Tree ovals - mean, gmean & hmean

[ValeryOchkov]

Tree ovals: mean - ellipse, gmean - Cassini oval & hmean - ???

Has the 3-d oval a name?

And what about  Generalized mean?

Solution without animation is not solution!

[ValeryOchkov]

d-ellipse

d=1 - normal ellipse

d=0 - Cassini oval

d=-1 - see above (hmean)

[ValeryOchkov]

With 3 focu

[ValeryOchkov]

Zoom - three petals of Bernoulli lemniscate

[ValeryOchkov]

One more animation - new lemniscates!

[ValeryOchkov]

Ellipse and Kolmogorov's ellipse

[ValeryOchkov]

plot 3/(1/sqr((x-1)^2+y^2)+1/sqr(x^2+(y-sqr(3))^2)+1/sqr((x+1)^2+y^2))=1.14668 - Wolfram|Alpha 

[ValeryOchkov]

An ellipse, a Cassini oval and Point oval

[ValeryOchkov]

2 foci and 7 Ovals - only rigth part.

[ValeryOchkov]

The end - we have three circles at d= -1 (the harmonic oval), d=-2 (no name) and d=-3 (no name too).

What curve we have at d=3? (d=2 - a circle, d=1 - an ellipse, d=0 - a Cassini oval)

[ValeryOchkov]

4 foci, 7 ovals. d=3, 2 (circle), 1 (elipse), 0 (Cassini oval), -1 (my oval), -2, -3.

[ValeryOchkov]

Nice too

Wolfram Alpha 

Can you do it with more than 4 foci - 5, 6, 7 Etc

[ValeryOchkov]

Seven ovals - d = 3, 2, 1, 0, -1, -2 & -3 - 4 foci

[ValeryOchkov]

New new ovals mean, gmean and hmean(L1, 2 L2)

[ValeryOchkov]

Triple lemniscate

[ValeryOchkov]

[ValeryOchkov]

Exponentiation oval

[ValeryOchkov]

[MaxNori · Accepted solution]

For the hmean (d=-1) I think it is the Cayley oval! 

https://mathcurve.com/courbes2d/cayleyovale/cayleyovale.shtml

[ValeryOchkov]

triginimetric ovals

[ValeryOchkov]

Exponential ovals