Showing results for 
Search instead for 
Did you mean: 
Showing results for 
Search instead for 
Did you mean: 

The PTC Community email address has changed to Learn more.

Portrait of roots of two equations

24-Ruby III

Portrait of roots of two equations

I announce a competition - who will create the most interesting and beautiful portrait of the roots of two equations. I will send a hard copy of my book (2⁵ Problems for STEM Education - 1st Edition - Valery Ochkov - Rout ( to the winner of the contest. The end date of the competition is April 1, 2022. I will personally determine the winner. Materials explaining the essence of the task, in attachment. In the picture - a sample of a work of science and art, which should be submitted to the competition.



Portrait not Find but Minimize function. f(x, y) is Himmelblau's function - Wikipedia


It looks like the for points in your picture (the right one) are not placed at their correct corresponding position as they are not collinear. You may compare with the pic below:


It may also be interesting to zoom in at specific positions. As far as I tried its chaotic but by far not as nice as doing it with the Mandelbrot.







Thanks, Werner! You have as we say in Russian "глаз-алмаз".



@ValeryOchkov wrote:

Thanks, Werner! You have as we say in Russian "глаз-алмаз".

Oh, yes! I am so sharp-eyed that sometimes I even see things that aren't there - but that's another story 🙂


I think it's better to have the points automatically entered into the image than to do it afterwards with an image editor.


Do you remember that one specific regular contributor in this forum always used to say that "a solution without an animation isn't a solution" ?



Но я думаю, что у нас должно быть 1, 2, 3 или 4 цвета для 1, 2, 3 или 4 реальных корней.

Или плюс один цвет - никакого решения!

Mathcad 15Mathcad 15PrimePrime

Yes, and my approach which you have seen in the animations and pics are just using as many numbers as we have zeros and an additional color (I usually use black) for "no solution found".
A contour plot as you use is far too less flexible concerning color choices and thats the reason I suggested using the image component in Mathcad (Prime can't do that).

@Werner_E wrote:


It may also be interesting to zoom in at specific positions. As far as I tried its chaotic but by far not as nice as doing it with the Mandelbrot.




We can open a contemporary art gallery!

@ValeryOchkov wrote:
We can open a contemporary art gallery!

Yes, but you first would have to change your sheet so you don't use a 3D plot but rather the image component. This gives a much better control over the colors and you can pick each of the five colors used individually from a palette of more than 16 millions different colors.

See here little animation using random colors


Of course you would have to use MC15 or below as Prime neither offers something like the image component nor does is offer any way to create animations. But you sure know that.

OK, think I have played around enough for now.

Here is a last animation showing symmetrical pictures (slope of line is zero). Its interesting to see that sudden appearance of the many black areas (no solution found) when the intercept gets positive.



Yes, it's time to finish! And then at night such a terrible muzzle dream. Some kind of mysticism begins.

See Medusa - Wikipedia



@ValeryOchkov wrote:

Yes, it's time to finish! And then at night such a terrible muzzle dream. Some kind of mysticism begins.

See Medusa - Wikipedia


I was rathetr thinking more of a demon. And he doesn't have to keep you awake if you penetrate his brain. Not sure if this is called brainstorming ;-). Anyway.. here for reassurance the psychedelic journey in and back again:


it wil be good to find good misic for this video

@ValeryOchkov wrote:

it wil be good to find good misic for this video

Here is another one you can lookup a suitable music for 😉


And here a less spectacular one


What we see at the and of those zooms sure is not the algorithm but rather the numerical limitations of the IEEE number format.

Music for this video Ride of the Valkyries - Wikipedia


Can you send my gif-files please.

@ValeryOchkov wrote:

Music for this video Ride of the Valkyries - Wikipedia



Can you send my gif-files please.

I guess you could download them directly from my postings above. Bute here they are attached in a zipped archive - have fun!


In case you have not meant the gif files but rather the avi file created by Mathcad - there are none. I used Mathcads animation feature just to let Mathcad save the single pics (you remember that I am using the image component and not a 3D plot) in (pretty old) BMP format and created the gifs from them using a freeware. This procedure has couple of advantages as I can recreate missing frames in case that Mathcad throws an error (happens quite often). Then I don't have to size the area for the video manually and I can use the full 16 mio colors and are not limited by the stone age old codec used by Mathcad. And I can easily create the continuous zoom-in zoom-out effect using the very same pictures.

BTW, this should be a way to create animations in Prime, too. Prime has no picture component but you can save a calculated picture using WRITERGB similar to real Mathcad. So you would have to create the pics blind as you cannot see and control them in Prime itself which is boring, but making the animation would be the same.
As you seem to like using Prime, this may be something you should give a try.

EDIT: No, sorry! Its NOT a way to make animations with Prime - at least not an easy one. As Prime does not offer the animation feature you could not create all the necessary pics in one go but would have to manually save each one yourself one after the other - sure not inviting. Its similar laborious like the Prime animation you did a long time ago where you manually took screenshots and turned them into a video. I wonder if this procedure could be automated using a software like Auto-It ??


EDIT: Attached the source (the single bmp pics) as well. Using an appropriate software you should be able to turn them into a mkv, mpg or avi and add your music in one go if thats really what you would like to do.

Thanks! But I need this


Here you are (but not mirrored like the pic you posted 😉


EDIT: In case you'd like to recreate the animation with larger resolution and more frames I attach the Mathcad file for your convenience

EDIT2: Also attach a more fluent and larger animation of the picture


Sorry, Werner! Can you send me the avi-file? please. I cannot creat myself.

Sorry, there is no avi-file!

I already explained above that the picture in the animation is the content of a picture component in Mathcad which I export (using Mathcads animation feature) in a series of 200 BMP picture files. These are copied and renamed to make a 400 pic series showing the zoom-in/zoom-out sequence which then is used to create an animated gif using the freeware "Any To Gif".
All I could do is to send you those picture files and you sure can find a software which creates a movie out of them.
BTW, what are you actually trying to achieve with those animations?

In the meantime some more pictures using a different set of functions.

EDIT: Just downloaded Bmp2Avi and converted the pics to an (uncompressed) avi. Hope it helps.





>BTW, what are you actually trying to achieve with those animations?

For my stdents! STEAM - education! A - Art!

From my old article



When the author has shown this picture to man knowledgeable in fine art, he was told, that it was… «Landscape at Auveres after the rain» by Vincent Van Gogh. The truth, this "artman" had not time (was in time) put on glasses, and the picture was shown to him have issued. But at more detailed comparison the similarity of colour scale and "invoices" (strip ridge, range) figures was marked. Whether or not it is so, we will let our readers will decide for themselves, having compared pictures.


Why not join infinity (lemniscate) and heart?

You sure will find some fitting real paintings 😉




The last request (or penultimate one) is to build a joint portrait of Tolstoy and Bernoulli

(see Prime in attach))



Similar in effect to my lemniscate & circle

Not very spectacular


unless you zoom in




or you destroy symmetry


The animation below does not show a zoom ride into a picture but rather shows the effect of different sized ovals.



Thank you. Last request. The lemniscate from below crawls onto Tolstoy's oval. Animation! I would like to receive an avi file.


An animation showing Mathcads 2D-plot would be a problem as Mathcad quite often fails on my machine and claims to be out of memory. But then the next frames would be rendered without problems(!). That way the animation would have empty white frames in-between which doesn't look nice.
Furthermore my machines are busy at the time with transcoding a couple of videos (no, not Mathcad) for the next few hours.
But if you are interested I can send you the Mathcad file and you change it accordingly and give it a try yourself.

Mars Attacks! Werner_E_1-1644864827146.pngWerner_E_0-1644864798864.png


Thanks! I have same problems with Mathcad-computer-memory. Send me please Mathcad-sheet.


And it is Leo Tolstoy


Here you are.

You may also rotate one or both of the graphs and/or translate them in any direction.
Scaling is only implemented for the oval (variable C) but it should be easy to scale the lemniscate as well.


Unfortunately Mathcads symbolic has problems when both graphs are rotated (or I simply was too impatient). I use the symbolic solve to get all zeros and then I filter to get only the real valued ones.

The sheet is not cleaned up and a bit of a mess. So some explanations concerning the variables used may be necessary.
vx1 and vy1 are the translation units in x- resp. y direction for the oval, phi1 its angle of rotation (counter clockwise) and C its scaling factor. Analog vx2, vy2 and phi2 for the lemniscate (no scaling factor here so far).

"cnt" contains the x- and y-coordinates of the center of the generated picture. "fkt" is a scaling factor for the x- and y-limits.

The ratio of the pic is always 3:4. fkt=1 means that the pic is 6 units height time times 8 units across. Smaller values for fkt zoom into the picture.
"width" is the pixel width of the image, the height is automatically 3/4*width.
The if-statement in the definition of width simply reduces the size of the pic significally if FRAME=1 as I was tired having to wait for the pic to finish calculating when an animation was finished and FRAME was set to 0 again. So animations should not use FRAME #0.

If you chose colorindex (cidx) to be 0, colors are generated at random. After a pic has been calculated you may change for another color set without having to recalculate the pic or you may select the region "Color := Col(cidx)" and press F9 for another random color set.
Have fun playing around with the sheet!


BTW, the animation above has no faulty frames (luck!) but a nasty change of color when the animation reaches the points where there are only 2 or only 3 common points. It would require to adopt the color selection algorithm so that the "outer" colors remain the same. A modification I leave up to you 😉