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24-Ruby IV
ValeryOchkov24-Ruby IV
24-Ruby IV
November 14, 2022
Solved

New oval - an ellipse^ellipse

  • November 14, 2022
  • 1 reply
  • 6651 views

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.

Ellipse-Ellipse-No-Error.pngEllipse-Ellipse-Error.png

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.

9-Ellipse-Hyperbola.png

Help please!

Best answer by Werner_E

 

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 🙂

Spielerei3.gif

🙂

 

1 reply

W
Werner_E25-Diamond I
25-Diamond I
November 14, 2022

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!

Werner_E_1-1668426685256.pngWerner_E_2-1668426753507.png

 

 

 

    V
    24-Ruby IV
    ValeryOchkov24-Ruby IVAuthor
    24-Ruby IV
    November 14, 2022

    Thanks, Werner!

    Can you send the correct Mathcad 15 sheet?

    W
    Werner_E25-Diamond I
    25-Diamond I
    November 14, 2022

    Here are the sheets corresponding which implement my suggestions #1 and #2 above.

    None of the two sheets is the "correct" (analytical) one, though. I guess in any of these sheets you can find situations where wrong results may show up. But at least they seem to work OK with your setup and S=3.7. Especially the first one - in the second one  the solve block seems to fail  quite often, hence the many gaps in the plot.

     

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