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The dilation is commutative, also given by = =. If B has a center on the origin, as before, then the dilation of A by B can be understood as the locus of the points covered by B when the center of B moves inside A. In the above example, the dilation of the square of side 10 by the disk of radius 2 is a square of side 14, with rounded corners ...
The Blaschke–Lebesgue theorem says that the Reuleaux triangle has the least area of any convex curve of given constant width. [19] Every proper superset of a body of constant width has strictly greater diameter, and every Euclidean set with this property is a body of constant width.
Dilation is commutative, also given by = =. If B has a center on the origin, then the dilation of A by B can be understood as the locus of the points covered by B when the center of B moves inside A. The dilation of a square of size 10, centered at the origin, by a disk of radius 2, also centered at the origin, is a square of side 14, with ...
Shrink the triangle to 1 / 2 height and 1 / 2 width, make three copies, and position the three shrunken triangles so that each triangle touches the two other triangles at a corner (image 2).
Among all shapes of constant width that avoid all points of an integer lattice, the one with the largest width is a Reuleaux triangle. It has one of its axes of symmetry parallel to the coordinate axes on a half-integer line. Its width, approximately 1.54, is the root of a degree-6 polynomial with integer coefficients. [17] [19] [20]
Sometimes it is desirable to have a triangulation with special properties, e.g., in which all triangles have large angles (long and narrow ("splinter") triangles are avoided). [3] Given a set of edges that connect points of the plane, the problem to determine whether they contain a triangulation is NP-complete. [4]
Lifting each point from the plane to its elevated height lifts the triangles of the triangulation into three-dimensional surfaces, which form an approximation of a three-dimensional landform. A polygon triangulation is a subdivision of a given polygon into triangles meeting edge-to-edge, again with the property that the set of triangle vertices ...
Measuring the height of a building with an inclinometer. Triangulation today is used for many purposes, including surveying, navigation, metrology, astrometry, binocular vision, model rocketry and, in the military, the gun direction, the trajectory and distribution of fire power of weapons. The use of triangles to estimate distances dates to ...