The Geometry of Numbers and Minkowski`s Theorem
... volume of this ball can be computed (either with cleverness or a brute-force quadruple integration) to be ...
... volume of this ball can be computed (either with cleverness or a brute-force quadruple integration) to be ...
Reconstructing a Simple Polygon from Its Angles
... The reconstruction of geometric objects from measurement data has attracted considerable attention over the last decade [7,11,13]. In particular, many variants of the problem of reconstructing a polygon with certain properties have been studied. For different sets of data this polygon reconstruction ...
... The reconstruction of geometric objects from measurement data has attracted considerable attention over the last decade [7,11,13]. In particular, many variants of the problem of reconstructing a polygon with certain properties have been studied. For different sets of data this polygon reconstruction ...
Scale factor
... 2. Graph triangle ABC and dilate it with a scale factor of 3. Record the coordinates of the dilated figure in the table below and graph the dilation. ...
... 2. Graph triangle ABC and dilate it with a scale factor of 3. Record the coordinates of the dilated figure in the table below and graph the dilation. ...
Reconstructing a Simple Polygon from Its Angles
... to retrace their movements alone already enables simple robots to reconstruct the visibility graph (when at least an upper bound on the number of vertices of the polygon is given), even though only an exponential algorithm was given [3]. Our result implies that measuring angles alone is also already ...
... to retrace their movements alone already enables simple robots to reconstruct the visibility graph (when at least an upper bound on the number of vertices of the polygon is given), even though only an exponential algorithm was given [3]. Our result implies that measuring angles alone is also already ...
Steinitz's theorem
In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the (simple) 3-vertex-connected planar graphs (with at least four vertices). That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs. Steinitz's theorem is named after Ernst Steinitz, who submitted its first proof for publication in 1916. Branko Grünbaum has called this theorem “the most important and deepest known result on 3-polytopes.”The name ""Steinitz's theorem"" has also been applied to other results of Steinitz: the Steinitz exchange lemma implying that each basis of a vector space has the same number of vectors, the theorem that if the convex hull of a point set contains a unit sphere, then the convex hull of a finite subset of the point contains a smaller concentric sphere, and Steinitz's vectorial generalization of the Riemann series theorem on the rearrangements of conditionally convergent series.↑ ↑ 2.0 2.1 ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑