Gaussian curvature of 2D surfaces

What you are seeing: a torus colored by Gaussian curvature: positive on the outer rim, negative on the inner rim, zero on the top/bottom circles. The sphere has constant $K = 1/R^2 \gt 0$ , the hyperbolic plane has constant $K \lt 0$ , the cylinder has $K = 0$ .

Figure 1. Gaussian curvature on a selectable surface: torus, sphere, cylinder, or saddle. Red marks K > 0, blue marks K < 0, grey marks K = 0.

Surface

R / r3.00

WHAT TO TRY

Switch between the four surfaces and watch how the curvature colouring and the K profile change.
Drag the canvas to rotate the surface; vary the torus aspect ratio R/r.