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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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What is a natural cubic "B-spline"?

I recently learned that natural cubic splines are strictly distinct from natural cubic B-splines while studying spline methods. It appears that natural cubic B-splines are obtained by adding ...
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Finding Basis for specific Spline Space

Let $S = \{s \in S: s'(a) = s'(b) = 0 \}$ be the spline space that holds all cubic splines with derivate at startpoint (a) and endpoint (b) =0. I want to find a basis for this vector space. I looked ...
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Closed B-spline formualtion wrapping knots

I am trying to implement the method for constructing a closed B-spline shown in $this website, in particular the "Wrapping knots" method, but I don't follow the explanation: Wrapping Knots ...
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Motion with parabolic blends problem. Seems easy but feels impossible!

I'm trying to calculate multi segment trajectories between points in a plane where the trajectory follows a linear function, but in order to keep continuous speed and position parabolic blends are ...
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Optimal knot placement for approximating this function with B-spline.

I have the following data points, which all lie along a smooth, unknown, function (it looks sigmoidal but might not be): I want to approximate it rather accurately with B-splines, either quadratic or ...
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Calculating an Inverse Matrix of a Matrix with variables

I am trying to understand a part of an article regarding quaternions spline interpolation, where the situation folded into the equation: $$ (\vec{a}\cdot\hat{e})\hat{e}+\frac{\sin\Delta\theta}{\Delta\...
BlueRevel 's user avatar
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truncated power basis elements expressed as linear combination of bsplines

Given the truncated power series basis, e.g. the following functions: $1, x, x^2, ((x-\eta_0)^{+})^{2}, ((x-\eta_2)^+)^2, \dots ((x-\eta_{l+1})^+)^2$ there is a set of bspline-functions $s_1, \dots $ ...
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Reducing the number of natural cubic spline interpolation points

Say we have cubic curve $\vec{C}(t)_ = (C_x(t), C_y(t), C_z(t))$ which approximates some parametric function $\vec{F}(t)$ within error less than $\epsilon$. The cubic curve is $C^2$ continuous and is ...
Donatas Šimeliūnas's user avatar
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Why is the middle segment of a 4 points cubic spline not matching a 100 points cubic spline?

Let's say I have x0, x1, ..., x99 and y0, y1, ..., y99 ...
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How to make arithmetic function continuous?

Suppose that we have an arithmetic function $f(x)$ defined as follows: What are the methods in the literature that will make this function continuous and differentiable? However, it should be noted ...
Severus' Constant's user avatar
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Quantifying the "curviness" of a spline

I'm developing an optimization problem that requires me to quantify the "curviness" of a spline. The spline is defined in a software library, and the only input to generate a spline with ...
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Singular Matrix with Closed B-spline Interpolation when Degree and Number of Data Points are Both Even

I have written an algorithm to perform closed B-spline interpolation on a set of $N$ data points for a given degree $p$. I first generate a cyclic, uniform knot vector, and also use uniform ...
Gary Allen's user avatar
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B-Spline with increasing knot distance

I'm trying to approximate a function $f(x)$ on $[0, M]$ that, in some sense, begins to rapidly "vary slower" as $x$ increases, i.e. its modulus of continuity (or the variation of its ...
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cubic spline interpolation - spline interval

The k function values are to be interpolated by a piecewise cubic spline. The k-1 cubic polynomials are defined on the intervals [xi,xi+1],i ∈ {1,...,k-1}. Indicate which of the following statements ...
Adelhard's user avatar
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Defining a Quad Spherical Cube Tile as a Uniform NURBS Surface?

I am trying to create NURBS surface that perfectly fits one face of a Quadrilateralized Spherical Cube (QSC) [also called a Cobb sphere in some contexts, I believe]. I have seen some visualizations of ...
Chaosoahc's user avatar
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Reparametrization of rational Bézier curve

I am trying to solve the following task Using rational Bézier curve find the control points and weights of one sixth of a circle $c_1$, such that $$c_1(0)=\{3,0\},c_1(1)=\{\frac{3}{2},\frac{3\sqrt{3}}{...
Weyr124's user avatar
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Why does the B-spline constructed from squaring its coefficients appear to converge to the square of the B-spline itself?

Why does the B-spline constructed from squaring its coefficients appear to converge to the square of the B-spline itself? To be clear, I am not sure if this occurs in all situations, but it appears to ...
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Interpolating a Sequence with B-spline of Order 1

Let $\operatorname{rect}(x)=\chi_{\left[-\frac{1}{2}, \frac{1}{2}\right]}(x)$ be the characteristic function of the interval $\left[-\frac{1}{2}, \frac{1}{2}\right]$ and $\operatorname{rect}^{(p)}(x)=$...
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Linear splines approximation error

i have some problems with the following task: Let $f\in C^2([a,b])$ and s the interpolating linear spline of f with the grid points $x_i=x_0+ih, i=0,...,n$ and $h=\frac{1}{n}(b-a)$. Proof for every ...
BananaHead's user avatar
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write cubic spline as a sum of the local splines times interpolation values

Not sure if this is the right place to ask. If so, please let me know. I am performing cubic spline interpolation to produce a globally $C^2([a,b])$ function $S(x)$ on the one-dimensional interval $[a,...
Simon's user avatar
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Construction of a curved grid in 2D space, based on a cubic spline interpolation

Please bear with me, this is my first question here and I hope it fits and is understandable. I want to construct a "curved" grid on the screen in a computer program. Visualization of the ...
dheller's user avatar
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Is it possible to have a quadratic spline and a cubic spline meeting and being C2 continuous at a point K?

I'm trying to interpolate some datapoints. Ideally I would like to have 3 splines: The first and third being quadratic, and the second one (in the middle) cubic. Is it possible to mix polynomials and ...
Hiperfly's user avatar
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Frequency domain spline approximation and B-Spline inverse transforms

In short, I would like pointers to closed-form formulas, or efficient algorithms for computing the inverse of ether sine, cosine, or Hartley transforms of the B-Spline basis. The motivation, as ...
Alex Shtoff's user avatar
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Intuitive understanding behind clamping of B-splines

I have been going through the implementation of B-splines, and I observe that whenever it comes to clamping, it's usually mentioned that we must repeat the end knots $p+1$ times for a spline of degree ...
Manish's user avatar
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Thin Plate Spline RBF Interpolation understanding

I have been looking at radial basis function interpolation: $f(x) = \sum w_i \phi_i(||x-x_i||)$ and examining the different kernels e.g. $\phi(r) = e^{-(\epsilon r) ^2}$ which are generally maximised ...
Governor's user avatar
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Is the Integral of a Spline multiplied with the Exponential Function of the Spline solvable?

I want to compute the integral, I have borders for $x$, however I would like a function of y as the result: $$ \int_a^b \quad \frac{\partial f(y,x)}{\partial y} \exp(f(y,x)) \quad dx $$ $f(y,x)$ is a ...
Matthias Herp's user avatar
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Is it possible to rewrite the B-Spline Basis functions nonrecusively

B-Splines have a recursively-defined basis function $N_{i,j}(t)$ as shown here: $$ \newcommand{\defeq}{:=} N_{i,j}(t)\defeq\frac{t-t_i}{t_{i+j}-t_i}N_{i,j-1}(t)\frac{t_{i+j+1}-t}{t_{i+j+1}-t_{i+1}}N_{...
CATboardBETA's user avatar
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Define position of an object as a function of time such that object travels in fancy ways to a given point within a given duration

For context, I am trying to write animation software and, in doing so, have encountered this problem. I ask that you not be too critical of the setup I have going on because I have no background on ...
Scene's user avatar
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Is it possible to construct a SQUAD quaternion spline in a piecewise fashion?

If I have a cubic Bezier spline that interpolates the points $p_1 ... p_n$, I can extend it to another point $p_{n+1}$ by just adding a new segment with the first two control points determined by the ...
john_stech's user avatar
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How to match an ellipse with a b-spline curve with C2 continuity?

I want to describe a Turbine Airfoil using B-Splines. For the leading edge (blue) of the airfoil I'm using an ellipse arc which is then followed by b-spline curve which describes the airfoil body (...
Adrian's user avatar
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To find a interpolating cubic spline to the points

My problem is to find a interpolating cubic spline to the points $$\left\{(-2022,8043), (-4, 1989), (-2,1983), (0, 1977), (1, 1974), (3, 1968),(2022,-4089)\right\}$$ I know I can use a formula here or ...
Andrea Kamil's user avatar
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choosing between Multivariate, Univariate and Spline interpolation

I have dataset of points with coordinates and temperature measured at each point. I would like to interpolate the points to generate a continuous image. I have checked Scipy and I have seen that there ...
Reut's user avatar
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Best approximation with unknown basis functions

Given an analytic real known function such as $f(x)=\exp(-x^2)$ defined on a finite interval such as $x \in [0,1]$, find the solution of the following problem $$\min_{c_k,\phi_k(x)} \int_a^b|e(x)|^2 ...
Hosein Javanmardi's user avatar
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One step beyond cubic spline interpolation, a fourth-order problem?

I am trying to fit a polynomial through three points, where I also know the derivatives at the two endpoints. I don't need a truly general solution. My specific problem is constrained as follows: <...
John Ladasky's user avatar
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Linear fitting of values on non-uniformly parametrized B-spline surface

I have a 2d (u,v) surface in 3d space (x,y,z) that is defined as a b-spline surface. The surface is not arc-length parametrized. Additionally, I have scalar values defined on the surface at given u,v ...
pheluxe's user avatar
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Whittaker smoother 2D, $C^0$ continuity

Suppose the 2D version of the Whittaker smoothing spline https://eigenvector.com/wp-content/uploads/2020/01/WhittakerSmoother.pdf minimizing $$ \phi=(X-X_{s})^{T}(X-X_{s})^{T}+(Y-Y_{s})^{T}(Y-Y_{s})+...
justik's user avatar
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1 answer
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Are B-spline basis functions also B-splines?

When you write a spline curve as a linear combination of b-spline basis functions, it's called a "b-spline". The basis functions are generated recursively by the deBoor-Cox algorithm, ...
Ronald's user avatar
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What exactly is the basis Function for B-splines?

I am using splines for 1D model in a research project. I found a reference for how to write the basis function that I put into my code but can no longer find it. Most guides and references on splines ...
Matt Daunt's user avatar
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Creating higher order cardinal B-Spline basis functions

I am currently trying to grasp how B-Spline functions work and seem to have hit an issue with its definition. One type of B-Spline basis function seems to be a cardinal B-Spline function, which I ...
Feirell's user avatar
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What is a general form cubic-spline approximation of an arc?

Say I have an arc from $P_0$ to $P_3$, that has unit tangents $t_0$ and $t_3$, respectively. Let $L$ denote the distance between $A$ and $B$. If $P_0 = (0, 1)$, $P_3 = (1, 0)$, $t_0 = (1, 0)$ and $t_3 ...
Matthew's user avatar
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Interpolation: cubic splines

Cubic splines Given $n+1$ data points $\left(x_i, f_i\right)$ such that: $x_i<x_{i+1}$ We want a function $y(x)$ such that this function $f(x)$ interpolates continuously (up to and including the ...
Ronald's user avatar
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An elegant way to verify the convexity of a spline with certain boundary conditions

It is known that, given constants $a, b$ with $a \geq 1$ and $-1 < b < 0$, one can construct on the interval $[0, 1]$ a cubic spline $$ g(x) = c_{0} + c_{1}(x-1) + c_{2}(x-1)^{2} + c_{3}(x-1)^{3}...
avs's user avatar
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What is the size of the basis for cubic B - splines

I have an exam questions which says we have mortality data observed between ages 20 and 60 with knots at 20, 30, 40, 50 and 60. Then the questions say we will use a set of 7 basis splines to fit a ...
Ronan Geraghty's user avatar
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Finding polynomial degree $1$ of $f(x)=\text{erf}(x-1)$ at $x=1$

Question: Find the polynomial of degree $1$ that has the highest possible order of contact with $f(x)=\text{erf}(x-1)$ at $x=1$. Plot the spline knotted at $(1,0)$ with $f(x)$ on the right and your ...
l0calgod's user avatar
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112 views

best high order interpolation method for set of points

what is the best way to interpolate a set of points in 2d, such that there is only one parameter to indicate position on the curve (like is the case for a Bézier curve)? one thing I know is that we ...
Hassan Ali's user avatar
1 vote
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78 views

Algorithm to transform a polyline into an equidistant polyline

I have a number of points in 3D space. These points represent the positions (the toolpath) for an industrial robot or a CNC machine. The points are calculated by a software program. There can be up to ...
user440957's user avatar
1 vote
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116 views

Choosing optimal sample points for cubic spline approximation of sinusoid with polynomial argument

My problem consists as follows. Say I have a sinusoidal function whose argument is a cubic polynomial: $$f(t) = \sin(at^3+bt^2+ct+d) \quad a, b, c, d\in \mathbb{R}, \quad t\in[T_1, T_2]$$ I want to ...
Donatas Šimeliūnas's user avatar
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1 answer
450 views

What is the big O notation of the cubic spline algorithm?

I'm trying to do a time complexity analysis of MATLAB's cubic spline algorithm. I have a 1000 x 1000 table and I want to know what is being done when I query a point that is in between two quantities ...
Kadhir's user avatar
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Rotation interpolation knowing values at specific points

Let's say we have an arc-length parametrized curve $\mathcal{C}(s)\in\mathbb{R}^{3}$. We want to find a frame $\mathcal{R}(s)=\left[\begin{array}{ccc} \hat{\mathbf{t}} & \hat{\mathbf{b}} & \...
StargazerDRK's user avatar
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256 views

Approximating an arc with NURBS

I am trying to find a way to approximate an arbitrary arc (given by an angle phi and a radius) with a degree 3 NURBS (ie at least 4 control points). This website: https://www.ibiblio.org/e-notes/...
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