Q.Why are dish antennas curved?

Step 1. A dish antenna receives very weak electromagnetic signals (radio or microwaves) arriving from a distant source as an effectively parallel bundle of rays, exactly like light from a star arriving at a telescope.

Step 2. By the image-tracing rule for a curved (parabolic, a limiting case of spherical for this purpose) reflecting surface, a ray travelling parallel to the axis reflects through a single common focal point $F$ -- this is true for every one of the many parallel rays striking the dish's curved surface, all converging to the same point.

Step 3. If the dish were flat instead, parallel rays would simply reflect off, still parallel, spread over a large flat area, with no concentration at all -- a receiver could only ever intercept the tiny fraction of signal striking its own small physical size.

Step 4. By curving the dish, essentially the entire incoming signal collected across the whole large dish area is redirected to arrive together at the single small focal point, where the actual receiving antenna/feed horn is placed -- giving a dramatically stronger, concentrated signal than a flat receiver of the same size could ever collect on its own.

Apply the parallel-ray-to-focus image-tracing rule for a curved reflecting surface to the large collecting area of a dish antenna.

• Thinking a flat reflector would work equally well, without recognising it cannot concentrate parallel rays to a single point.