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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Alternatives / Extensions to the Thin Plate Splines method

Thin Plate Splines are a great method to find a smooth interpolating surface given scattered data. Essentially, the method involves calculating weights for a radial basis function centred around each ...
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a smooth bandlimited and short-lived B-spline function?

Is there exist a smooth Spline function which lives only between $[-1, 1]$ and is also bandlimited? I have not found any! I will need to replace the $\text{sinc}$ function/interpolator which this ...
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12 views

How to apply a time shift to a pulse-shape, spanned with spline functions?

I have a sampled pulse shape: $ h = [1, 0.5]$ and I do not know what is its real underlying continuous-time pulse. I want to compute the samples of $h(t-\Delta t)$. If I write the continuous pulse ...
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14 views

Definitions of BSpline

I can think of 4 ways of defining BSpline with separate, equally spaced knots: recursion convolution sum divided difference. Methods 1 & 2 start with the indicator function of the unit ...
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16 views

Irregular time intervals

This is a fairly general question which I can not find a simply-put answer to anywhere. My mathematics are limited/rusty in this area. I have written a Runge-Kutta integrator code, which I am ...
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24 views

Nurbs parametric coordinate span

I am using the Nurbs definition of Wikipedia. I might have missed something in the definition but I cannot understand how to know on which interval does the parametric coordinate span. Particularily ...
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1answer
29 views

Deriving uniform cubic B-spline matrix

The page 6 of this paper says: Condition 1: $p(0) = q(1)$ – Symmetry: $p(0) = q(1) = 1/6(\pi-2 + 4 \pi-1 + \pi)$ Condition 2: $p’(0) = q’(1)$ – Geometry: $p’(0) = q’(1) = 1/2 ((\pi – \pi-1) + ...
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52 views

Measure of curve smoothness

Could someone please give me the intuition behind using integral of squared second derivative as a measure of curve smoothness? I was thinking that since curvature measures how fast a curve changes, ...
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19 views

Finding the curvature from a set of datapoints

I have a set of 1. 1-d 2. 2-d data. I want to find the curvature at each single point. Till now I was using difference technique to find out the curvature, i.e, central difference at middle and ...
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19 views

Quartic spline explanation

I need to use a quartic spline (natural spline) for a robotics application. However, most of the documents explaining splines focus on quadratic and cubic splines. Could you please give me some ...
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16 views

How to insert a knot in NURBS if it coincides with the first knot?

I want to insert a knot to the knot vector. Currently I use the algorithm from the NURBS book, but it has an assumption that U={0,...0,u_{k},u_{k+1}...,1,...1}, the first knot and the last knot repeat ...
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How to select the number of nodes in a spline interpolation?

I am writing a program to test the precision of different methods for imputing missing data in a time series. One of the methods I am going to test is a natural cubic spline interpolation. I'll be ...
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44 views

Number of points for local spline interpolation

I have a large scattered data set of (x,y = f(x)) points and I want to interpolate them to regularly spaced grid points. To do this I have chosen to use cubic splines as my interpolation method. The ...
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2answers
36 views

Wich kind of splines are the 3DS Max graph editor splines?

I'm trying to reproduce the splines from the program , and I have the correct point data, but my representation of Bezier Splines using the same anchor point data fails to give me a correct curvature. ...
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1answer
29 views

Cubic Uniform BSpline surface interpolation

I want to understand cubic BSpline surface( very hard for me to figure out). I prefer matrix form which presented here. Equation 4.12 in page 33, describes how data point should be presented ...
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1answer
24 views

What is the difference between a rational spline and a “regular” spline?

I'm pretty comfortable with Bezier curves (not as much with b-splines, nurbs, hermite, catmull rom, etc), such that i know how to generate a bezier curve of any degree using the bernstein polynomials ...
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29 views

Interpolating a set of GPS points on ellipsoid earth model.

I have a set of GPS (latitude, longitude) co-ordinates along with the time at which the coordinates were collected. Additionally, I have the speed and the heading of the vehicle at those coordinates. ...
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23 views

3D BSpline approximation

I read a lot about Bspline but I am confused. Hope someone can help me here. I have 3D control point matrix (104x54x104), and i have 3D Grid which if I am not mistaken it is considered as my ...
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87 views

B-Spline: How to generate a closed curve using Uniform B-Spline curve?

Given $n+1$ control points; $P_0,P_1,...,P_n$ (where all are 2-dimensional points), and $k$ (which defines the order of the polynomial, and hence its degree; $k-1$) the B-Spline curve is defined by: ...
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174 views

Centripetal Catmull–Rom spline

What is "t" in this short and simple example below? There are 4 points Pn[xn,yn] in 2D space: A[1,6] B[3,1] ...
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136 views

How to find tangent at any point along a cubic hermite spline

I have a cubic hermite spline path that I am using to move sprites around on the screen (2D). I use two end points and two tangents to define the curve and then I use the basis functions for ...
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19 views

My Restricted Cubic Spline Equation is Wrong?

I am trying to fit a restricted (ie natural) cubic spline to toy data, attempting to follow Hastie, Tibshirani, Friedman 2nd Ed, Section 5.2.1 pp.144-146, Equations 5.4 and 5.5. Data: Is basically a ...
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25 views

Find the clamped spline with the initial and final derivative are equal, and the derivative at $x = 3$ is set to $0$.

The set of $x$ and $y$ are $x=\{0,1,2,3,4,5\}, y=\{0,1,2,3,4,5\}$ I want to do a clamped spline with the two end derivatives equal and the derivative at $x=3$ is $0$. I tried splitting the data into ...
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Generating volume constrained splines

Suppose I have a set of points in $\mathbb{R}^3$, $\{\vec{r}_1,\vec{r}_2, ...,\vec{r}_n\}$, suppose between points $\vec{r}_i$ and $\vec{r}_{i+1}$ there is an associated volume $V_i$. I want to ...
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Does the condition $S \in C^2[x0, x2]$ lead to a meaningful solution, when conctructing quadratic spline

we have three data $x_0 , x_1 , x_2$ we want to find the quadratic interpolation . $$S_0(x) = a_0 + b_0(x − x_0) + c_0(x − x_0)^2 on [x_0, x_1]$$ $$S_1(x) = a_1 + b_1(x − x_1) + c_1(x − x_1)^2 on ...
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cubic interpolation and data on a straigh line

Suppose the data we want to interpolate lie on a straight line. What can be said about the natural cubic spline ? i think i should show that the cubic spline in fact becomes a line . so if the spline ...
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31 views

how to learn the fast method of finding the cubic splines

in finding cubic splines if we have n points we get system of equations of magnitude 4(n-1) in the naive approach . in more sophisticated approach one only solves a system of equations with (n-1) ...
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29 views

solving $c_j$ system of equations for cubic splines?

the problem is like this : There are $N$ points $(x_0,y_0),(x_1,y_1),\dots,(x_{N-1},y_{N-1}) \in \mathbb{R}^2$ where $x_0 < x_1 < \cdots < x_{N-1}$. Cubic spline interpolation should give ...
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1answer
102 views

What is the difference between nodes and knots in interpolation?

I have been reading literature about polynomial interpolation (Lagrange) where the principles are described around nodes. The literature I have read about spline interpolation, however, talks only ...
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28 views

Re-initialize a 3D spline surface using different control points

I have a 3d spline surface is that is modeled with 65 points where the x,y,z position of each point is known. I want to keep the surface shape, but have different x,y positions for my control ...
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About minimal curvature of splines

I am given a the following problem set: Let $s$ be a natural cubic spline that interpolates a function $f \in \mathcal{C}^2 ([a,b])$ at points $a = x_0 < x_1 < \ldots < x_n =b$ with ...
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1answer
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How to determine if an equation represents a cubic spline?

Given the equation $$ f(x) = \left\{ \begin{array}{lr} 2x^3+x^2+4x+5 & : 0 \le x \le 1\\ (x-1)^3 + 7(x-1)^2 + 12(x-1)+12 & : 1 \le x \le 2 \end{array} \right. ...
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How much does convolution with a compact C^m kernel increase the order of continuity.

Let $f \in C^n$ and $g \in C^m$, with $g$ compactly supported and integrable. How much does the convolution $f\star g$ of $f$ with $g$ increase the order of continuity? Statement: I think that, under ...
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Why Are Fresnel Functions Used For Splines?

Why are Fresnel functions still used in the research and implementation of clothoid splines? They cannot represent curves of constant curvature, which has led to a lot of research/implementation ...
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1answer
69 views

intuition about cubic splines vs quadratic splines (degree 3 vs degree 2).

my intuition about quadratic(degree 2) splines is that by the help of its three variables (in each sub-interval) you can make a piecewise differentiable function on the whole interval. in the process ...
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1answer
80 views

Cubic splines better than quadratic splines?

I have read in a number of places that cubic splines are of more practical use than quadratic splines in general (there are exceptions of course). Anyone know specifically why they are more ...
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1answer
49 views

Interpolating discrete points along a spline

I have 2 lines which i need to connect with a spline (or some other curve, but not the circle; it has to gradually increase its turning angle). The lines cross each other so the curve should make a ...
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38 views

Selecting nodes for spline interpolation

Is there a general method to determine the best sample points for spline interpolation (whether for piecewise linear or piecewise cubic Hermite) given $x$, $f(x)$, and estimating $f^\prime(x)$? Does ...
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1answer
19 views

Need help for construct DLSQ-spline in $B$-form

In Schumaker, Larry L. "Computing bivariate splines in scattered data fitting and the finite-element method." Numerical Algorithms 48.1-3 (2008): 237-260 (Link to journal, link to author page for ...
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Hermite Spline curvature

I have a doubt about the Hermite Spline. Is it the interpolation with the minimum value of curvature among the possible interpolative functions between two points? Is it possible to demonstrate it? ...
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43 views

$L^2$ vs $L^{\infty}$ norm for interpolation

Under what circumstances should I consider utilizing the $L^2$ norm instead of $L^{\infty}$ when interpolating a function based on sample points? Probably related: Does the answer significantly ...
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21 views

Empirical Mode Decomposition: How to make a suitable spline interpolation when the number of extrama is small?

I am doing an EMD(Empirical Mode Decomposition) project and I am getting a problem with the spline step (find the upper and lower envelopes) because there is always a big deviation in spline ...
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1answer
31 views

How can I approximate a function that is not derivable with derivable ones?

Suppose that I have a function whose graph has many angles (i.e. my function is not derivable). How can I approximate this function with derivable ones? Thank you!
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1answer
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A Practical Guide to Splines (De Boor) - Proof of Leibniz formula

In De Boor's A Practical Guide to Splines (1978) Leibniz' formula is defined as follows (p.5): If $f = gh$, i.e. $f(x) = g(x)h(x)$ for all x, then $$ [\tau_i, ..., \tau_{i+k}]f = ...
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Bernoulli monosplines

Please help me with Bernoulli monosplines. Let's consider $2\pi$-periodic cubic spline, which is consist from $N$ ranges $0<x_1<x_2<\cdots<x_N<2\pi$. We can introduce a periodic ...
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3answers
236 views

Program to find closest function to fit arbitrary data

I've wanted this for years, but have never come across anything; a program for Windows to find the closest function to fit arbitrary data. The data I feed it is simple: A table with two columns ...
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1answer
48 views

Derivates of periodic parametric cubic splines

My Problem is sort of solved, I overlooked, that paameters $B$ to $D$ are dependent on $x$ and $y$ one question remains, see bottom of question. I implemented a periodic parametric cubic spline, and ...
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1answer
63 views

Interpolation with a constrained range between given control points

I am trying to create an algorithm that creates smooth color gradient functions, given control points in the red, green, and blue components. Mathematically, each curve would have a domain [0, 1] ...
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1answer
66 views

spline derivation

Assume the following representation for cubic splines with $T$ interior knots is given. Let $g(Y)=\sum_{j=0}^3 \alpha_j Y_j+\sum_{t=1}^T \gamma_t (Y-\zeta_t)_{+}^{3}$ where $(Y-\zeta_t)_{+}:= ...
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64 views

Reverse spline interpolation

Say I have a number of sets $(x, y)$ for $x \in \{0, 1, \dots, 255\}$. I want to find the least number of points to reproduce the set with a certain accuracy using linear interpolation. What is the ...