# 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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### Computing cyclic cubic splines

Cubic splines (piecewise interpolator with $C_2$ continuity) are well-known to be computable from a tridiagonal system of equations that give estimates of the second derivatives at the interpolated ...
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### Roots/zero crossing of additive piecewise linear splines

Suppose we have a function $f$ in p-variables $x_1,...,x_p$ which has the form $f(x_1,...,x_p) = f_1(x_1)+...+f_p(x_p)$. Assume further that each function $f_p$ is a piecewise linear spline. What I ...
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### Interesting $2\text{D}$ Poisson equations to approximate via interpolations?

I am in need of at least one "interesting" $2\text{D}$ Poisson equation. I am working on interpolation of equations, and in particular on the magnitude of errors of the approximated ...
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### What is a basis function in spline regression?

I'm learning about spline regression right now and there's this notation of basis function. $$f(X)=\sum^M_{m=1}B_mh_m(X)$$ This is the notation using is The Elements of Statistical Learning - Data ...
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### Matrix Representation for Clamped b-spline

I wanted to know how can I write a matrix representation of a clamped b-spline. Following this document, https://ieeexplore.ieee.org/document/731996, I get matrix representation to write b-spline ...
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### generalization of cubic spline and thin plate spline

Let us consider some interpolation problems: $\renewcommand\phi\varphi$ We have some points $n$ points $x_i \in \mathbb R^d$ along with corresponding values $y_i \in \mathbb R$. We'd like to find a ...
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### Convert cubic spline to Bézier curve and get control points

There is a cubic spline represented by the standard equation: $$f(x) = a + b (x - x_0) + c (x - x_0)^2 + d (x - x_0)^3$$ and 2 endpoints: $P_0~ [x, y]$ - starting point $P_1~ [x, y]$ - end ...
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### Convergence of quadratic b-spline interpolant

Given a function $f \in L_2([0, 1])$ and a b-spline basis on equispaced knots in $[0, 1]$, what can be said about the convergence in the $L_2$ norm of the spline interpolant to $f$ as the number of ...
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### How to create line with 6th order spline?

I am dealing with spline interpolation and what I do is basically interpolating $6$th order ($7$ control points) spline through some discrete points. Curve-based part of my algorithm is done, however, ...
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### Surface generation using 4 cubic parametric boundary curves

I'm thinking about a way of generating a 2d surface from a region defined by 4 splines, each spline being a simple parametric 1d cubic curve. Ideally I maybe can do something similar to the Tensor ...
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### Curve Fitting of Nautical Tabular Data

I have sets of data describing hull forms. The data is in $(X, Y, Z)$ format except that the $Y$ and $Z$ coordinates have been rounded to either $1/16"$ or $1/32"$ of an inch. $X$ is always a multiple ...
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### Why do we still need subdivision surface modeling when we can compute B-spline directly?

By using a recurrence formula due to deBoor, we can directly compute B-spline based on the control polygon. Then why do we still need subdivision surface modeling such as catmull-clark?
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### Prove you can't make a natural spline from a B-spline, from a polynomial of third degree.

How can I prove the following: Given a polynomial $f(x)$ of third degree, suppose you have another polynomial $S(x)$ which is a B-spline representation of $f(x)$. Prove that it is not possible to make ...
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### Is Catmull–Clark subdivision surface a refinement scheme?

To do the knot refinement, one important thing is to not change the geometry. Also, it is done in the parametric domain. However, for Catmull–Clark subdivision surface, it is done on the physical ...
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### In B-spline, why is the number of knot spans equal to the number curve segments?

I will show one example to explain my questions. Let's say we have 11 control points, degree 3 and 15 knots with first four and last four knots clamped. The number of knot spans is equal to 8. The ...
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### Should a natural cubic spline ensure 3rd derivative on the boundary to be 0?

Say we have K interior knots (t_1 to t_K). We can form K + 1 piecewice polynomials. If we want to fit a natural cubic spline s(x) In regression, we would enforce ...
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### Advantages of using cubic splines to expand functions

I've recently determined the deuteron's binding energy using cubic B-Splines to expand the system of coupled differential equations I obtained for my problem. This method of expanding functions using ...
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### Finding the first and second derivative of a spline (only from data, function unknown)

I have received data values for a spline (which was already fit to some ndvi data). I just have only the data points of the spline and do not know the function that the spline follows. My goal is to ...
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### Prove the null space of K is spanned by functions linear in X for a smoothing spline

I am reading The Elements of Statistical Learning (Hastie et. al.) and, having already derived the Reinsch form $S_\lambda = (\mathbb{I} + \lambda K)^{-1}$ for a smoothing spline, we are asked to: ...
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### What kind of proof should I be creating for the C^p-1 continuity of B-Splines?

So I'm working on my project and trying to prove this theorem: $B_{j,p,t}(x)$ has $C^{p-1}$ continuity at each simple knot where $p\geqslant1$ is the degree of the polynomial. My first idea was to ...
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### What does a spline transformation do?

I'm not a mathematician and new to splines. I'm searching to understand basics of spline transformations in a practical way (i.e. by example). Does anyone know a good place to begin?
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### Dimesnion of Spline vector space

Definition: $S: [a, b] \rightarrow \mathbb{R}$ is Spline function for $\Delta = \{t_0, ..., t_n\}$ if: $S$ is polynomial of degree $k$ in $[t_i, t_{i+1})$, $i=0,...,n-1$. $S$ is at least $(k-1)$ ...
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### Trajectory interpolation with known coordinates and heading angles.

I have a set of $n$ tuples $(t_i,x_i,y_i,r_i)$ for $i \in \{0,...,n\}$, where $t_i$ is a timestamp, $x_i$ and $y_i$ location coordinates and $r_i$ a heading angle of a moving object in 2D space. Note ...
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### Confusion about the order of a B spline

I am reading the wiki page on b splines and I am very confused about the order. In particular the page states: "A spline of order {\displaystyle n}n is a piecewise polynomial function of degree {\...
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### basis representation of cubic spline interpolant

If i do cubic spline interpolation (periodic), lets say on $n$ pairs $(x,f(x))$, i will get $n-1$ piecewise polynomials s_i(x) = a_i(x-x_i)^3+b_i(x-x_i)^2+c_i(x-x_i)+d_i \;\; \text{on} \;\;[x_i,x_{...