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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Knot sequence for a natural cubic (B-)spline interpolant

say I am given $n+1$ data points $(x_i,y_i)$ with $0\leq i \leq n$ and $x_0 < x_1 < \dots < x_n$. I want to interpolate these with a natural cubic spline $s(x)$ ($C_2$ continuous at knots ...
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Compact set of function involving BSpline functions.

Let $X = \left\{x_0,...,x_{n-1}\right\}$, $x_i-x_{i-1} = h$ for $i=1\ldots n-1$ and $$ \phi(x;X,\vec{\beta}) = \sum_{j=0}^n \beta_j \phi_{j,2}(x;X) $$ where $\phi_{j,2}$ is the $j$-th BSpline (of ...
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16 views

B spline - Partition of unity

I need the Laplace form of the second order B splines over logarithmically-spaced knot. I have read a paper in which it is mentioned that ; " Heaviside function can be represented as the sum of all ...
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31 views

Cubic spline interpolation results

I have a set of data points on which i am trying to do cubic spline interpolation. Below is the snapshot of the curve with the input data points marked in green color. And the red color marked point ...
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15 views

Estimating the curvature of a discretized curve in 3d with cubic splines

I have a computer simulation in which I'm modeling a physical curve by discretizing it and updating the locations of these points. I want to find/estimate the location of the maximum curvature of the ...
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1answer
26 views

B-Spline approximation deviates a lot while increasing the number of control points???

I'm dealing with a problem to approximate some data points with B-Spline. I follow the method and implemented the algorithm from this site: Curve Global Approximation. 1) The first step is to ...
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1answer
56 views

Cubic Spline Interpolation

My problem is to find a interpolating cubic spline to the points $$\left\{(0,0), \left(\frac{\pi}{2}, 1\right), \left(\pi,0\right), \left(\frac{3\pi}{2}, -1\right),(2\pi,0)\right\}$$ I did as ...
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19 views

Curve Conversion

I have a curve that is kind of spline. It is defined as a sequence of polynomial segments. It is defined by order, knots, and coefficients of the polynomials. How can I transform or convert this ...
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19 views

Interpolate with smoothing parameter

I need to implement in C++ interpolation with smoothing parameter. To the non-familiar with this function: The smoothing parameter gets a value from 0 to 1. 0 brings absoulte linear interpolation ...
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13 views

Example of a “abrupt function”

I need example of a simple function to show that cubic spline gives better result than Lagrange's interpolation in case of some special functions. Thank you
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What equation produces this curve?

I'm working on an engineering project, and I'd like to be able to input an equation into my CAD software, rather than drawing a spline. The spline is pretty simple - a gentle curve which begins and ...
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14 views

Finding the equation of a quadratic with 2 points and a known slope. (SPLINES)

Sketch the spline of degree 2 with value 0.5 at x = 2.5 and the values 1, 1, 0, 0 at t0, . . . , t3, respectively. (t0=0, t1=2, t2=3, and t3=5) What is the value of the spline at x = 1 and 4? What I ...
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1answer
32 views

Cubic Spline for a function

I have the function $f(x)=x^3$ and I need to find the cubic spline. The given points are: $\{-1, 0, 1\}$. What is the cubic spline for this function and what would a demonstration to this be? I would ...
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1answer
18 views

n-order B-splines interpolation

I am wondering if the following statements are correct: (1) zero-order B-splines interpolation is equivalent to nearest-neighbor interpolation. $C^0$ continuity thus is not differentiable. (2) ...
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26 views

Deriving a tridiagonal system for cubic spline interpolation

Can anyone explain how $B_{i-1} = 1/4$ and $B_{i+1} = 1/4$ were chosen in line 6 of the picture, just above the matrix? I'm trying to understand cubic splines but this result seems like it came ...
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1answer
17 views

B-splines locally controlled

I have read that in contrast of the thin-plate splines, B-splines are locally controlled, which makes them computationally efficient even for large number of control points. I didn't understand what ...
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39 views

Interpolation by splines: how to set up the equation system for finding the coefficients of the spline (in a B-spline basis)

Problem. I want to interpolate a function $f$ in some equidistant points $x_0<x_1<x_2<x_3<x_4$ using a quadratic spline. My attempt. I assume that we can use the interpolation points as ...
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33 views

Addition of two B-spline curves

Suppose I have two B-splines, both with the same degree, $p$, and uniformly distributed knots, but with different numbers of knots and control points. Is it possible to sum the two splines to obtain ...
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13 views

Enforce Spline Section Always Greater Than/Equal Zero

I'm attempting to do some cubic spline interpolation to make incomparable discrete datasets comparable by naively making them continuous. Its largely histogram based data with different bin thresholds ...
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1answer
8 views

Normalized vs Non Normalized Bernstein

I know what Bernstein polynomials themselves are, and am intimately familiar with one of their usage cases - Bezier curves. However, I recently came across someone mentioning of "Not Normalized ...
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21 views

Gaussian process regression from predictions

Can any body provide example of a gaussian process regression being used to generate confidence intervals?
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45 views

The quadratic spline

I'd like to fit the data in table as blow x f(x) 3.0 2.5 4.5 1.0 7.0 2.5 9.0 0.5 when $x=5$, I want to find value of $f(x)$ by using ...
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25 views

Bspline matrix form?

I understand how bezier curves can be expressed in matrix form: you have a matrix multiplied by a vector containing the power series of t, and also multiply be a vector containing the control points. ...
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14 views

Does this operator produce an interpolating spline of it's argument?

I'm studying a bit of spline functions theory: In this book, chapter 6, a lot of error bounds are given when spline functions are used to approximate functions belonging to a specific function spaces. ...
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28 views

Geometric Meaning of Modulus of Smoothness

Could you explain me the "geometric meaning" of the following definition (it's taken from a book on spline functions theory)? Definition: Given $1 \leq p \leq \infty$ a positive integer $r$, and ...
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71 views

Curve-fitting using circles

I'm working for a firm, who can only use straight lines and (parts of) circles. Now I would like to do the following: imagine a square of size $5\times5$. I would like to expand it with $2$ in the ...
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1answer
27 views

C2 continuous interpolation on a 4-dimensional dataset

I am currently coding up a project where interpolations must be performed such that C2 continuity be preserved along the length of the whole set. The end result ought to look like a line (which will ...
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1answer
28 views

Terminology: Spline interpolation

I have read two different definitions of Splines: A differentiable piecewise polynomial. A piecewise polynomial. If I build a piecewise polynomial using cubic polynomials, it's ...
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41 views

Calculating B-Splines and dimension of spline space

I've got the following assignment: Let $S$ be the space of piecewise polynomials of degree $3$ on the intervall $[-1;1]$ with knots $x_i = -1+\frac{i}{2}, 0 \leq i \leq 4$. (a) Calculate a basis of ...
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52 views

The dimension of the space of continuous functions that are piecewise polynomials of degree $k$

I am trying to calculate $$ \dim( \{ f\in C^0 ([a,b]) : f_{|[x_{j-1},x_j]} \in \mathcal{P}_k, j = 1,...,m \}) \text{ with }m,k \in \mathbb{N} $$ Is $m(k+1)$ correct? My thoughts: I have $m$ ...
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1answer
109 views

Error Bounded Cubic B-Splines with fewest segments

I have some odd constraints in my project. Suppose we want to use Cubic BSplines to approximate a set of Points. There is two Constraints: error value should be an input to the algorithm (error ...
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46 views

Spline terminology

I am reading up on splines and as a beginner I have a basic question - Does it make sense to say - "I will fit a cubic b-spline to the data". As b-spline is just a representation of spline in terms ...
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34 views

How sensitive is a natural cubic spline?

I am interested in using natural cubic splines to generate possible replacement values in the quality control of data. I would like to do this as close to real-time as I can. That is, I would like ...
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1answer
33 views

Which Properties of a Natural Cubic Spline does the following function possess and not possess

I need to determine which of the properties of a natural cubic spline the following function possesses or does not possess: $$f(x) = \begin{cases} (x+1)+(x+1)^{3}, & x \in [-1,0] \\ ...
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Catmull-Rom: spline and filter

On this website, the author gives this definition for Catmull-Rom splines (slide 10): $$catmullRom(t) = \frac{1}{2}\left\{\begin{array}{ll} t^3 + 5t^2 + 8t + 4 & \text{if } -2 \le t \lt -1\\ ...
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1answer
40 views

Natural Cubic Spline Problem - what do I do next?

Related to this similar problem. I am attempting to solve the following: Find a natural cubic spline function whose knots are $-1$, $0$, and $1$ and that takes on the values $S(-1)=5$, $S(0)=7$, ...
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2answers
42 views

Which interpolation method for complicated, smooth curves?

Which interpolation method should I use for complicated "smooth" curves such as $\frac{sin(x)}{x}$ for $x>0$.
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1answer
66 views

Signal processing : future values prediction

Let $f : \mathbb{R}^+ \rightarrow \mathbb{R} $ be a continuous function. Do you have some references (books or online resource) about techniques that allow to predict $f(x_{n+1})$, knowing $f(x_0), ...
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38 views

Let S be a cubic spline that has knots t0 < t1 < · · · < tn.

Let S be a cubic spline that has knots $t_0 < t_1 < · · · < t_n$. Suppose that on the two intervals $[t_0, t_1]$ and $[t_2, t_3]$, S reduces to linear polynomials. What does the polynomial S ...
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28 views

Convergence theorems of periodic and natural cubic Splines

I have this question on the topic convergence theorems of cubic splines, for cubic splines which theire first derivatives at the start and end points are equal to first derivatife of the $f$ function ...
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1answer
33 views

Bézier Curve and b spline curves.

Well I am learning about curves. I have come across Bézier and Spline curves. I want to know which one should be learned first? Are their concepts independent? or I need to know about one before ...
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46 views

second order interpolation with positive weights

I have a function evaluated at predefined grid points $x_i=\frac{i}{N},\ i=0,...,N,\ f(x_i)=y_i$ I am looking for an interpolation scheme satisfying $\|I(f)-f\|_{L^\infty} = ...
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22 views

Evaluating a Hermite Quaternion Curve

I have a set of fixed poses (position and orientation) and want to interpolate C1 continuous between the orientations. I tried to follow A General Construction Scheme for Unit Quaternion Curves with ...
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2answers
23 views

Find real constants $c$ and $k$ such that $y=cx^k$ passes through point $(a, b)$ with slope $m$

In the Cartesian plane, can a power function of the form $y=cx^k$ (where $c>0$ and $k>1$, not necessarily an integer) be found such that its graph passes through any arbitrary point $(a, b)$ ...
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39 views

Cubic spline. What is symmetrical form and why?

I'm trying to understand the algorithm for cubic spline from Wikipedia. It says the polynomial can be written in symmetrical form: A third order polynomial $q(x)$ for which $q(x_1)=y_1$, $(x_2)=y_2$, ...
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Forecasting electricity load based on various parameters using neural networks.

I have a task of predicting the electricity load for 2 months of a region based on various parameters, like avg. rainfall, avg. solar radiation, winter or summer. etc. There are two sets of data ...
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32 views

Quadratic spline?$f(x)=x$ when $x\in (-\infty, 1]$, $f(x)={-1\over 2}(2-x)^2+{3\over 2}$ when $x\in [1,2]$, and

Determine whether this is a quadratic spline $f(x)=x$ when $x\in (-\infty, 1]$, $f(x)={-1\over 2}(2-x)^2+{3\over 2}$ when $x\in [1,2]$, and $f(x)={3\over 2}$ when $x\in [2,\infty)$. I think this is a ...
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15 views

Computing splines using Hermite interpolants

The form for a cubic Hermite interpolant has the form $p_i(x)=a_i+b_i(x-x_i)+c_i(x-x_i)^2+d_i(x-x_i)^3$ according to the following conditions: $p_i(x_i) = y_i$ $p_i(x_{i+1}) = y_{i+1}$ ...
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32 views

Spline interpolation explanation

I'm trying to learn about spline interpolation, and I'm struggling to understand what h_i^3 is in the second and third equation. I don't understand how they derived that equation. I'm trying to ...