Height threshold50%
Level sets20
Rotate30°
—
Height functions on surfaces reveal topology. Critical points — minima, maxima, saddles — count the holes via the Morse inequalities and Euler characteristic.
Morse inequality: #minima − #saddles + #maxima = χ (Euler characteristic). Torus: 1−2+1=0. Sphere: 1−0+1=2. Each level set crossing a saddle changes connectivity — a handle is attached. This builds CW-complex decompositions from smooth functions.