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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12 views

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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57 views

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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41 views

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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CubicSpline interpolation with $x = (0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4},1)^T, y = (1, 0 , 2, -1, 1)^T $

$x = (0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4},1)^T, y = (1, 0 , 2, -1, 1)^T$ Here is what I tried, but it looks like I have too many equations in comparison to unknowns. $$\left[0, \frac{1}{4}\right]...
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How do I determine the curvature of an arc length parameterized curve in the $xy$-plane? [closed]

I have a 2D curve in the $xy$-plane, which was arc length parameterized numerically, and fitted by cubic splines for both $x$ and $y$. If one of the segments of the cubic spline is: \begin{align} x&...
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Derivative estimation from noisy measurements subject to magnitude constraint on second derivative

Consider a smooth function $f(x)$. A set of noisy measurements of this function is observed: $ y_i = f(x_i) + e_i $ A method is sought to estimate $f'(x_i)$, the derivative of this function at $\{...
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Slice 3D curve into series of Bézier curves

I've been stuck on this for 2 weeks and have not had much luck asking else where either. I have a curve, which is an arc, at each end of the arc there is tangent vector which is sloped in some ...
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How to add constraints to a spline?

I am using the geomdl python library to build 3D splines, either by fitting a set of points, with underlaying NURBS or ...
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How to measure the accuracy of a spline?

If I fit a cubic spline to data points, isn't the spline function forced to go through all given points? How is it possible to measure the accuracy (as squared deviations) if all points are already ...
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Ideal spline type for software optimized well-defined interpolation over time domain

I'm working on a software program which currently utilizes cubic Bézier splines for generating continuous well-defined function output (only 1 output value per input) in the time domain with varying ...
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Catmull-Rom intersection at a given X

I'm using centripetal Catmull-Rom to interpolate keyframe values for animation. Each keyframe represents an animation time $x$ and a channel value $y$. I am trying to write a function that is ...
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I need a spline type that interpolates through knot points and produces a single-valued y=f(x)

I have a set of $(t, f(t))$ points that describe a function of time. I need to interpolate those with a spline that goes through all the given points, and doesn't have any loops, so it creates a ...
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explicit formula for calculation of coefficients in piecewise polynomial representation of uniform cardinal B-spline

I want to compute the monomial coefficients of a uniform cardinal B-spline of order d explicitly. Can someone point me to a useful source? I found many sources for evaluation of B-Splines and I know ...
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can fixed point iteration apply to spline?

For a polynomial function, y = f(x), I can use fixed point iteration to find x0 given y0 such that y0 = f(x0). However, if y = f(x) is a spline function, can I use the same method? x maybe be in ...
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Is this semiparametric multiplicative regression model identifiable?

I am considering a semiparametric regression with a multiplicative model as below: $$Y_i=m_1(X_{1i})+m_2(X_{2i})+g_1(X_{1i})*g_2(X_{2i})+U_i, \ i=1,...,n, \quad (1)$$ where $\{Y_i,X_{1i},X_{2i}\}_{i=...
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How to interpolate tangent with Cubic Hermite Spline

I can find the interpolation of position with Cubic Hermite Spline in Wikipedia: https://en.wikipedia.org/wiki/Cubic_Hermite_spline On the unit interval (0,1), given a starting point $p_0$ at t=0 ...
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How to take the partial derivative of a line integral where the line is a B-spline contour in a 2D image?

Short Version How do I apply this partial derivative to a line integral along a contour in an image? The contour is a B-spline and the term I am trying to resolve is below. The detailing of each part ...
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Construction of a piecewise linear function

Given some continuous probability distribution $\rho$ over $\mathbb{R}^3$-tuples $(a, w, b)$, define the function $$y(x) = \int a \max(0, wx+b) \rho(da,dw,db)$$ Can we construct a continuous ...
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Is there a way to find the range and domain of a cubic spline?

Wolfram alpha is telling me there isn't. I guess this reduces down to finding the inflection point in some way and testing against points A and D. Here's the equation I'm using. $$ x=(1-t)^3*A_{x}...
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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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Intuition behind the definition of normed B-Splines

Definition: The normed B-Splines $N_k^m$ of degree $m\geq 1$ are recursively defined as $\hspace{4,5cm}N_k^0(x)=\begin{cases} 1, & \text{if}\ x\in[x_k,x_{k+1}) \\ 0, & \text{...
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Bspline values at end points

Using for example Wolfram alpha definition of Bspline basis functions, the basis functions of order p=3 (cubic) and knotvector = [0,0,0,0, 0.2, 0.5, 0.5, 0.5 , 0.8, 1,1,1,1] looks like this: As you ...
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What is the exact definition of a spline?

I know that a spline basically involves joining several functions together. I am hoping to get a more precise definition so I know exactly when to use the word "spline". I have a few questions ...
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Spline questions (Degrees of freedom of cubic spline)

How many degrees of freedom has a cubic Spline? And how to calculate it? I know it has to do something with the degree of the polynomial, so in cubic the $n=3$, and also let's say we have $k-1$ ...
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Does the cubic Hermite spline interpolation of a monotone data with known slopes guarantee the monotonicity in interpolation segments?

I am trying to interpolate monotone data with known data values and also known first derivative values at knots. If I used these values with cubic Hermite spline interpolation, Can I guarantee the ...
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how to perform spline interpolation on GPS coordinations?

this may look like a programming problem but actually it have to do with math more than programming. I have GPS coordinations in a csv file that I predict it using a regression model, just two ...
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least square approximation in spline space

I am confused about how to obtain the least squares estimator of this problem: $$ \inf\{||s-\lambda||_2:s\in S_{k,v}\}=||P_k\lambda-\lambda||_2\tag{1} $$ where $||\lambda||_2=\{\int_0^1\lambda^2(x)dx\}...
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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 ...
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Good polynomials to use that can fit closed complex 2D shapes

Are there any good polynomials that can be used to fit closed complex 2D shapes? For example, the silhouette of everyday objects or people. I am aware I can use a spline for example, however, they ...
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Need just a little bit of help proving this graph is not a natural cubic spline

So this is my graph I am aware that we can see its not a cubic spline because $p_1(x), 0≤x≤1$ is linear. But I wanted to prove it and got stuck. From what I know, For it to be a natural cubic ...
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sketch third derivative of cubic spline interpolation

Let S be a cubic spline interplant defined on the nodes $x_0=1, x_1=2, x_2=3$. Make a sketch of a typical $S'''(x)$

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