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The degree of a map between general manifolds was first defined by Brouwer, [1] who showed that the degree is homotopy invariant and used it to prove the Brouwer fixed point theorem. Less general forms of the concept existed before Brouwer, such as the winding number and the Kronecker characteristic (or Kronecker integral ).
How the Earth is projected onto a cylinder. The projection was invented by the Swiss mathematician Johann Heinrich Lambert and described in his 1772 treatise, Beiträge zum Gebrauche der Mathematik und deren Anwendung, part III, section 6: Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten, translated as, Notes and Comments on the Composition of Terrestrial and Celestial Maps.
Gimbal lock occurs because any map T 3 → RP 3 is not a covering map. In particular, the relevant map carries any element of T 3, that is, an ordered triple (a,b,c) of angles (real numbers mod 2 π), to the composition of the three coordinate axis rotations R x (a)∘R y (b)∘R z (c) by those angles, respectively.
In mathematics, topological degree theory is a generalization of the winding number of a curve in the complex plane. It can be used to estimate the number of solutions of an equation, and is closely connected to fixed-point theory. When one solution of an equation is easily found, degree theory can often be used to prove existence of a second ...
In mathematics, the Leray–Schauder degree is an extension of the degree of a base point preserving continuous map between spheres (,) (,) or equivalently to boundary-sphere-preserving continuous maps between balls (,) (,) to boundary-sphere-preserving maps between balls in a Banach space: ((), ()) ((), ()), assuming that the map is of the form = where is the identity map and is some compact ...
The Gauss map can be defined for hypersurfaces in R n as a map from a hypersurface to the unit sphere S n − 1 ⊆ R n. For a general oriented k-submanifold of R n the Gauss map can also be defined, and its target space is the oriented Grassmannian ~,, i.e. the set of all oriented k-planes in R n. In this case a point on the submanifold is ...
The projection found on these maps, dating to 1511, was stated by John Snyder in 1987 to be the same projection as Mercator's. [6] However, given the geometry of a sundial, these maps may well have been based on the similar central cylindrical projection, a limiting case of the gnomonic projection, which is the basis for a sundial. Snyder ...
The map also first showed the Pacific Ocean, separating the Americas from Asia. [2] The map is drafted on a modification of Ptolemy's second projection, expanded to accommodate the Americas and the high latitudes. [3] A single copy of the map survives, presently housed at the Library of Congress in Washington, D.C.