I was wondering about a generalization of conic sections and if ways of drawing them translate well into the generalization. I'm thinking these are like k-ellipses, but only in special cases.
There's the classic way to draw an ellipse. Put a rubber band around two pegs. Keep the rubber band taut as you move clockwise, or counter clockwise. You'll trace out an ellipse. This works because theirs a constant amount of tension in the band between the point on the curve and the two pegs plus the tension between the two pegs.
The sum of the distances of a point on the curve to the two foci are held constant because the sum of the distance of the two foci, plus the distance to the two foci are held constant.
What happens if you add another "focus"? What sum rule still applies?
For example, If your three foci are arranged as vertices in an equilateral triangle and you trace out a clock wise (or counter-clockwise) curve while keeping the band taut, you'll still trace out an ellipse, right?
What if that initial triangle is Isosceles? Scalene?
When dealing with tension, I'm thinking this isn't an n-ellipse problem, but I'm not sure. The interest isn't in the total distance between the three points, but the distance to two of the points and their distances to each other.
Suppose your foci are $(x_1, y_1),(x_2, y_2),(x_3,y_3)$.
The 3-ellipse equation is $\sqrt{(x-x_1)^2+(y-y_1)^2}+\sqrt{(x-x_2)^2+(y-y_2)^2}+\sqrt{(x-x_3)^2+(y-y_3)^2}=d$
The "Band Equation" for what I have in mind is more like :
$\sqrt{(x-x_a)^2+(y-y_a)^2}+\sqrt{(x-x_b)^2+(y-y_b)^2}+\sqrt{(x_a-x_c)^2+(y_a-y_c)^2}+\sqrt{(x_b-x_c)^2+(y_b-y_c)^2}=T$
Where $(x,y)$ is a point on the curve, $(x_a,y_a)$ and $(x_b,y_b)$ are the points adjacent to $(x,y)$ in the quadrilateral shape the band takes on. $(x_c,y_c)$ is the point not adjacent on the quadrilateral.
The values of $a,b$ and $c$ are the same $1,2$ and $3$ as the 3-ellipse case, but the values differ for them different over the curve.
Suppose the 3 focal points are $(0,2L), (-L,0),$ and $(L,0)$.
The $n-ellipse$ equation is $\sqrt{x^2+(y-2L)^2}+\sqrt{(x-L)^2+y^2}+\sqrt{(x+L)^2+y^2}=d$
The Band Equation in the first quadrant would be $\sqrt{(x-L)^2+y^2 }+\sqrt{x^2+(y-2L)^2}+\sqrt{L^2+4L^2}+2L=T$. For the II quadrant would be $\sqrt{x^2+(y-2L)^2}+\sqrt{(x+L)^2+y^2}+2L+\sqrt{L^2+4L^2}=T$ These are the same by symmetry.
and in quadrants $III$ and $IV$: $$T=2\sqrt{L^2+4L^2}+\sqrt{(x-L)^2+y^2}+\sqrt{(x+L)^2+y^2}$$
So the Band Ellipses are 3 partial 2-ellipses with each ellipse corresponding to its 2 adjacent foci. The 3-ellipse looks to be something different since its the sum of three terms depending on x and y while the Band Ellipse is is the sum of two lengths adding to a quadrant dependent value.
In the two foci case, the Band Curve and the 2-ellipse are the same because one of the 4 square roots goes away, and the third is a constant resulting in the equation for a usual ellipse.
Are these really different in the foci case, or am I missing something? Is there a technical name for what I'm calling "Band Curves"?
Further evidence they are different. The n-ellipse is a smooth curve, the Band Curves don't necessarily have continuous derivatives.
Picture from Desmos Graphing Calculator. Red is n-ellipse, the others or 3 components of a Band Curve.
