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where: α and β are the two greatest valence angles of coordination center; θ = cos −1 (− 1 ⁄ 3) ≈ 109.5° is a tetrahedral angle. When τ 4 is close to 0 the geometry is similar to square planar, while if τ 4 is close to 1 then the geometry is similar to tetrahedral.
Raising and lowering is then done in coordinates. Given a vector with components , we can contract with the metric to obtain a covector: = and this is what we mean by lowering the index. Conversely, contracting a covector with the inverse metric gives a vector:
Geometric constraint solving is constraint satisfaction in a computational geometry setting, which has primary applications in computer aided design. [1] A problem to be solved consists of a given set of geometric elements and a description of geometric constraints between the elements, which could be non-parametric (tangency, horizontality, coaxiality, etc) or parametric (like distance, angle ...
A subgroup H of finite index in a group G (finite or infinite) always contains a normal subgroup N (of G), also of finite index. In fact, if H has index n, then the index of N will be some divisor of n! and a multiple of n; indeed, N can be taken to be the kernel of the natural homomorphism from G to the permutation group of the left (or right ...
A vector treated as an array of numbers by writing as a row vector or column vector (whichever is used depends on convenience or context): = (), = Index notation allows indication of the elements of the array by simply writing a i, where the index i is known to run from 1 to n, because of n-dimensions. [1]
In the example of circles, for instance, the moduli space is not just an abstract set, but the absolute value of the difference of the radii defines a metric for determining when two circles are "close". The geometric structure of moduli spaces locally tells us when two solutions of a geometric classification problem are "close", but generally ...
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices.
In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general case situation, as opposed to some more special or coincidental cases that are possible, which is referred to as special position. Its precise meaning differs in different settings.