In
computational geometry, the source unfolding of a
convex polyhedron
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the w ...
is a
net
Net or net may refer to:
Mathematics and physics
* Net (mathematics), a filter-like topological generalization of a sequence
* Net, a linear system of divisors of dimension 2
* Net (polyhedron), an arrangement of polygons that can be folded u ...
obtained by cutting the polyhedron along the
cut locus
The cut locus is a mathematical structure defined for a closed set S in a space X as the closure of the set of all points p\in X that have two or more distinct shortest paths in X from S to p.
Definition in a special case
Let X be a metric s ...
of a point on the surface of the polyhedron. The cut locus of a point
consists of all points on the surface that have two or more shortest
geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
s to
. For every convex polyhedron, and every choice of the point
on its surface, cutting the polyhedron on the cut locus will produce a result that can be unfolded into a flat plane, producing the source unfolding. The resulting net may, however, cut across some of the faces of the polyhedron rather than only cutting along its edges.
The source unfolding can also be continuously transformed from the polyhedron to its flat net, keeping flat the parts of the net that do not lie along edges of the polyhedron, as a
blooming
Bloom or blooming may refer to:
Science and technology Biology
* Bloom, one or more flowers on a flowering plant
* Algal bloom, a rapid increase or accumulation in the population of algae in an aquatic system
* Jellyfish bloom, a collective ...
of the polyhedron. The unfolded shape of the source unfolding is always a
star-shaped polygon
In geometry, a star-shaped polygon is a polygonal region in the plane that is a star domain, that is, a polygon that contains a point from which the entire polygon boundary is visible.
Formally, a polygon is star-shaped if there exists a poi ...
, with all of its points visible by straight line segments from the image of
; this is in contrast to the
star unfolding, a different method for producing nets that does not always produce star-shaped polygons.
An analogous unfolding method can be applied to any higher-dimensional
convex polytope, cutting the surface of the polytope into a net that can be unfolded into a flat
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hype ...
.
References
{{Mathematics of paper folding
Polygons
Polyhedra
Computational geometry