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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23 views

Name for non-polynomial generalization of B-splines?

(1-dimensional) Splines are functions defined over contiguous intervals delimited by knots. Between those knots, a spline is a low-degree polynomial function - but not the same function in every ...
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24 views

How to compute the quadric form matrix of the B-spline norm $\int_{k_1}^{k_n}f(t)^2dt$?

Let $f:[k_1,k_n]\to\mathbb{R}$ be a degree $d$ B-spline function with real knots $\{k_1,\ \ldots\ ,\ k_n\}$ and real coefficients $\{c_1,\ \ldots\ ,c_{n-d-1}\}$. The norm $\ \int_{k_1}^{k_n}f(t)^2dt\ ...
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666 views

Best spline method for closed curves

What is the best spline method I can use to obtain a closed curve for the following data? Note that I will need to obtain the derivative as in every point there will be a vector that is ...
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1answer
74 views

Will any linear combination of degree 3 b-splines be C2 continuous?

Question is as in title. I have read that natural cubic splines are always C2 continuous, but am unsure whether all cubic splines are.
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54 views

How does Raph Levien's Spiro choose angles for the ends of a path?

I've read Raph Levien's paper on splines (http://www.levien.com/phd/phd.html), and think I mostly understand chapter 8, the nuts and bolts of fitting a piecewise polynomial spiral to a sequence of ...
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837 views

Cubic Spline Interpolation - Solve X from Y

I'm a programmer, not a mathematician, but I've got a real-world problem I'm trying to solve that's out of my league, and my Google skills so far have failed me. I have an analog waveform that's been ...
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45 views

Integral of two zero-order spline basis functions

I try to derive the formula for the discrete convolution from the continuous convolution using piecewise constant interpolation. Whereby the zero-order spline basis function is given as: $$\phi^0(x) ...
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82 views

Formula for the partial derivative of a bivariate tensor-product spline on a grid of points

I was reading the documentation for the spline interpolation routines in the Rogue Wave IMSL Fortran Numerical Library and found the following formula in the description of ...
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2k views

C2 continuous Bezier contour.

I am searching a way to make c2 continuous CLOSED bezier spline via given points. I found a perfect description for OPENED spline here. Briefly it is: If were describe path needed as $i$ segments, ...
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49 views

Book suggestions on B-spline method for solving differential equations

I am a beginner level at B-splines method. I want to solve ODEs and PDEs by this method(or other splines). Could you suggest me an elementary book for solving partial (and ordinary) differential ...
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35 views

Evaluation the interpolation polynomial at $x$

I have to approximate the second derivative of a function $f : \Bbb R \rightarrow \Bbb R$ at the point $x$. There we evaluate the function for a $h > 0$ at the points $t_0 = x-h $, $t_2 = x$ and $...
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336 views

How to construct a B-spline from nodal point in Matlab?

I need to work with B-spline in Matlab using its nodal point as variable of my optimization system. So I would like to know how to obtain the B-spline from the nodal points to work with it. I need ...
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1answer
127 views

Quadratic B-spline curve tangency to its control polygon [closed]

Hello I've been asked to prove that a quadratic B-spline curve with simple knots only is tangent to each edge of its control polygon. In the second part of the question I need to show the relation ...
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25 views

Can the relocation of one control point of a NURBS curve be compensated by an adjustment of some weights?

Can the relocation of one control point of a NURBS curve be compensated by an adjustment of some weights? How to show that there exists NURBS curves for which even the modification of all weights does ...
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135 views

inhomogeneous coordinates to homogeneous coordinates

I will appreciate your help to solve this question. For a function $fi:R^2 → R$ in inhomogeneous coordinates, we obtain a function $fh : R^2$ × ($R$ \ {0}) $→ R$ in homogeneous coordinates as: $fh(...
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115 views

Almost locality of cubic spline interpolation

The natural cubic interpolating spline is the unique $C^2$, interpolating cubic spline, endowed with two extra boundary conditions. Obtaining this spline, denoted by $s(x)$, involves the inversion of ...
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155 views

Approximate spline equation with Wolfram Mathematica

Well, I'm using Wolfram Mathematica for a universitary project, as the title says. I need to define the trajectory of an object, which moves in a tri-dimensional space. I chose eight control points ...
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2k views

Find control points to produce a given curve

I've been reading all possible papers about splines for a couple of days now and couldn't answer my own question. All papers I was able to find start the definition of Bezier Curve by either ...
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1answer
139 views

Quadrature rule for spline interpolation

Consider an integrable function $f$ on $[-1,1]$. We denote $\left(x_j\right)_{-N}^{N}$ the equally spaced grid on $[-1,1]$, and wish to compute the integral $I = \int\limits_{-1}^{1} f(x) \, dx$ using ...
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183 views

How to Determine a clamped B-spline curve passes through a given point q

Let P be a clamped B-spline curve of degree two defined by the control points, The control points are : $\binom{-2}{-2},\binom{-2}{0},\binom{0}{2},\binom{2}{2},\binom{2}{0},\binom{0}{-2} $ and ...
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2answers
368 views

How to prove that a B-spline curve is a Bezier curve

Let P be a quadratic B-spline with three control points $ p_{0},p_{1},p_{2}, $ and knot vector $\tau = (0,0,0,1,1,1) $. How to prove explicitly that P is a Bezier curve?
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73 views

Relation between B-spline curve and its control polygon if either of them would be convex

would you please discuss the following statement : "If control polygon of a B-spline curve is convex then B-spline curve is also convex".
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55 views

Question on smoothing spline for two point boundary value problem

In short... I am solving a two point boundary value problem using B-spline. There are some numerical issues during solving a quadratic programming problem with bad conditioned hessian. The details ...
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1answer
819 views

Which spline interpolation method can incorporate slope information at the support points?

Let be given a set of measurements $\left\{(x_1,y_1), (x_2,y_2),\ldots, (x_n,y_n)\right\}$. For these points, we further are given the slopes $y'_i$ measured at the support points $x_i$ for $i \in \{1,...
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103 views

Solving interpolation problem with B-splines

I have a sequence of $N$ equispaced knots defined in the interval $[t_0,t_f]$ and a number of values, $x(t_i), i=1...N+1$, that requires perfect matching at these knots. The resulting interpolating ...
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1answer
323 views

Multiplicity and continuity issues for infinite knot vector of Bspline

I want to investigate the continuity issues at different knot which is given as a sequence of (infinite) knot vector $\tau = (0,1,2,2,3,4,5,6,...)$. What is the multiplicity of above knots? how to ...
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147 views

Quadratic and cubic spline

Can somebody help me with this? I have to find the quadratic spline $s_Q$, when I know: $$ i = 1 \quad 2 \quad 3 \quad 4 $$ $$ x_i = −1 \quad 1 \quad 2 \quad 3 $$ $$ y_i = −2 \quad 0 \quad −1 \...
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68 views

where Bspline curve is not continuous

Consider the (infinite) knot vector $ \tau $ := ($t_0, t_1, t_2, t_3, t_4, ...$) with $t_0=0, t_1=1, t_2=t_3=2\ and\ t_j = j-1 $ for all $\ j \in N\backslash \{ 1,2,3 \} $. Identify all (permissible) ...
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1answer
257 views

Is Monotonicity-preserving cubic spline interpolation continuous to the second derivative?

I need to use cubic splines to interpolate between data points (sets of x-y-coordinate pairs). The problem is that there is the well-known "overhooting" of the spline that occurs every now and then (...
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245 views

Derivative of B-spline basis functions for degree 2

Lets first derivative of $N_{i,p}(t)$ (i-th Bspline basis function) is as follow: $N'_{i,p}(t)=\frac{p}{t_{i+p}-t_{i}}N_{i,p-1}(t)+ \frac{p}{t_{i+p+1}-t_{i+1}}N_{i+1,p-1}(t)$ Now Let's consider a ...
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230 views

What are Thin-Plate-Splines?

I am not sure I understand what Thin-Plate-Splines are. I thought it was the name of a regularization technique for b-splines surface fitting (i.e. approximation smoothing, not exact interpolation). ...
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76 views

find local min and max of natural cubic spline

My question is simple: is it possible to calculate the local maximums and minimums for data fitted with a natural cubic spline? If yes, how would I approach this problem? I work on large genome wide ...
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41 views

B-spline surfaces fitting references

I am looking for reference papers / publications regarding b-spline / NURBS surfaces fitting (I think I have noticed NURBS fitting is not very widespread so I guess most references will be related to ...
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1answer
53 views

How to find the smoothest average curve?

I have an array structure $S$ of length $n$ in each element $S\{i\}$ there are $k$ scalars saved, and $k \in [1,10]$. So $k$ can be different for each element $S\{i\}$. There is another array $v$ ...
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2answers
216 views

What is the formula for a quadratic curve with defined crossing and maxima points?

This is a problem that I've been wrestling with on and off in the process of creating quadratic splines for a game I'm building. Given crossing points $x_0$ and $x_1$, and a maximum point $m$, how ...
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1answer
161 views

Regularization term nurbs surface fitting

When fitting b-spline curves $c(u)$ to points, it is usual to write a cost function such as: $$ \sum_{k=1}^{n} \left( [(p_k-c(u))^T\cdot n_k]^2 \right) $$ The function is then derived so as to ...
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38 views

How to interpolate figure with splines?

Lets say I have a set of coordinates which when correctly (directly - a straight line) connected form a shape (a simple curve if you will). Think of star-shape, important is that a xalue x might have ...
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52 views

How to solve a system of non linear equations

I have the following system of non linear equations, and I want to solve it using fsolve but how do I do that? ...
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1answer
224 views

Convexity of a functional on a Sobolev space

Denote a specific Sobolev space by $$W^{2,2}(a,b) = \left\lbrace x\in L_2(a,b) : x'\in AC[a,b],\quad x''\in L_2(a,b) \right\rbrace $$ where $AC[a,b]$ is the class of absolutely continuous functions on ...
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122 views

When is the derivative of a quadratic spline a spline itself?

Let the quadratic spline $s_2$ interpolate $f\in\mathcal{C}^2([a,b])$ in the points $a=x_0<x_1<\cdots<x_n=b$. Is $s_2'(x)$ a linear spline? $s_2'$ is certainly a linear function and ...
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269 views

Construct a cubic spline for equally spaced nodes.

I understand what a cubic spline is, but our professor likes to give us graduate level assignments for our undergraduate class. So, im given $f(x) = \frac{1}{1+x^2}$, Let $-5 = x_0 < x_1 < ... &...
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318 views

Smoothing vs Regularization

I am working on inverse problem - calibration of local volatility (financial application). This inverse problem is ill-posed. The optimization algorithm takes a $ \sigma $ surface as input and solves ...
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1answer
49 views

Finding x/y translation between two parametric curves with partial overlap

I have two parametric curves $\mathbf{c}_1 = (x_1(u), y_1(u))$ and $\mathbf{c}_2 = (x_2(v), y_2(v))$ which are defined for $u,v\in[0, 1]$. Assuming one curve (say $\mathbf{c}_2$) is a translated ...
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1answer
299 views

Constrain the first and second derivative of a cubic bezier spline connecting multiple points

I wanted to generate a cubic bezier spline which connects specified points with constraints on the first and second derivatives of the cubic spline (with first and second derivatives less than some ...
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1answer
137 views

Envelope of a continuous function

For real valued $x$, and $a>0$, I'm looking for an (upper) envelope of the function $$f(x)=\frac{\sin(a \pi x)}{\pi x},$$ such that the property $f(0)=a$ is still preserved. The function $$g(x)=\...
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3k views

Motivation of Splines

What is the motivation of splines, in particular cubic splines. For example, why does it matter that they have any type of smoothness at the knots.
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105 views

Combining Quintic Spline Interpolation with Mollifier Functions

I am using quintic splines for creating smooth contours especially around the edges. Although it gives me continuous third derivatives the values are too big. Would it be possible to limit it by ...
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1answer
117 views

Finding Bernstein coefficients for spline

Thoughts: Since we have information up to the second derivative, we could use linear, quadratic, and cubic splines to solve the problem. I am stumped on how to find the coefficients, however. (More ...
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1answer
168 views

Deriving custom cubic spline

A cubic Bezier curve can be expressed in the following form: $$P(t) = GBT(t) = \left[ {\begin{array}{*{20}{c}} {{P_1}}&{{P_2}}&{{P_3}}&{{P_4}} \end{array}} \right]\left[ {\begin{array}{*{...
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1answer
139 views

Interpolating a three point curve at any angle using cubic splines

I'm trying to interpolate a curve using cubic splines and three points in the x-y plane. I have some troubles finding the equation for the middle point such that the normal vectors in point P0 is ...

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