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A biarc is a smooth curve formed from two circular arcs. [1] In order to make the biarc smooth ( G 1 continuous ), the two arcs should have the same tangent at the connecting point where they meet. Biarcs are commonly used in geometric modeling and computer graphics .
A smooth structure on a manifold is a collection of smoothly equivalent smooth atlases. Here, a smooth atlas for a topological manifold M {\displaystyle M} is an atlas for M {\displaystyle M} such that each transition function is a smooth map , and two smooth atlases for M {\displaystyle M} are smoothly equivalent provided their union is again ...
It produces smooth surfaces, which are infinitely differentiable. There are no free parameters that need manual tuning. It has closed-form solutions for both warping and parameter estimation. There is a physical explanation for its energy function. However, note that splines already in one dimension can cause severe "overshoots".
Common smooth functors include, for some vector space W: [2] F(W) = ⊗ n W, the nth iterated tensor product; F(W) = Λ n (W), the nth exterior power; and F(W) = Sym n (W), the nth symmetric power. Smooth functors are significant because any smooth functor can be applied fiberwise to a differentiable vector bundle on a manifold.
Similarly, if is a smooth function on an open set in , then the same formula defines a smooth function on the open set (). (In the language of sheaves , pullback defines a morphism from the sheaf of smooth functions on N {\displaystyle N} to the direct image by ϕ {\displaystyle \phi } of the sheaf of smooth functions on M {\displaystyle M} .)
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.
The chord function is defined geometrically as shown in the picture. The chord of an angle is the length of the chord between two points on a unit circle separated by that central angle. The angle θ is taken in the positive sense and must lie in the interval 0 < θ ≤ π (radian measure).
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.