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Smoothstep is a family of sigmoid-like interpolation and clamping functions commonly used in computer graphics, [1] [2] video game engines, [3] and machine learning. [ 4 ] The function depends on three parameters, the input x , the "left edge" and the "right edge", with the left edge being assumed smaller than the right edge.
A B-spline function is a combination of flexible bands that is controlled by a number of points that are called control points, creating smooth curves. These functions are used to create and manage complex shapes and surfaces using a number of points. B-spline function and Bézier functions are applied extensively in shape optimization methods. [5]
Slerp has a geometric formula independent of quaternions, and independent of the dimension of the space in which the arc is embedded. This formula, a symmetric weighted sum credited to Glenn Davis, is based on the fact that any point on the curve must be a linear combination of the ends.
Non-analytic smooth function – Mathematical functions which are smooth but not analytic; Quasi-analytic function; Singularity (mathematics) – Point where a function, a curve or another mathematical object does not behave regularly; Sinuosity – Ratio of arc length and straight-line distance between two points on a wave-like function
A smooth curve that is not closed is often referred to as a smooth arc. [6] The parametrization of a curve provides a natural ordering of points on the curve: () comes before () if <. This leads to the notion of a directed smooth curve. It is most useful to consider curves independent of the specific parametrization.
Consider the standard CW-decomposition of the n-sphere, with one zero cell and a single n-cell.From the point of view of smooth manifolds, this is a degenerate decomposition of the sphere, as there is no natural way to see the smooth structure of from the eyes of this decomposition—in particular the smooth structure near the 0-cell depends on the behavior of the characteristic map : in a ...
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If a smooth manifold comes with additional structure it is often natural to consider a subbundle of the full frame bundle of which is adapted to the given structure. For example, if M {\displaystyle M} is a Riemannian manifold we saw above that it is natural to consider the orthonormal frame bundle of M {\displaystyle M} .