I would like to plot (to animate) an oval - an ellipse with one focus as a point and second - an ellipse.
See about this problem Symmetry | Free Full-Text | A New Ellipse or Math Porcelain Service (mdpi.com)
But I have problems - tee pics and Mathcad 15 in attach.
One solution in SMath
So, an ellipse and a hyperbola can be specified either through a focus with a directrix, or through a pair or more foci. But the focus can be not only a point, but also... a circle, and an ellipse and a hyperbola can be crossed and get an ellipsohyperbola.
Figure shows a family of closed curves, which can be called these same ellipsohyperbolas. These are ellipses, in which the first focus is a traditional point, and the second focus is a circle. Outside the circle we have ordinary ellipses, and inside the circle we have branches of a hyperbola. This can be verified not only visually, but also mathematically - inside the circle, addition (ellipse) turns into subtraction (hyperbola). The ellipse (oval) on the border of the circle turns into a hyperbole at an angle and can finally bring harmony into the soul of the poet - see the divertissement epigraph. If you smoothly change the S parameter, you get an animation - a small green snake (a straight line segment between the left focus and the left edge of the circle) predatory (see the second name of the divertissement) swallows the circle in much the same way as a boa constrictor swallows a rabbit. At the end of this process, the ellipsohyperbola becomes a rather plump ellipse.
Help please!
ACCEPTED SOLUTION
Accepted Solutions
Ah yes, I now remember your post on porcelain manufacturing. There may be easier ways to create irregular ovals, but why not.
It's a little questionable, though, whether creating plates for the Fields award ceremony is a viable business model 😉
Maybe it could be made more interesting for a broader market by personalization using a silhouette 🙂
🙂
The problem lies in he numeric solve block you use to determine the shortest distance of an arbitrary point to your ellipse.
I can think of three solutions but I guess only the third one would be a real solution:
1) use a different kind of solve block with S.tc(....)=0 and MinErr(x.c,y.c)
2) provide guess values to your solve block which depend on x an y as well, not only the center of the ellipse.
3) Find an analytic way to determine the shortest distance
When using #1 or #2 I would also suggest to catch any errors in the solve block with an "on error" statement which returns a NaN vector in case of the solve block not able to find a solution.
Furthermore you can speedup the calculation abit if you don't call F(x,y) twice in your function f(x,y).
Good luck!
I see. Instead of failing, because there is NO solution, MinErr returns an inaccurate result.
But failures like this must be expected in one way or another as long as you use numeric algorithms. Only cure would be an exact symbolic solution.
But when you use Minerr, you can "weight" your constraints and if "Point on ellipse" is more important than "sum of lenths = S", you may weight it more like shown here:
EDIT: I have integrated all three suggestions offered so far into a single worksheet for selection. So you can play around with them and add your own methods ... If you find something more efficient as #1, let us know.
I may be wrong but didn't you deal with that kind of modifications of the ellipse definitions already in the past? Especially the second where you use two points and a constant product of lengths. BTW, how about seeing your Cassini plot as a contour plot and creating the appropriate 3D shape?
But .... can you see any practical use for these curves?
And then another idea. Why are you limiting yourself to just one ellipse as focal element and still one point (I am talking about your initial question here where the sum of the shortest distances is constant)?
You could use any shape for any of the two (or more?) "focal elements". Again I don't see any use for it but its funny to play around with them 😉
This time the gaps seem to stem from the implicitplot2D function and not from determining the shortest distance. The plots were made with a grid of N=55 resp. N=75 only to save calculation time.
>But .... can you see any practical use for these curves?
One example in my article:
Fig. 11. Ellipse with three foci: triangle, square and circle
The oval in Figure 11 is the outline of a dish (a slightly skewed “blue-rimmed saucer”) with an avant-garde geometric pattern (with our unusual tricks), which could be 3D printed at the porcelain manufactory in Meissen. This production, we recall, was founded by Count Tschirnhaus, a researcher of three-focal ellipses. A service made of such dishes would be very appropriate at a banquet on the occasion of the Fields or Abel Prizes, which are considered the most prestigious prizes in mathematics. They are often called the Nobel Prizes for this queen of sciences. There are many guesses why Alfred Nobel himself did not establish a prize in mathematics, replacing it with a rather controversial and scandalous peace prize.
Ah yes, I now remember your post on porcelain manufacturing. There may be easier ways to create irregular ovals, but why not.
It's a little questionable, though, whether creating plates for the Fields award ceremony is a viable business model 😉
Maybe it could be made more interesting for a broader market by personalization using a silhouette 🙂
🙂
@ValeryOchkov wrote:
Whose profile is this? Who is it?
Its Johann Joachim Kändler (1706 – 1775). -> https://en.wikipedia.org/wiki/Johann_Joachim_K%C3%A4ndler
I got inspired and triggered by "porcelain" 😉
But I admit that using a Mozart silhouette used by Augarten Porzellan would have been a more patriotic choice. On the other hand it would have been much more cheesy and clichéd. 😉
https://www.interismo.at/augarten/schaelchen-mozart
You can try any shapes you like.
Mathcad is dispassionate about this, but I prefer musicians like Strauss Jr., Mozart and Haydn instead of a warmonger.
