Geodesic nets

Media

---

Description

---

The concept of isometry lies at the core of the study of surfaces. Loosely speaking, two surfaces are isometric if one can be obtained by bending but not stretching the other. The deforming map is then called an isometry, and the properties of a surface that are invariant to isometries are called intrinsic properties.
Surfaces that are locally isometric to a plane are called developable surfaces. Since these surfaces can be obtained in our physical world by bending thin sheets of material, they are of particular interest for manufacturers, architects and artists.
Smooth developable surfaces are well-studied in differential geometry and are often characterized as surface with vanishing Gaussian curvature, or as ruled surfaces with constant normal along each ruling.
Given a smooth developable surface a natural discretization can be obtained as it can be locally represented by a single curve and its orthogonal rulings. This representation however has limitations when it comes to interactive modeling of a developable surface.
Another way to explore developable surface is made possible by discretizing a lesser-known local condition for them: the existence of an orthogonal geodesic parametrization.

Discrete orthogonal geodesic nets (DOGs) are a quad mesh analogue of developable surfaces.
The shape space of DOGs generally is locally a manifold of a fixed dimension - apart from singularities - implying that DOGs are continuously deformable.Smooth flows can be constructed by a smooth choice of vectors on the manifold’s tangent spaces, selected to minimize a desired objective function under a given metric. Such vectors can be computed by solving a linear system. Using new findings a geometrically meaningful way to handle the singular points was devised.

References

---

Authors

See also