QB-Complex

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 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 !

Most menu items in the program should be clear by themselves. You pick a figure and a rotation, and a pseudoanimation will be shown. However, as this program dates back from Amigabasic (!) where each image would need a painstaking minute or so to build, the sole "animation" effect, the default number of steps for a 360° rotation is but 9. You should first increase this in the menu Preferences>More>Number of images. Also, the line scanning option in the menu Preferences>Parameter curves, helpful after the slow buildup in the early systems, is now outdated (or should be re-tuned...).

Click pictures below to show full size, Animate for an animated GIF

Circle-Hyperbola X^2 + Y^2 = R^2 OR Y = 1/X :

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.

Exponential Y = e^X :

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

The exponential and its reverse combine, their asymptot blades disappear resulting in a double screw, where both meet appears the (co)sine curve. The hyperbolic sine and cosine are typical scan curves of the double blade.

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.

Tan Y = tan(X) :

Animate

description of complex.bas, complex.exe	tips
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 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 !	Most menu items in the program should be clear by themselves. You pick a figure and a rotation, and a pseudoanimation will be shown. However, as this program dates back from Amigabasic (!) where each image would need a painstaking minute or so to build, the sole "animation" effect, the default number of steps for a 360° rotation is but 9. You should first increase this in the menu Preferences>More>Number of images. Also, the line scanning option in the menu Preferences>Parameter curves, helpful after the slow buildup in the early systems, is now outdated (or should be re-tuned...).
Click pictures below to show full size, Animate for an animated GIF
Circle-Hyperbola X^2 + Y^2 = R^2 OR Y = 1/X : Animate	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.
Parabola Y = X^2 : Animate	A 4D paraboloid.
Exponential Y = e^X : Animate	Progressive rotation of an exponential curve resulting in an asymptotic blade X=0, and a screw-form blade. The function is periodic, this is one period, repeating itself along the imaginary X-axis.
Cosine Y = cos(X) : Animate	The exponential and its reverse combine, their asymptot blades disappear resulting in a double screw, where both meet appears the (co)sine curve. The hyperbolic sine and cosine are typical scan curves of the double blade.
Cosec Y = cosec(X) : Animate	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.
Tan Y = tan(X) : Animate	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. (A less 'fluid' animation here, as this item causes an overflow bug for some rotation/step choices in the qb-program:-)
Double Hyperbola Y = 1/X^2 : Animate	A "square" Circle-Hyperbola, 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.

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: On-Line Mathematics	some beyond the third dimension
Thomas Banchoff's Home Page	a start page
Websites related to "Visual Complex Analysis"	a little portal
Vis96-Contour_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