I'm trying to calculate the points of a certain shape based on a given height $h$, width $w$ and radius $r$. The shape can be described as a bar with rounded ends that has been bent in half at a certain angle along a fixed inner radius (very similar to the profile of a name plate stand). The shape looks as follows, where I have marked the relevant parameters.
Diagram of curved shape with parameters
My approach so far has been to extend the shape to include another point above the curved section, and use that to calculate the angle of the curve, as shown below. My goal is to be able to calculate the coordinates of the points highlighted in yellow.
Diagram of shape with highlighted points (my attempt)
However, while trying to solve for the highlighted points, it leads me to the following expression which I can't seem to solve:
$$ H + 2r - \frac{2r}{sin(arctan( \frac{w - 2r}{2(H - 2r)} ))} = h $$
I wish to solve for $H$. However I can't seem to rearrange it to get that. I got to this expression by having $H$ be the full height of the extended shape, then calculating the coordinate of the center of the curved section of the path based on the angle being made, and then adding on $3r$ to get to the height of the shape. I need to rearrange for H because the only parameter I can depend on is the original height of the shape.
Perhaps there is a better way though to calculate the angles involved. I would also like to figure out if there is a generalized approach that can be applied to similar shapes.
I'm sorry if this isn't clear, but I'm happy to clarify and provide more information if required.