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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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NURBS circle without all the double knots?

I've been looking at various examples of a circle parametrized as a degree-2 NURBS curve, e.g.: NURBS circle example on wikipedia Philip Schneider's "NURB Curves: A Guide for the Uninitiated" David ...
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Integrals of products of B-splines and positive definiteness [on hold]

Let $B_i(x), i = 1, \dots, n$ denote a B-spline basis function of order $k$ defined with some knot vector $\mathbf{t}$. Then is it true that the $n \times n$ matrix $$ C_{ij} = \int_{-\infty}^{\infty} ...
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Drawing De Boor's algorithm

Assume we have 4 control points $[c_0, c_1, c_2, c_3]$ and uniform knot sequence $[0,1,2,3]$ If we were to draw an quadratic bezier we would be forced to use only 3 of the control points and then we ...
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Would the use of cubic splines increase the number of data points to interpolate from result in smaller error and avoid Runge's phenomena?

I am implementing the cubic spline method to interpolate the function: $$f(x)=\sin(x);\ -π≤0≤π$$ I am using this paper by Antony Jameson to construct my code, with a few modifications to correct the ...
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Confusing myself with Cox de boor's algorithm

I am using the recursive definition to understand the algorithm, mainly: $$ B_{i,0}(t) = \begin{cases} 1 & \text{if} & t_i \leq t < t_{i+1} \\ 0 & \text{otherwise} \end{...
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Polynomial representation: Marsden's Identity.

Marsden's Identity states that: For every $\tau$ in $\mathbb{R }$: $(\cdot -\tau)^{k-1}=\sum_j\Psi_{j,k}B_{j,k,t}$ with: $\Psi_{j,k}=(t_j-\tau)\times...\times(t_{j+k-1}-\tau)$ Following de Boor'...
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Representing rectangular function using divided differences.

I was reading the chapter 9 of A practical guide to splines (de Boor, C.) were the autor defines a B-spline of order 1, basically a rectangular function i.e. a function taking value $1$ if the ...
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Finding B-spline for space spanned by Multi-dimensional Spline.

It's known that a B-spline of degree $p$ , $B_j^p(x)$ is completely determined by a knot vector $(u_j,u_{j+1},...,u_{j+p+1})$. One could define it as: $B_j^p(x)=[u_j,u_{j+1},...,u_{j+p+1}](\cdot - x)^...
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Approximate a ($y²-x²=1$) hyperbola with line segments and elliptic ($a(x-x_0)²+b(y-y_0)²=1$) arcs

On some IT graphic systems, you have tools which draw line segments and circle or elipse arcs, but which do not draw parabolas or hyperbolas, and in many cases, those system keep track of graphical ...
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Adaptive knot selection for B-spline fitting.

When fitting a B-spline for regression purposes I've seen a lot of cases where knots are fixed uniformly ,but in some situations this could lead to poor estimations because the behaviour of the curve ...
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Aggregate and interpolate overlapping time-series data

I'm trying to aggregate counter data from two different types of measurements. The first type of measure gives an exact value of the counter on a given day. ...
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Interval Spanned by $(n+1)$ points in $\mathbb{R}$

Suppose that we have a real valued function $f(x)$ that has local support, i.e. it's non-zero just for some values of $x$. If you are familiar with B-splines, this function $f$ can be interpreted as a ...
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Fitting a spline: find coefficients using Fourier Transform?

I came up with a idea to estimate the coefficients of a B-spline fit by using the Fourier Transform but I don't know if it makes any sense to estimate them in this way. Given that $$s(x)=\sum_kc(k)\...
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What does this notation mean? Double arrow with $z$ above

I am reading a paper about digital filtering (for the very first time) and I found this notation (double arrow with $z$ above) which I do not quite understand. Could you please give me some hint? ...
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“fastest” curve through n points

I'm programming an AI for a race game, where my car has to drive through some checkpoints. If I let it drive straight in direction of the checkpoints, it has to slow down and make a huge turn after ...
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Order of convergence for spline interpolation in a Sobolev norm

We are interested in a result stating the order of convergence of a spline interpolation for a given function in $W^{k,p}_{\text{loc}}(\mathbb{R})$. Let us be more precise: Let $h\in \mathbb{R}_{>...
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$[t_j,t_{j+1},…,t_{j+k}]f$ Divided Difference on B-splines.

While reading Moments and Fourier Transform of B-splines (Neuman,E.) I found a different notation for the B-splines. The author define $M_{j,k}(x)$ with knots $t_j<t_{j+1}<...<t_{j+k}$ as: $$...
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Conditions for two B-Splines to represent the same curve

What will be the weakest constraint for two B-Spline curves $S_1(t)$ and $S_2(u)$ to represent the same curve in space? Assume that their orders are $k_1$ and $k_2$ respectively, knot vectors are ($...
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B splines recursion

Given that by definition the i-th B-spline of order k is: $$B_{i,k}=w_{i,k}B_{i,k-1}+(1-w_{i+1,k})B_{i+1,k-1}$$ where $w_{j,k}=\frac{x-t_j}{t_{j+k-1}-t_j}$ We can define the spline space as $$S_{k,...
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Monotonic and smooth interpolation between three points

The problem I have is the following: Given three $x,y$ points, $(-1, -y_{0})$, $(0,0)$ and $(1,y_{2})$, where $y_{0} \geq 0$ and $y_2 \geq 0$, I want to interpolate smoothly between them with a ...
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Discrete norm approximation of the $L^p$ norm for spline functions

In Theorem 5.2 in Lynche (1988) "A data reduction strategy for splines with applications to the approximation of functions and data", a bound for the difference between the $(l_2,t)$ and $L^2$ norms ...
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How to refine NURBS mesh in isogeometric analysis?

Let we have a coarse description of 2D domain using NURBS. That is, we have two sets of knot vectors $\{\xi_1, \dots, \xi_n\}$, $\{\eta_1, \dots, \eta_m\}$, the set of control vectors $\{B_{ij}\}$, ...
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B and P splines bibliography recommendation.

I am following a course in Statistical Learning and we are using Elements of Statistical Learning(Hastie et. al.) for a very first introduction to smoothing splines, B-spines and P-splines, but ...
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Fitting a cubic spline model with two knots in $R$

We are given data from a set of data from an example given by: battery voltage drop in a guided missile motor observed over the time of missile flight. The set of data has three columns of data: ...
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How to calculate smoothing spline coefficients

I am attempting to calculate smoothing spline coefficients based on the description in Reinsch's 1967 paper, but I'm having some trouble. The first derivative is not continuous. Here are the ...
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Strong convex hull property of B-spline

B-spline curve is contained in the convex hull of its control polygon. Here is one example, the degree of the curve is 5, The first segment is $u\in [u_5,u_6)$ . The control polygon is from $P_0 ... ...
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Convert continuous Bezier curve to B-Spline

Is there an algorithm/process for converting a sequence of bezier curves into a b-spline? I've found much discussion of the reverse, but nothing for this. I'm attempting to make a spline editor in ...
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Is a B-Spline always made up of Bezier curve segments?

According to what I have read, a B-Spline curve is made up of segments, with each segment controlled by 'k' control points (where k is the order of the curve). Also, a B-Spline curve can be formed by ...
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Smoothing algorithm for unequal variances in data

I have data in the form of a histogram with bins that each have their own error bars. I'm interested in finding a smoothing algorithm that fits the data while taking the error bars into account (e.g. ...
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Are lagrange polynomials equal to B-spline polynomials?

I am having a discussion with a friend about whether or not the set of B-splines contains the set of lagrange interpolants or not. I think it is not the case because the basis for B-splines are ...
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Cubic Smoothing Splines and Eigenvalues

1) Based on the following linear smoothing matrix for cubic smoothing splines, how can one show that the first two eigenvalues are equal to one and the others $\in (0,1]$ of $S_{\lambda}$ (given $d_k$ ...
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Convert two B-splines into one?

Let's say I have two second-order, non-periodic B-splines which touch at one endpoint (that is, I have an array of control points, weights, and knots for each). My task is: can you make a single ...
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Approximating the intersection of a line and the iPhone X screen as well as its normals

I am creating a simulation where little, fast moving, particles need to intersect with the edges of the iPhone X screen. Previously I have had no difficulty with my collisions. The particles have ...
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$\chi^2$-test for smoothing splines: degrees of freedom

Suppose we are given raw data, e.g. raw mortality rates $\widetilde{q_x}$, which are graduated by a smoothing (cubic natural) spline $S$. That is, we obtain smoothed rates by setting $q_x:=S(x).$ Let ...
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Convert multiple Bézier segments to a nurbs curve

I have multiple cubic Bézier curve segments which are contiguous and G1 (they are the result of the fitting of many curve samples). Now I would like to transform these Bézier segments into a single ...
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Parametric derivatives of curves in three or more dimensions

I want to model a number of points in multidimensional (e.g. three dimensional) space by a one dimensional parametric curve. My ansatz was to take cubic spline interpolation functions between each of ...
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Interpolation method that does never overshoot

for implementing a system that will control hardware, I need an interpolation between points on a graph that does never overshoot. By overshooting I mean that between two points there may be no y-...
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Average radius of curvature of set of points on a road?

I have a set of points (lat, lng) on a serpentine road. What would be a good way to calculate average radius of curvature of the road? Also even without considering (lat, lng) points, just on a 2D ...
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Giving a point on a uniform bspline curve, how to get the tangent vector of this point

A bspline curve of order $k$ is given by $$C(t) = \sum_{i=0}^n P_i N_{i,k}(t).$$ where $P_i$ are the control points and $N_{i,k}(t)$ a basis function defined on a knot vector $$T = (t_0,t_1,...t_{n+k})...
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How to compare B Spline Surfaces?

We can define b spline surfaces over a set of points. However, is it possible to compare two set of b-spline surfaces fitted over different set of points ? The objective is to identify if the curves ...
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Iterated backward difference quotient from splines

I am working on a certain problem: say we have a function $f$ defined on an interval $[a,b]$ and we want the spline $Sf$ of order $k$ (and knots needed for the B-splines are in $[a,b]$) to agree to $f$...
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Minimization problem with latent function and splines

I have a dataset consisting of pairs $(x_i, y_i)$. I want to determine the function $f$, so $$ f(x)f(y) = 1 $$ with the constraint that $f(x) \leq x$, $f'(x) \geq 0$ and $f''(x) \geq 0$. I was ...
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How to get new temporary control points in deboors algorithm

I am trying to implement deboor's algorithm in c++ to make a b spline. I'm trying to follow the Wikipedia page of it (https://en.wikipedia.org/wiki/De_Boor%27s_algorithm). Here is the implementation ...
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“Swivel plane” tangent vector in Séquin circle spline

I'm trying to implement the circle spline scheme described here. I feel like I have a solid working understanding of the procedure, but I'm generally pretty rusty when it comes to geometry. A big ...
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252 views

The advantage of B-spline compared to Bézier if the number of control points is very small

If the number of control points is n+1, and the degree of the basis function is p If n = p, B-spline is as same as Bézier curve. Suppose I have a chance to increase the number of control points say ...
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Is it cheat to use spline to interpolate data from experiment or computational results.

Say we got data from the experiment or computational results. However, instead of using the line to connect data one by one. We use the spline to interpolate the data to make it look more smooth. ...
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Doubts on Oslo's Algorithm

I have a simple doubt on the Oslo's algorithm which is used for knot insertion in a knot vector for a B-spline. Here alpha = entries in Refinement Matrix Tau = original knot vector t = New knot ...
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Relation between spline and $C^d$ curve

Spline including NURBS, B-spline, etc can provide the $C^d$ continuous curve, which d is based on the degree of the spline. However, can all $C^{d-1}$ continuous curve be expressed by d degree Spline? ...
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What is the purpose of having repeated knots in a B spline?

A primer on the cpr package in R (page 2 of https://arxiv.org/pdf/1705.04756.pdf) writes the following about B-splines. A B-spline basis matrix is defined by a polynomial order $k$ and knot ...
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What is the relation between b-splines and polyharmonic splines?

I am working on image warping and both b-splines and polyharmonic splines (like thin-plate spline) are mentioned but never in the same text. Is one a subset of the other?