl3VCg5p3Ema52hEG&index=5
> https://www.desmos.com/calculator/ijcs47qmaz?lang=nl (Desmos2D)
> https://www.geogebra.org/calculator/truptem5 (Geogebra)
> https://www.desmos.com/3d/q9vhspfqq7?lang=nl (Desmos3D example of w=cos 
> z, haven't done parabola yet)
> 
> *Another interesting family of 3D surfaces you won't encounter elsewhere 
> is that of "true curve" surfaces, ie curves belonging "as such" (courbes 
> vraies = "telles quelles") to the 4D function surface. I've only this 
> year 'rediscovered' them (my first ever attempts were drawing 3D curves 
> belonging to the 4D surfaces). See my Desmos page above for examples.
> 
> Feel free to explore, and welcome to the interested ;-)
> 



thanks!    i think i thought about this when i was younger...
             Haven't thought about it for 30+ years.

Graph of  Y= X^2   ( where X=a+bi )

           Y= X^2  has no imaginary part only when  a=0 or b=0.

For this clip (below,  the 2nd half  "animate..." ),
            are you just ignoring the imaginary part of Y ?


Is your surface the same as this one?
https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/5667/2021/09/23134416/4-7-3.jpeg


https://www.youtube.com/watch?v=oIyGTf1ZKCI

Visualization of Complex Functions: the Parabola Y = X ^ 2

                          3,276 views          Jul 3, 2017


For Y = y1 + i y2 and X = x1 + i x2, the function Y = Y(X) is a 4D 
surface in space (X,Y) ~ (x1, x2, y1, y2). Let us project this "simply" 
unto our 2D screen ...