ParaInscribedEllipse3

Parallelograms inscribed in Ellipse -3

Given an ellipse with axes a and b, construct all parallelograms EFGH inscribed in the ellipse and having the same area E < 2*a*b. Then find those that have maximum/minimum perimeter.

[functions] y1 = rsqrt(b² + (a²-b²)(cos(x))²) + sqrt(1-r²)sqrt(a²-(a²-b²)(cos(x))²)
[rparameters] r = 0.5, a = 3.0, b = 4.0
[nparameters] nrPts = 190
[intervals] [0.0, 2.0*pi]
[nrofpoints] nrPts
[comments] Function giving 1/4 of the perimeter of a parallelogram inscribed in an ellipse with axes a, b.

The construction of all the inscribed parallelograms with constant area has been done in ParaInscribedEllipse2.html . The parallelograms result by applying an affinity to rectangles inscribed in a circle. For convenience the circle can be taken to be the unit circle and the affinity given by the diagonal matrix diag(a,b), where a, b are the axes of the ellipse. The perimeter of the parallelogram is four times the length |O*J*|+|O*K*| = |f(OJ)|+|f(OK)|. If J = r(cosu, sinu), then K = sqrt(1-r^2)(-sinu, cosu) and applying f (i.e. the matrix), we get: J* = r(a*cosu, b*sinu), K* = sqrt(1-r^2)(-a*sinu, b*cosu). The problem reduces to that of finding the minimum/maximum of the function h(u) = r*sqrt(a^2*cos(u)^2 + b^2*sin(u)^2) + sqrt(1-r^2)*sqrt(a^2*sin(u)^2 + b^2*cos(u)^2) = = r*sqrt(b^2 + (a^2-b^2)*cos(u)^2) + sqrt(1-r^2)*sqrt(a^2 - (a^2-b^2)*cos(u)^2). The red line gives the graph of the function h(u) in dependence of the angle u = angle(IOJ). The minimum occurs when u is a multiple of pi, the sides of EFGH being parallel to the axes. For values of u equal to multiples of pi/2 the sides of EFGH are again parallel to the axes but the area takes the position of a local minimum and not a global maximum as one may expect. Finally the maximum is obtained at a value of u that can be calculated by equating the derivative of h(u) to zero. (Exercise: show that this value satisfies cos(u) = sqrt( ((a^2+b^2)*r^2 - b^2)/(a^2-b^2)) . For values of r smaller than r0=b/sqrt(b^2+a^2) there is no solution to the previous equation and the graph becomes sinusoidal, providing a maximum and minimum as expected, attained at rectangles with sides parallel to the axes. ) By the above construction the maximum occurs at values of u for which the diagonals of ABCD coincide with the x-achis. Switch to the selection on contour-tool (press CTRL+2), catch and move point J and watch the corresponding parallelograms inscribed in the circle and their images, inscribed in the ellipse. Watch also the dependence of point L(u,f(u)) on J, on the graph. Finally change the parameter r to values less than r0 to see the graph changing behaviour, as stated in the exercise above.

Parallelograms inscribed in Ellipse -3

See Also