1. Cut the boards. Cut the front, back, bottom, and ends to size from 3/4-inch-thick redwood or cedar. The back piece should be roughly the width of your window plus an inch or two.
2. Drill holes. To mount the planter to the wall later, drill countersunk mounting holes through the back piece now, centering the holes over studs in the wall. Bore 1/2-inch drain holes in the bottom piece.
3. Craft the front. Mark the opening on the front, drill a hole to insert blade, and cut the arch-top opening. Rout a rabbet on the inside face of the frontpiece and cut a piece of 1/4-inch exterior plywood to fit the rabbeted opening.
4. Assemble the box. Butt-join the ends to the back and miter the front corner joints. Glue all joints, using water-resistant glue. Reinforce the butt joints with 1-1/2-inch deck screws and the mitered joints with finish nails. Counterbore the holes so the screw heads are slightly recessed. You can putty over them later. Set the nails.
5. Make the brackets. Enlarge the gridded bracket pattern, and transfer it three times to 2x10 stock. Cut the three brackets to shape. Using an exterior latex or enamel, paint the planter box, brackets, and plywood panel.
6. Attach the brackets. Screw the brackets to the bottom of the planter. Glue the painted plywood panel into the arch-top rabbeted opening.
8. Hang the box. Secure the planter to the house with 3-1/2-inch deck or lag screws. Hit studs when possible.