Technical Papers
The Shape Space of Discrete Orthogonal Geodesic Nets
Event Type
Technical Papers
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TimeWednesday, 5 December 20182:57pm - 3:18pm
DescriptionDiscrete orthogonal geodesic nets (DOGs) are a quad mesh analogue of
developable surfaces. In this work we study continuous deformations on
these discrete objects. Our main theoretical contribution is the characterization
of the shape space of DOGs for a given net connectivity. We show that
this space is locally a manifold of a fixed dimension, apart from a sparse set
of 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. We show how to compute such vectors by solving a linear
system, and we use our findings to devise a geometrically meaningful way
to handle singular points. We base our shape space metric on a novel DOG
Laplacian operator, which is proved to converge under sampling of an analytical
orthogonal geodesic net. We further show how to extend the shape
space of DOGs by supporting creases and curved folds and apply the developed
tools in an editing system for developable surfaces that supports
arbitrary bending, stretching, cutting, (curved) folds, as well as smoothing
and subdivision operations.