Why does a parabolic dish focus distant signals?

A large radio dish gathers a weak signal over a broad area and directs it toward a much smaller receiver. Its useful shape is not just any bowl. A paraboloid has the reflecting geometry needed to bring waves arriving parallel to its axis toward a common focus.

The illustration takes a two-dimensional slice through an ideal dish. Its curve is y = x²/(4f), with the vertex at (0,0) and focus at (0,f). Here f is two length units. Parallel rays travel downward and meet the curve at different horizontal positions.

At each meeting point, the angle of incidence equals the angle of reflection, measured from the local normal: the line perpendicular to the surface. The parabola's slope changes continuously with position. Those changing slopes redirect all the pictured axial rays through the same focus, even though their surface points and reflected directions differ.

For example, at x equal to four the surface height is two, level with the focus. That downward ray reflects horizontally toward (0,2). A ray striking closer to the vertex reflects upward and inward instead. Rotating this section about its axis produces the dish surface.

In an actual antenna, radio waves have finite wavelengths, so diffraction and surface errors prevent an infinitely sharp focus. The receiver and its feed structure also matter. The dish concentrates intercepted energy; it does not amplify that energy by geometry alone. Accurate pointing is essential because signals arriving away from the axis do not share this same ideal focus.