Wugi's QBasics QBasic gecodeerd gepeins  Qbasics in codex Guido "Wugi" Wuyts @ Dilbeek, Belgium, Europe, World, Solar System, Milky Way, Local Cluster, ... 



Wiskunde Maths
2D weergave van 4D
objecten beantwoordend aan complexe functies Y=F(X)
****** 2D rendering of 4D objects
representing complex functions Y=F(X)
A Complex
Function Y = Y(X), or its real equivalent
(y1, y2) = F(x1, x2), may be rendered by a 2D
object, or surface, in a 4D space.
This not being
accessible to our seeing, it may in turn be reduced
to a 3D projection, even a 2D projection. That's what this program
does, mainly for some quadratic functions ("conics") but
also for some others.
This page
shows snapshots and animation for some of them. See explanatory
intro and some menus in Complex
intro.pps
I 'discovered' this
description of complex space back in highschool, when
unsatisfied with the there treatment of 'complex
coordinates' and the absurd theorems about 'isotropic
straight lines' (being perpendicular to themselves and
having distance zero among their points). My then teacher
was amazed with my first tentative descriptions and
drawings of 3D parametric lines belonging to such objects.
My later university prof, typically for math wizzes, didn't deem the topic worth
much bothering about. However, I was already given Banchoff as
a
reference back in 1979 !
See some examples
of my first complete drawings (after tentative sketching
and with the help of little HP calculator programs:), and
an Amigabasic screenshot in Complex
past.pps
By now there is quite some material available on the web,
on graphic rendering of complex space. See for instance
some examples in complex
net.pps. However, I still miss the basic
approach and picturing you'll find here: if there exist
similar pages I'll be glad to receive reference at them !
QBasic: description of 
Other tools 1 
Other tools 2.1 
Other
tools 2.2


Only recently, ie
2016, I discovered this nice free smartphone app:
Math Grapher 3D Using the projection trick X = x1 + A y1 + B y2 Y = x2 + C y1 + D y2 Z = E y1 + F y2 with coefficients A...F corresponding with projection angles of y1 and y2 axes onto the XYZ system, this app is able to render some of the complex functions considered here. It produces some nice results, amazing for a smartphone tool. See the graphs in this column. 
The smartphone
app having its limitations, eg, with entering long
or repetitive formulas, I went looking further. So,
more recently even,
I discovered a tool that has been there already for a number of years, witness it being host of the first of the references on the net down this page, since at least 2008 (!) ... It's called the
and has proved to be a powerful
tool, and easy to learn and use.
A free Viewer can be downloaded there. See the corresponding graphs in this column for each object. The animations are even more impressive, but due to limited storage capacity I cannot upload them to my site *. So, downloading the Grapher files (they are actually text files name.gcf) and installing the free Viewer you will be able to open and explore them that way! * I made some animation videos using AVS4YOU screen capture, and uploaded them to YouTube. The ones available have an active link mentioned below... 
Only after
rethinking the 3D graphics of the adjoining
column, I realised it must also be possible to use the
same Graphing
Calculator 4.0
to make a 2D representation similar to the
QBasic pictures.
The results are in this column, to be compared with the leftmost column. (Bye bye QBasic ?;) Download 2D grapher files with diverse animated rotations (1 zip file) here. 
Notice the equivalence of the circle and hyperbola equations, representing a same object otherwise oriented; the asymptotic planes (here X=0 and Y=0); and the presence of both circle and hyperbola curves on this object. 

Grapher file ; Youtube link 



Grapher file ; Youtube link 

Progressive rotation of an exponential curve resulting in an asymptotic blade X=0, and a screwform blade. The function is periodic, this is one period, repeating itself along the imaginary Xaxis. 

Grapher file ; Youtube link 



Grapher file ; Youtube link 

A half period of the function, with a half cosec curve in the real plane, and bordered by the asymptotic planes Y=0 and X=0, by cosech curves at one side, and a sech curve at the other. 

Grapher file ; Youtube link 

Tan Y = tan X A half period of the function, with a half tan curve in the real plane, and bordered by the asymptotic planes Y=0 and X=0, by cotanh curves at one side, and a tanh curve at the other. 


Double Hyperbola Y = 1/X^{2} A "square" CircleHyperbola, with a double bladed asymptot Y=0, a "squared" hyperbola in the real plane and a minimal closed centre curve somewhat like a doublewinding circle. 


(5D) 









24 Views of the Complex Exponential Function  see complex net.pps 
"complex functions"+4D  Google zoeken  Google lookup 
The Complex Exponential and Complex Logarithm  see complex net.pps 
Robert_Liebo_Final.pdf (application/pdf Object)  see complex net.pps 
meshview 4Dim figures  see complex net.pps 
banchoff: OnLine Mathematics  some beyond the third dimension 
Thomas Banchoff's Home Page  a start page 
Websites related to "Visual Complex Analysis"  a little portal 
Vis96Contour_Meshing.pdf (application/pdf Object)  see complex net.pps 
Thinking Like a Mathematician  talking about a book on complex space and its 50 or 60 pictures, but no pictures 
Dr William T. Shaw  "Complex Analysis with Mathematica" 
Davide P. Cervone (CV/Art): Exponential Tetraview  referenced by Th. Banchoff's page 
Tetraview  Wikipedia, the free encyclopedia  referenced by Th. Banchoff's page 
exp z 4D  see complex net.pps 
sitov_sergei.pdf see fig 4.8  see complex net.pps 
wikipedia : Clifford_torus  The
Clifford torus and its 3D Cyclide explained 
https://www.youtube.com/watch?v=8oZhO9asPxI  A 3D Cyclide 
https://www.youtube.com/watch?v=oVryLr9rm8E  The 4D Clifford torus itself 
wikipedia : ThueMorse_sequence  About the endless and nonperiodic abba
baab sequence 
Isthereanywaytointuitivelyvisualizeafunctionof4variablesinacomputerasinfR4rightarrowR?  The question of visualising z = z (x,y,u,v) 
https://en.wikipedia.org/wiki/ 
Visuals like mine, of the Clifford torus,
not withheld for "not understood" 
https://www.youtube.com/watch? 
The 3Sphere explained and visualised 