next curve  previous curve  2D curves  3D curves  surfaces  fractals  polyhedra 
MULTIFOCAL ELLIPSE or EGGLIPSE
Curve studied by Tschirnhaus in 1686, and Maxwell in
1846.
Other names: multiellipse, polyellipse, hyperellipse, Tschirnhaus egg. Sites : English Wikipedia (Nellipses) English Wikipedia (Generalized conic) French Wikipedia (point de FermatWeber) http://www.aimath.org/WWN/convexalggeom/AIM.pdf On the Approximation of Convex, Closed Plane Curves by Multifocal Ellipses book by Vincze text by Maxwell article on the algebraic side on the triellipses http://www.journaloftheoretics.com/Articles/66/MFCC.pdf?q=foci on the Fermat point Gyula Sz.Nagy (June 1950). "Tschirnhaus'sche Eiflachen und EiKurven". Acta Mathematica Academiae Scientiarum Hungarica 1 (2): 167–181. 
The multifocal ellipses, with n foci, are the loci
of the points of the plane for which the sum of the distances to n
points is constant.
Multipolar equation: .
The case n = 1 gives, of course, the circles.
The case n = 2 gives the classic ellipses. More
precisely,
is, at the minimum, AB and when ,
gives the ellipse with foci A and B and major axis a,
reduced to the segment line [AB] when a =AB.
For n = 3, the minimal value of
is reached at a unique point F, called the Fermat point of the triangle
ABC, and the ellipses with three foci, called triellipses,
form concentric ovals around F.
If the angles of the triangle are acute, then the Fermat point is defined by the fact that, from it, the sides are seen under 120° angles (example opposite). If one of the angles is greater than or equal to 120°, then the Fermat point is the corresponding vertex; in the flat case with distinct vertices, the Fermat point is the middle vertex. If two points coincide, they also coincide with the Fermat point, and we get a Cartesian oval. The minimal value of the sum is where , and a,b,c are the sides of the triangle. As we can see opposite, the triellipses have a tangent at all point except when they pass by the vertices of the triangle. 

Generalisation of the gardener's ellipsetracing method for a triellipse, proposed by Maxwell. 

Opposite, the case of an equilateral triangle. 

Since the product
of the eight terms
is a polynomial function of ,
the curve with equation =
0
(called HeltonVinnikov curve) is algebraic of degree 8, and contains the
triellipse
.
For example, the equation of the algebraic curve corresponding
to the triellipse passing by the vertices of the previous equilateral triangle,
represented opposite, is:
In the other cases, the triellipse is only an oval connected component of the algebraic curve, that can contain at most four of them. 

As for the ellipse, the tangent to the ellipse is constructed perpendicularly to the vector . 

For ,
the minimal sum is always obtained at a unique point, called the FermatWeber
point.
The nellipse is included in an algebraic curve of degree , that can be lowered to when n is even. Opposite, the quadriellipse passing by the vertices of a square, and the associated algebraic curve, of degree 16 – 6 = 10. 

Compare to the Cassinian
curves, with equation .
Animation of the complete triellipse associated to an
equilateral triangle, starting from a = 0.
For large values of a, the curve tends to become a circle centred on the barycentre of . 

next curve  previous curve  2D curves  3D curves  surfaces  fractals  polyhedra 
© Robert FERRÉOL 2017