1. Surface Energy Functionals
An Overview

2. Functional #1: Bending Energy
Minimize total curvature
Proposed minimizers:
Lawson Surfaces
Average normal curvature at a given surface point
Lawson Surfaces for
genus = 2, 3, 1
is a topological constant, ignore!

3. Functional #1: Bending Energy
Bending Energy ≡ Willmore Energy
(topological constant)

4. Functional #1: Bending Energy
Special case: open surfaces with position constraints on boundary:
Bending Energy <=> Surface Area
Minimize surface area
→ Zero mean curvature everywhere
(“minimal surfaces”)
Soap films:
natural minimal surfaces

5. Functional #1: Bending Energy
Consider a functional that favors umbilics
Bending Energy
Bending Energy
≡ “how umbilic is a surface”

6. Bending Energy as an Aesthetic Functional

7. Functional #2: MVS Minimum Variation Surfaces
Minimize change in principal curvatures
along respective principal direction
Curvature variation along lines of curvature
(function of κ1, e1, κ2, e2)
Introduced by Moreton and Séquin (1992)

8. Functional #2: MVS Minimum Variation Surfaces
Bending Energy Minimizer
MVS Energy Minimizer

9. Functional #2: MVS Minimum Variation Surfaces
Produces cyclides (lines of curvature are circles)
spheres, cylinders, cones, tori, Horn cyclides …all have MVS energy = 0
MVS cannot distinguish between cyclides!
image © Jürgen Meier (

10. Functional #3: MVScross (MVS + cross terms)
Minimize change in principal curvatures
along both principal directions
Gradient of principal curvatures in Riemannian metric
Therefore, more complete than MVS
MVScross Energy = 0 for spheres and cylinders
Best approximation of a sphere or cylinder

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