The spherical reflector is based on a circle, with its focal point being in the center. But if the sphere were complete, no light would get out.
We then place the source of the light (the filament of the lamp) in the focal point of the reflector.
When the source is properly aligned with the focal point, the reflector doubles the amount of light passing through the front of the instrument.
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An elliptical reflector is based on the ellipse, a shape that has two focal points.
By placing the filament (light source) at one focal point, the reflector will redirect the light to the second focal point (the conjugate focal point).
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That second focal point of the reflector is then at the focal point of the lenses in a focused leko.
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When a lens is moved in relation to the focal point, the light passing through the lens either converges or diverges. This shift is what is happening when we "run the barrel" in a leko. We are moving the lenses away from the focal point, which is why the light can have a "soft" edge or a "hard" edge (in focus or out of focus).
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Parabolic reflectors return projected light in parallel rays, throwing a very directional beam.
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Asymmetrical reflectors are used for diverting light away from the source in a less even spread. Primarily used for lighting drops in units called Far Cycs, they help the light spread down the drop in more of a column style.
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