I would like to cut a straight piece of moulding into wedged shapes that, when assembled, will form an ellipse. The goal is to wrap this moulding around the perimeter of an elliptical arch that is installed between two porch columns. I need to know the miter saw angles for cutting each wedge.
I can measure distances a and b on the ellipse (see illustration below) where the width is 2a and the height is 2b. I would like to cut moulding wedges where the outside edge is a fixed width w. There will be wedges 1 through i cut from the straight piece. The moulding has height h.
How can I generate a spreadsheet giving the cut angles for wedge i?
I have to apologize for not giving a mathematical answer, but are you quite sure that those things are ellipses, or do they just look like ellipses? Do you have to do a lot of them? If so, are they all exactly the same? As a practicing woodworker I would never trust a "theoretical prediction", even if it was easy to do (which it would be) because errors in measurement and execution are cumulative. I would cut two "wedges" with square ends, offer them up and measure the gap.
SUPPLEMENT Then I guess you could define a set of points on the ellipse by $$x_i=a\cos(\theta_i),y_i=b\sin(\theta_i)$$ Then the angle that each wedge makes with the horizontal is $$\arctan[(y_{i+1}-y_i)/(x_{i+1}-x_{i})]$$ and from these angles you easily get the angles between the wedges and hence the cut angles. The length of each wedge comes from Pythagoras.
The remaining question is how to choose the angles $\theta_i$. The wedges can be longer closer to the center of the arch, but just how much longer probably doesn't matter. A simple choice is to make the angles $\theta_i$ equally spaced and this somewhat mimics planetary motion in which equal arcs are swept out in equal times. Otherwise this is a good use for a drawing board, since the only criterion is what looks good.
As a practical suggestion, on the job I would work in from each end and fit the last wedge by offering it up and marking it, so as to cancel any errors in the cutting. Good luck!