Remove exterior siding. Frame a rectangular window opening using 2-by-4s to make studs, a sill and a transom. The frame opening should be a half-inch taller and wider than your semi-circle window. Leave the plywood or laminate exterior intact.
Use a pencil to trace the outline of your window on a piece of cardboard. Make another outline that is a quarter-inch larger than the one you traced. Cut around the outer line with a utility knife. The curved cardboard in the shape of your window will be a template for installation.
Tape the cardboard template to the inside of the plywood or laminate wall sheathing and trace around the outline with a pencil.
Bore an access hole in the sheathing large enough to insert the blade of a reciprocating saw. Cut around the outline.
Place a 2-by-4 diagonally against the top left corner of the frame. Hold a layout square against it. Mark a 45-degree angle so that when you cut and install it, the inside of the 2-by-4 will fit against outside of the window curve. Measure a right corner diagonal the same way.
Cut the top corner 2-by-4s and nail them into place with 8d nails. Their bottoms fit against the frame studs and their tops fit against the header. The top of the frame will now look like half of an octagon. The frame is 1/2-inch wider than the bottom, sides and curved top of your window.
Install the window in this space following manufacturer’s instructions.
Scribe the lengths of new exterior siding with a compass so they will fit the curve of the window and maintain the correct overlap. Cut the curves with a jigsaw. Hold the lengths in place. If they fit the curve, mark the ends so you can cut them to the correct length.
Apply a bead of caulk where the siding fits against the edge of the window. Cut the siding to the correct length. Fit the curved end of the siding over the caulk and nail it in place with galvanized nails. Caulk around the window when you finish applying the siding.