TL;DR

Where map ellipse’s parametric representation to the one used in SVG.

In Ellipses (for SVG): parameter and angles we took a quick look at a few possible representations for an ellipse on the $XY$ plane, namely the implicit, the explicit and the paremetric representations.

For our bounding box purposes, the best one is the parametric, because it allows us to separate the two coordinates and find maxima and minima independently. So, that is the representation we are aiming for. This is also quite useful to represent arcs of ellipses, because it suffices to restrict the parameter $t$ to a suitable interval (as opposed to considering the full $[0,2\pi[$ range).

Alas, SVG paths come with a different representation for an arc of ellipse, based on the following parameters:

• a source point $\mathbf{P}_1$ (on the ellipse);
• an destination point $\mathbf{P}_2$ (on the ellipse);
• the two radii $r_x$ and $r_y$ that we already know from our representations (we are lucky with them);
• a rotation angle (counter-clockwise) $\phi$ (same as that in our parametric representation, so we’re lucky again but we have to consider that it’s expressed in degrees);
• two flags (taking values of either $0$ or $1$) to indicate:
  • $f_A$ which ellipse should be chosen (the equations usually allow two distinct ellipses, which are mirror images with respect to a line through $\mathbf{P}_1$ and $\mathbf{P}_2$);
  • $f_S$ which arc between $\mathbf{P}_1$ and $\mathbf{P}_2$ should be considered (i.e. the longer or the shorter one).

So, starting from the parameters above, we have the goal to find:

• the center of the ellipse $\mathbf{C}$;
• the parameter $t_1$ corresponding to point $\mathbf{P}_1$;
• the parameter $t_2$ corresponding to point $\mathbf{P}_2$;
• the parameters $t_{begin}$ and $t_{end}$ from $t_1$ and $t_2$ so that $[t_{begin}, t_{end}]$ is a contiguous interval representing the arc of ellipse we are after.

As you might have already guessed, our next move when we have these parameters will be to find the minimum and maximum values for the two coordinates when $t$ varies in the $[t_{begin}, t_{end}]$ range, and this will give us the bounding box.

Stay tuned!