# 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 Catmullā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 Catmullā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_{...
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### 3D Spline interpolation

I have 3D data i need to interpolate via Spline. My data is a set of (x,y,z) on an irregular grid. I have to find the z of some x,y not on the grid. I can't use any library since i have to manually ...