Bill's idea is an analog method for the drafting technique I was proposing. But essentially, if you draw out a matching half-circle the same size (diameter) as your sphere part with the 'seam' oriented vertically then draw an offset line parallel to the 'seam' line & the same 'height' as the bubble (this represents the 'edge' side view of the flat projection plane.) Then draw a 90 degree line that connects to the top of the vertical plane line and then draw another 1/2 circle that is offset the same distance as the original. You'll end up with two half circles that are rotated 90 degrees from each other and facing the vertical & horizontal reference lines. This is the basic setup to map the opening.
Now, if possible using a copy of the image showing the frontal aspect of the helmet & faceplate that is a matching size (diameter) as your bubble, overlay it with a regular grid of squares. This will provide a reference coordinate system that you can then "map" onto the vertical and horizontal lines of your drawing. Basically, your "1/2 circle" drawing gives you a top and side view of the '2D image' so you can mark the width and height of the faceplate opening. Just mark the reference lines with points then draw horizontal and vertical lines from those points to the 1/2 circles that represent the side and top views of the bubble. Then you can measure the point of the intersection with the circles from the 'seam' line, or simply lay the bubble half on the drawing and mark the lengths along the bubble edge to use to directly transfer to the other bubble half. I'd recommend marking the faceplate half with an X & Y axis so you can transfer the points accurately. This is easily done by drawing a full circle with both a vertical and horizontal lines through the center point, 90 degrees apart, then using the intersections to transfer to the faceplate bubble half's edge. Then just "connect the dots" with thin strips of tape.
Measuring the map points from your image is just a matter of taking the distance at the center of the faceplate opening between the top (and bottom) edge of the helmet to the top (and bottom) of the opening. This distance is marked onto the reference lines. Then repeat for the side points. The result is the 'boundaries' of the opening.
Doing the corners is a little trickier. On the picture, you have to mark a line connecting opposite corners and then measure the angle between that line and the vertical lines of your grid. This same angle then is used to angle off from your drawing's vertical reference line (could be connected to the bottom endpoint since the top one is already used for the top view. Then it is just again drawing another 1/2 circle offset the same distance as the other reference lines followed by the same mapping procedure.
If the opening is symmetrical about both axes, then you only need a mask (template) for one of the four quadrants which then can just be flipped about for the other three.
I know this seems complex, but its just in the explaining, not in the 'doing'. Main point is, if you 'project' the front view onto a flat plane, you get the image (picture) you are working from. To map this 'projection' back onto the 3D surface of the sphere just requires taking the image's points in any one plane (for example, the 'side view' is really a vertical slice on a plane parallel to the plane of the paper...and so forth) and connecting them back to the sphere's cross section in that same plane. By rotating the cross-sectional plane, you can develop enough points to map the correct shape of the opening. This is easily done in this case because the bubble is a regular sphere and hence of the same cross section at any point that passes through it's center point. So long as you have a drafting triangle and a compass, you should be able to do it very accurately. If you have drawing software on the computer, even more so.
Mastering this draftsman's technique gives you a useful tool for other applications were you have an image (i.e., 2D) of something that is of a 3D surface not completely parallel to the image plane (as in 'flat'.) For example, I wish the artists at Microscale who created the B-26 bomber nose art on one of their decal sheets had used it to remove the 'distortion' in the images they used for reference instead of just 'tracing' the image!
Regards, Robert