computational geometry Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems ar ...

, 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 wo ...

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 up ...

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

p

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

p

. For every convex polyhedron, and every choice of the point

p

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 n ...

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 poin ...

, with all of its points visible by straight line segments from the image of

p

; 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 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 wo ...

, 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 hyper ...

References

{{Mathematics of paper folding Polygons Polyhedra Computational geometry