Define a surface, choose a point, and see the same level set in both the xy-plane and on the 3D surface.
Try: x^2+y^2, x^2-y^2, sin(x)+cos(y), x*y, 3*exp(x)*sin(y)
pi/2 works
sqrt(2) works
3D surface and the same level set
The thick blue curve satisfies f(x, y) = f(P). It lies on the surface and in the horizontal plane z = f(P).
surface
level curve f(x, y) = f(P)
tangent plane / normal
Level-set view in the xy-plane
Click anywhere to move P. The highlighted contour here is the same curve drawn on the 3D surface.
Move P and f(P) changes. The same level curve moves in both pictures.
Geometry at P
P =
c = f(P) =
∇f(P) =
c = f(P) =
∇f(P) =
The level curve has tangent direction T satisfying ∇f · T = 0. In 3D, the tangent plane is the first-order approximation of the surface at the same point P.