# Questions tagged [spline]

A smooth piecewise-defined curve formed by joining segments together, end-to-end. The segments are usually described by polynomial or rational functions. Splines are typically used for approximation or data fitting.

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### Uniform interpolation on a Cubic Hermite Spline

I have a 3D spline with points $p_0,p_1,...,p_n$ and tangents $m_0,m_1,...,m_n$. I'm using the formula described in this page. $p(t) = (2t^3-3t^2+1)p_0+(t^3-2t^2+t)m_0+(-2t^3+3t^2)p_1+(t^3-t^2)m_1$, ...
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### Piecewise quadratic interpolation in a symmetric manner

I am trying to derive a reasonable symmetric interpolant for quadratic $C^0$ interpolation. For odd degrees I have no trouble since things are symmetric. For example for a piecewise cubic interpolant ...
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### Calculate length of a cubic spline, given it's vertices in 3d space?

Given a cubic spline defined by $n$ number of vertices in a 3d space, how would one calculate the length of this spline? (Attached picture is just for illustration, to explain what I mean by vertices, ...
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### Determine whether a spline curve describes an elliptical arc

I'm currently writing software which analyses STEP files to hunt for engineered parts with particular shapes. The software uses Open Cascade which presents me with topological and geometrical objects ...
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### Proving the integration of piecewise cubic functions as 0

Given that $g(x)$ is a piecewise cubic function, and $h(x) = \bar g(x)-g(x)$. I am struggling to show that $\sum_{i=1}^{N}\int_{x_{n-1}}^{x_n}g'''(x)h'(x)dx = 0$ I understand that $g'''(x)$ will be a ...