While we're playing around with ideas this is another option to consider. The numbers are different than those I posted yesterday.

The larger ellipse contacts the collar tie (WP1), the post (WP3) and the rafter at 8½ feet below the peak (WP2). The rafter at this point of contact is not tangent to the curve of the ellipse.

The heights (values on the y-axis) are measured from the upper surface of the tie beam:
WP1 = (0, 109)
WP2 = (–106.779505, 78)
WP3 = (–160, 0)

The dimensions for the Elliptic Arch Calculator are:
Width 1 = 2 × 106.779505 = 213.55901
Height 1 = 109 – 78 = 31
Width 2 = 2 × 160 = 320
Height 2 = 109

Solving the ellipse semi-axes:
Semi-Major Axis on the x-axis = 161.10074
Semi-Minor Axis on the y-axis = 123.40069
Note that the x-axis is now 14.40069 below the upper surface of the tie beam.

The axes of the smaller ellipse are 8 less than values above.



Note: This post has been edited. I had the x and y values of WP2 bass-ackwards.