My Aunt replaces windows for a living. She has to replace the window in the arch of a doorway, and she sent the window company this diagram.
They sent back a window that has a greater curve than what was needed. How do we measure the window so that they are able to send us the right curve? In this reference image the cardboard is an exact cutout of the glass, and the incorrectly sized window they sent is underneath.
Do we need to measure from the bottom left corner to the center of the curve and send them that measurement? How do we find the center of the curve?
Let us follow your title by assuming that the shape is an arc of an ellipse with horizontal and vertical axes intersecting at the bottom right of the figure. Let us work with the implicit equation
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\tag{1}$$
where $b$ is known: $b=28 \tfrac34 = 28.75$ and $a$ is determined by the fact that if $x=32 \tfrac38 = 32.375$ then $y=9 \tfrac58 = 9.625$ giving the following constraint for $a$ :
$$\frac{32.375^2}{a^2}+\frac{9.625^2}{28.75^2}=1$$
yielding $a=34.3576$.
Now you have all the elements for plotting the corresponding curve, in particular by turning (1) into the explicit form for the quadrant of interest for you:
$$y=b \sqrt{1-\frac{x^2}{a^2}}$$