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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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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38 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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47 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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29 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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2answers
24 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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17 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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28 views

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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34 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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The Elements of Statistical Learning: How does this nature cubic spline have K basis functions with K knots with the given solution? See Details.

I'm referring to this found in chap 5: Picture from ELS If K=2 (as in Sec 5.2), then we have N1, N2, N3 and N4 basis functions. So 4 basis functions and 2 knots. I know that a natural cubic spline ...
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38 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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44 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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36 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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15 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
27 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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43 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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13 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
21 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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31 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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2answers
52 views

Find the natural cubic spline function whose knots are $-1$, $0$, and $1$ and that takes the values $S(-1)=13$, $S(0)=7$, and $S(1)=9$.

Find the natural cubic spline function whose knots are $-1$, $0$, and $1$ and that takes the values $S(-1)=13$, $S(0)=7$, and $S(1)=9$. I'm not sure how to go about this. Any solutions/hints are ...
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17 views

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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31 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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13 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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26 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 ...
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63 views

Can I make this numerical integration continuously differentiable?

Suppose I have the discrete values $f(x_i)$ for every $x_i$ greater than some $\varepsilon$, and I want to numerically calculate the following integral: \begin{equation} n = \int_\varepsilon^\infty ...
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31 views

How to set control points for spline curves

I've written a program that calculates points on spline curves (including Hermite, Bezier, and B-splines) and plot the curve on the screen (the curve is plotted on an html canvas using javascript). ...
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1answer
20 views

Tangent of Cubic Hermite curve

I have created cubic curve using CatmullRom Spline or Akima spline. From those, I obtain $a, b, c, d$ parameters. To get point on the curve, I do this $f(t) = a + bt + ct^2 + dt^3$ How to get ...
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16 views

Why do bsplines of order 3 go through the endpoints only when the endpoints have multiplicity 3

Hi I'm learning about different types of splines and interpolating curves. I wrote a script to generate cubic bsplines. I noticed that the bspline does not go through the endpoints if I give a knot ...
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33 views

Can monotone cubic interpolation be implemented explicitly in B-spline form?

I have been interpolating cubic splines to some data, but it is now clear that I need my curves to be monotonic. Wikipedia and StackExchange sources describe how to impose the monotonicity condition ...
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25 views

Does cubic spline interpolation preserve both monotony and convexity?

I have a question. Let's say i have a function $f(\cdot)$ such that $Dom(f) = [a,b]$. The function is at least of class $C^2$ and it is both strictly monotone and convex. My question is, does a ...
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42 views

Linear, Bilinear and B-spline interpolation

I've read that the linear interpolation isn't differentiable everywhere and it would be better to model a continuous-space image using quadratic or cubic B-spline interpolation because is ...
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21 views

Spline interpolation degrees of freedom

When using cubic spline interpolation, we have to solve $n-1$ equations with $n+2$ unknowns. What we can do is set $z_0 = z_n = 0$, which gives the natural cubic spline. But could we also set some ...
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41 views

How to get normal of Catmull Rom spline

I have a piece wise track made out of Catmull Rom splines. I originally crossed the tangent vector with (0,0,1), then the result of that crossed with the tangent to get the normal vector. However, ...
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2answers
59 views

Cubic spline with clamped boundaries

I have a cubic spline interpolation problem to work through. I think I understand what is required of the question, but my biggest concern is the nature of clamped versus natural boundaries. All the ...
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1answer
14 views

Is cardinal $B$-spline of order $n$ really piecewise Bezier order $n$ curve?

Is cardinal $B$-spline of order $n$ really piecewise Bezier curve $n$? I think I saw this in some lecture notes, but I can't recall where.
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24 views

Numerical integration with limited support

This is somewhat of a crossover between mathematics and programming, but I suspect the core idea I'm missing is mathematical. Starting with the following integral: \begin{equation} \tilde{F}(k) = ...
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21 views

Derivative of multivariate splines using tensor products

I am trying to compute the derivative of a multivariate spline, in fact bi-variate I use a b-spline univariate to create a basis, for the first $x_1$ and second variable $x_2$, then I use the tensor ...
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1answer
74 views

Subdividing a Bézier curve into N curves

NOTE: I am only concerned with quadratic Bézier curves. So, dividing a Bézier curve into two is remarkably easy; just interpolate between start and control points by $t$, and get the end point for ...
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1answer
41 views

Extrapolation and Splines

If you have a smooth curve, and at a certain point in time you want to predict the next turning point, and you assume it is a non-periodic, stationary, smooth process, then what would be the best way ...
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49 views

B-spline weighted least squares fit

Can someone please point to an easy to read source for Bspline curve fitting with weighted least squares. Basically I want to fit a function, and I have some points which are more important then ...
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1answer
85 views

How is the B-Spline definition constructed?

I'm trying to understand how the B-Spline definition is constructed. That is, where did the knot vector and the basis functions and their recursive definition come from. The definition can be seen ...
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26 views

How to Find the Error for Spline Interpolation Without the Original Function?

Most of the literature (e.g.: http://pages.cs.wisc.edu/~amos/412/lecture-notes/lecture11.pdf) I have consulted thus far indicates how one would determine the error of a cubic when the original ...
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1answer
102 views

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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17 views

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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14 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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16 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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44 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
77 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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121 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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40 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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29 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 ...