Finding the focal point of a Reflector

mistake

The so-called "focal point" of a spherical reflector (like the IKEA) is a point that is half the distance of the center of curvature from the bottom of the dish. At this point, a lamp will give you a lot of light as a roughly parallel ray beam. (A true parabolic reflector would work much better for this.) But that is not what you need to send to a condensor fresnel!

Put the lamp at the center of curvature (equidistant from all points on the reflector surface). Then any light hitting the reflector will bounce right back through the lamp arc. The condensor fresnel will "see" this as a bright point source. If the distance from the lamp arc to the condensor fresnel matches the fresnel's focal length, then you will get mostly parallel rays out the other side of the fresnel. The second fresnel will take these parallel rays and focus them into your projection lens. That gives you a bright image.