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From: guido wugi <wugi@brol.invalid>
Newsgroups: sci.lang,sci.math
Subject: Re: f(x) = (x^2 + 1) --------- strange (curved Surface) Graph
Date: Mon, 29 Jul 2024 23:30:57 +0200
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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 ;-)

-- 
guido wugi