### Minimum Perimeter Elliptical Hull of a Triangle

This post continues the last post.

Problem: given a triangle, find the ellipse of least perimeter and containing it\

observation: it looks pretty clear that such a minumum perimeter elliptical hull ought to pass thru all three points of the triangle.

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Just as a circle has three parameters (coordinates of its center and radius), an ellipse has 5 (coordinates of center, angle of orientation and the major an minor axes).

Sub-Question: given a triangle (3 points), we need to find ellipses passing thru all 3 and also being aligned along the X and Y axes (ie, its axes are parallel to the coordinate axes). this indicates that the locus of the centers of the possible ellipses will lie along some curve. what is this locus?

If the three pts are (x1,y1) (x2,y2),(x3,y3) and the center is (x0,y0) then they satisfy the eqns for ellipse:

(x1-x0)^2/a + (y1-y0)^2/b = 1

(x2-x0)^2/a + (y2-y0)^2/b = 1

(x3-x0)^2/a + (y3-y0)^2/b = 1

One could use mathematica to eliminate a,b from above and the collect the terms involving (x0,y0)- Thanks, Arun!

Since there is an x0y0 term and linear terms in x0 and y0, the locus seems to be a hyperbola.

In the above, we only found ellipses aligned to the coordinate axes. Intuitively, if we seek ellipses thru all 3 points and with arbitrary orientation, we should get a one parameter family of hyperbolas as the locus of the centers of all these ellipses – a stack of hyperbolas.

Now, for any point that lies on any one of this stack of hyperbolas, we have the center, orientation (given by which parabola from the stack we are on) and the three triangle vertices, we have an ellipse fully determined. For this ellipse, we could use the approximate expression known (or number crunching) to find its perimeter, then repeat for all possible ellipses and find the min perimeter.

Known facts: many of the centers of the triangle lie on the so-called Euler line. The incenter does not, in general. If we are looking for the containing ellipse with least area (not perimeter), its center is always the centroid of the triangle and the ellipse is called the Steiner circumellipse.

Speculation: The minimum perimeter ellipse containing the triangle appears to have its center necessarily inside the triangle. That the perimeter of an ellipse has no known closed form expression need not necessarily imply that we cannot say anything precise about the center of the minimum perimeter elliptical hull. It is conceivable that say, it lies on the line joining the incenter and circumcenter.

Problem: given a triangle, find the ellipse of least perimeter and containing it\

observation: it looks pretty clear that such a minumum perimeter elliptical hull ought to pass thru all three points of the triangle.

-------------------

Just as a circle has three parameters (coordinates of its center and radius), an ellipse has 5 (coordinates of center, angle of orientation and the major an minor axes).

Sub-Question: given a triangle (3 points), we need to find ellipses passing thru all 3 and also being aligned along the X and Y axes (ie, its axes are parallel to the coordinate axes). this indicates that the locus of the centers of the possible ellipses will lie along some curve. what is this locus?

If the three pts are (x1,y1) (x2,y2),(x3,y3) and the center is (x0,y0) then they satisfy the eqns for ellipse:

(x1-x0)^2/a + (y1-y0)^2/b = 1

(x2-x0)^2/a + (y2-y0)^2/b = 1

(x3-x0)^2/a + (y3-y0)^2/b = 1

One could use mathematica to eliminate a,b from above and the collect the terms involving (x0,y0)- Thanks, Arun!

Since there is an x0y0 term and linear terms in x0 and y0, the locus seems to be a hyperbola.

In the above, we only found ellipses aligned to the coordinate axes. Intuitively, if we seek ellipses thru all 3 points and with arbitrary orientation, we should get a one parameter family of hyperbolas as the locus of the centers of all these ellipses – a stack of hyperbolas.

Now, for any point that lies on any one of this stack of hyperbolas, we have the center, orientation (given by which parabola from the stack we are on) and the three triangle vertices, we have an ellipse fully determined. For this ellipse, we could use the approximate expression known (or number crunching) to find its perimeter, then repeat for all possible ellipses and find the min perimeter.

Known facts: many of the centers of the triangle lie on the so-called Euler line. The incenter does not, in general. If we are looking for the containing ellipse with least area (not perimeter), its center is always the centroid of the triangle and the ellipse is called the Steiner circumellipse.

Speculation: The minimum perimeter ellipse containing the triangle appears to have its center necessarily inside the triangle. That the perimeter of an ellipse has no known closed form expression need not necessarily imply that we cannot say anything precise about the center of the minimum perimeter elliptical hull. It is conceivable that say, it lies on the line joining the incenter and circumcenter.