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In differential geometry, the Gauss map of a surface is a function that maps each point in the surface to a unit vector that is orthogonal to the surface at that point. Namely, given a surface X in Euclidean space R 3 , the Gauss map is a map N : X → S 2 (where S 2 is the unit sphere ) such that for each p in X , the function value N ( p ) is ...
The Euclidean algorithm was probably invented before Euclid, depicted here holding a compass in a painting of about 1474. The Euclidean algorithm is one of the oldest algorithms in common use. [27] It appears in Euclid's Elements (c. 300 BC), specifically in Book 7 (Propositions 1–2) and Book 10 (Propositions 2–3). In Book 7, the algorithm ...
Animation showing an application of the Euclidean algorithm to find the greatest common divisor of 62 and 36, which is 2. A more efficient method is the Euclidean algorithm, a variant in which the difference of the two numbers a and b is replaced by the remainder of the Euclidean division (also called division with remainder) of a by b.
This algorithm consists of replacing of the input (a, b) by (b, r), where r is the remainder of the Euclidean division of a by b, and repeating this operation until getting a zero remainder, that is a pair (d, 0). This process terminates, because, at each step, the norm of the second Gaussian integer decreases.
This norm maps the ring of integers of a number field K, say O K, to the nonnegative rational integers, so it is a candidate to be a Euclidean norm on this ring. If this norm satisfies the axioms of a Euclidean function then the number field K is called norm-Euclidean or simply Euclidean.
Thus in order to compute the linking number of the diagram corresponding to v it suffices to count the signed number of times the Gauss map covers v. Since v is a regular value, this is precisely the degree of the Gauss map (i.e. the signed number of times that the image of Γ covers the sphere). Isotopy invariance of the linking number is ...
As with the Euclidean algorithm, the method is iterative; at each step the larger of the two vectors is reduced by adding or subtracting an integer multiple of the smaller vector. The pseudocode of the algorithm, often known as Lagrange's algorithm or the Lagrange-Gauss algorithm, is as follows:
Cobweb plot of the Gauss map for = and =. This shows an 8-cycle. This shows an 8-cycle. In mathematics , the Gauss map (also known as Gaussian map [ 1 ] or mouse map ), is a nonlinear iterated map of the reals into a real interval given by the Gaussian function :