Abstract: A curved flapper and seat is disclosed for use in a subsurface safety valve. The flapper is biased to a normally closed position to prevent fluid flow through the wellbore. The curved flapper has a sealing surface for engaging a corresponding sealing surface on a seat when the flapper is in its closed position. The sealing surface of the flapper is configured to contact the sealing surface of the seat along a sinusoidal sealing line, or seam, such that the reactive force from the seat is normal to the sinusoidal seating line. In one aspect, the sealing surface of the flapper has a convex spherical configuration relative to the seat. The sealing surface of the seat, in turn, has a concave conical shape relative to the flapper. When well conditions dictate, a resilient soft seat may optionally be used, and is disposed on the seat proximate the sinusoidal seating line.