angular defectsの例文

• Its difference from 180?is a case of " angular defect " and serves as an important distinction for geometric systems.
• The discrete analog of curvature, corresponding to curvature being concentrated at a point and particularly useful for polyhedra, is the Descartes'theorem on total angular defect.
• Topologically, a regular 2-dimensional tessellation may be regarded as similar to a ( 3-dimensional ) polyhedron, but such that the angular defect is zero.
• The polyhedron can be thought of as being folded from a sheet of paper ( a homeomorphic ( topologically equivalent ) to a sphere, and locally Euclidean except for a finite number of cone points whose angular defect sums to 4.
• A converse to this theorem is given by Alexandrov's uniqueness theorem, according to which a metric space that is locally Euclidean except for a finite number of points of positive angular defect, adding to 4?, can be realized in a unique way as the surface of a convex polyhedron.
• It states that if a metric space ( " X ", " d " ) is geodesic, homeomorphic to a sphere, and locally Euclidean except for a finite number of cone points of positive angular defect summing to 4, then ( " X ", " d " ) can be represented as the development of a convex polyhedron.
• The icosahedron has the largest number of faces and the largest dihedral angle, it hugs its inscribed sphere the most tightly, and its surface area to volume ratio is closest to that of a sphere of the same size ( i . e . either the same surface area or the same volume . ) The dodecahedron, on the other hand, has the smallest angular defect, the largest vertex solid angle, and it fills out its circumscribed sphere the most.