Focal surface
![Focal surface of Monkey saddle. The two surfaces (upper and lower) are antisymmetric and go to infinity at the central point (which has zero Gaussian curvature).](/Images/godic/202501/13/Focal_surface4147.png")
![A surface with an elliptical umbilic, and its focal surface.](/Images/godic/202501/13/Elliptical_umbilic_focal_surface4147.png")
![A surface with a hyperbolic umbilic and its focal surface.](/Images/godic/202501/13/Hyperbolic_umbilic_focal_surface4147.png")
For a surface in three dimension the focal surface, surface of centers or evolute is formed by taking the centers of the curvature spheres, which are the tangential spheres whose radii are the reciprocals of one of the principal curvatures at the point of tangency. Equivalently it is the surface formed by the centers of the circles which osculate the curvature lines.