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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327 views

B-Spline how to create control points for a curve to pass through knot values

I want to create a b-spline curve that will pass through all the (knot) points I give it. How do I construct it? Do I need to find the control points for that curve? And if so - how?
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Fitting splines to famous curves programmatically

this may be a bit of a naive question. I am looking for a way to input a Cartesian description of a famous curve and map that to some spline say NURBS to make spline paths. Is this at all possible? ...
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91 views

“Standard methods of the calculus of variations” or “Do you read German?”

I'm trying to understand an article of Reinsch (1967), on smoothing spline functions. The author uses some rules that unfortunately I couldn't find. The minimized functional is: $$ \int_{x_0}^{x_n} g'...
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295 views

NURBS circle without all the double knots?

I've been looking at various examples of a circle parametrized as a degree-2 NURBS curve, e.g.: NURBS circle example on wikipedia Philip Schneider's "NURB Curves: A Guide for the Uninitiated" David ...
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517 views

Drawing De Boor's algorithm

Assume we have 4 control points $[c_0, c_1, c_2, c_3]$ and uniform knot sequence $[0,1,2,3]$ If we were to draw an quadratic bezier we would be forced to use only 3 of the control points and then we ...
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124 views

Would the use of cubic splines increase the number of data points to interpolate from result in smaller error and avoid Runge's phenomena?

I am implementing the cubic spline method to interpolate the function: $$f(x)=\sin(x);\ -π≤0≤π$$ I am using this paper by Antony Jameson to construct my code, with a few modifications to correct the ...
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318 views

Confusing myself with Cox de boor's algorithm

I am using the recursive definition to understand the algorithm, mainly: $$ B_{i,0}(t) = \begin{cases} 1 & \text{if} & t_i \leq t < t_{i+1} \\ 0 & \text{otherwise} \end{...
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86 views

Polynomial representation: Marsden's Identity.

Marsden's Identity states that: For every $\tau$ in $\mathbb{R }$: $(\cdot -\tau)^{k-1}=\sum_j\Psi_{j,k}B_{j,k,t}$ with: $\Psi_{j,k}=(t_j-\tau)\times...\times(t_{j+k-1}-\tau)$ Following de Boor'...
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135 views

Finding B-spline for space spanned by Multi-dimensional Spline.

It's known that a B-spline of degree $p$ , $B_j^p(x)$ is completely determined by a knot vector $(u_j,u_{j+1},...,u_{j+p+1})$. One could define it as: $B_j^p(x)=[u_j,u_{j+1},...,u_{j+p+1}](\cdot - x)^...
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Approximate a ($y²-x²=1$) hyperbola with line segments and elliptic ($a(x-x_0)²+b(y-y_0)²=1$) arcs

On some IT graphic systems, you have tools which draw line segments and circle or elipse arcs, but which do not draw parabolas or hyperbolas, and in many cases, those system keep track of graphical ...
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60 views

Adaptive knot selection for B-spline fitting.

When fitting a B-spline for regression purposes I've seen a lot of cases where knots are fixed uniformly ,but in some situations this could lead to poor estimations because the behaviour of the curve ...
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61 views

Interval Spanned by $(n+1)$ points in $\mathbb{R}$

Suppose that we have a real valued function $f(x)$ that has local support, i.e. it's non-zero just for some values of $x$. If you are familiar with B-splines, this function $f$ can be interpreted as a ...
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Fitting a spline: find coefficients using Fourier Transform?

I came up with a idea to estimate the coefficients of a B-spline fit by using the Fourier Transform but I don't know if it makes any sense to estimate them in this way. Given that $$s(x)=\sum_kc(k)\...
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1answer
57 views

What does this notation mean? Double arrow with $z$ above

I am reading a paper about digital filtering (for the very first time) and I found this notation (double arrow with $z$ above) which I do not quite understand. Could you please give me some hint? ...
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127 views

“fastest” curve through n points

I'm programming an AI for a race game, where my car has to drive through some checkpoints. If I let it drive straight in direction of the checkpoints, it has to slow down and make a huge turn after ...
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1answer
26 views

B splines recursion

Given that by definition the i-th B-spline of order k is: $$B_{i,k}=w_{i,k}B_{i,k-1}+(1-w_{i+1,k})B_{i+1,k-1}$$ where $w_{j,k}=\frac{x-t_j}{t_{j+k-1}-t_j}$ We can define the spline space as $$S_{k,...
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62 views

Monotonic and smooth interpolation between three points

The problem I have is the following: Given three $x,y$ points, $(-1, -y_{0})$, $(0,0)$ and $(1,y_{2})$, where $y_{0} \geq 0$ and $y_2 \geq 0$, I want to interpolate smoothly between them with a ...
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150 views

How to calculate smoothing spline coefficients

I am attempting to calculate smoothing spline coefficients based on the description in Reinsch's 1967 paper, but I'm having some trouble. The first derivative is not continuous. Here are the ...
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1answer
498 views

Convert continuous Bezier curve to B-Spline

Is there an algorithm/process for converting a sequence of bezier curves into a b-spline? I've found much discussion of the reverse, but nothing for this. I'm attempting to make a spline editor in ...
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257 views

Is a B-Spline always made up of Bezier curve segments?

According to what I have read, a B-Spline curve is made up of segments, with each segment controlled by 'k' control points (where k is the order of the curve). Also, a B-Spline curve can be formed by ...
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164 views

Convert two B-splines into one?

Let's say I have two second-order, non-periodic B-splines which touch at one endpoint (that is, I have an array of control points, weights, and knots for each). My task is: can you make a single ...
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Approximating the intersection of a line and the iPhone X screen as well as its normals

I am creating a simulation where little, fast moving, particles need to intersect with the edges of the iPhone X screen. Previously I have had no difficulty with my collisions. The particles have ...
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116 views

Convert multiple Bézier segments to a nurbs curve

I have multiple cubic Bézier curve segments which are contiguous and G1 (they are the result of the fitting of many curve samples). Now I would like to transform these Bézier segments into a single ...
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1answer
715 views

Interpolation method that does never overshoot

for implementing a system that will control hardware, I need an interpolation between points on a graph that does never overshoot. By overshooting I mean that between two points there may be no y-...
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729 views

Average radius of curvature of set of points on a road?

I have a set of points (lat, lng) on a serpentine road. What would be a good way to calculate average radius of curvature of the road? Also even without considering (lat, lng) points, just on a 2D ...
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202 views

Giving a point on a uniform bspline curve, how to get the tangent vector of this point

A bspline curve of order $k$ is given by $$C(t) = \sum_{i=0}^n P_i N_{i,k}(t).$$ where $P_i$ are the control points and $N_{i,k}(t)$ a basis function defined on a knot vector $$T = (t_0,t_1,...t_{n+k})...
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54 views

How to compare B Spline Surfaces?

We can define b spline surfaces over a set of points. However, is it possible to compare two set of b-spline surfaces fitted over different set of points ? The objective is to identify if the curves ...
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30 views

Iterated backward difference quotient from splines

I am working on a certain problem: say we have a function $f$ defined on an interval $[a,b]$ and we want the spline $Sf$ of order $k$ (and knots needed for the B-splines are in $[a,b]$) to agree to $f$...
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198 views

Minimization problem with latent function and splines

I have a dataset consisting of pairs $(x_i, y_i)$. I want to determine the function $f$, so $$ f(x)f(y) = 1 $$ with the constraint that $f(x) \leq x$, $f'(x) \geq 0$ and $f''(x) \geq 0$. I was ...
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87 views

How to get new temporary control points in deboors algorithm

I am trying to implement deboor's algorithm in c++ to make a b spline. I'm trying to follow the Wikipedia page of it (https://en.wikipedia.org/wiki/De_Boor%27s_algorithm). Here is the implementation ...
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31 views

“Swivel plane” tangent vector in Séquin circle spline

I'm trying to implement the circle spline scheme described here. I feel like I have a solid working understanding of the procedure, but I'm generally pretty rusty when it comes to geometry. A big ...
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1answer
2k views

The advantage of B-spline compared to Bézier if the number of control points is very small

If the number of control points is n+1, and the degree of the basis function is p If n = p, B-spline is as same as Bézier curve. Suppose I have a chance to increase the number of control points say ...
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54 views

Is it cheat to use spline to interpolate data from experiment or computational results.

Say we got data from the experiment or computational results. However, instead of using the line to connect data one by one. We use the spline to interpolate the data to make it look more smooth. ...
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1answer
111 views

Doubts on Oslo's Algorithm

I have a simple doubt on the Oslo's algorithm which is used for knot insertion in a knot vector for a B-spline. Here alpha = entries in Refinement Matrix Tau = original knot vector t = New knot ...
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Relation between spline and $C^d$ curve

Spline including NURBS, B-spline, etc can provide the $C^d$ continuous curve, which d is based on the degree of the spline. However, can all $C^{d-1}$ continuous curve be expressed by d degree Spline? ...
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442 views

What is the purpose of having repeated knots in a B spline?

A primer on the cpr package in R (page 2 of https://arxiv.org/pdf/1705.04756.pdf) writes the following about B-splines. A B-spline basis matrix is defined by a polynomial order $k$ and knot ...
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60 views

Extrapolation of points by a function

I have got a positive series convergent series $w_j$, which is convergent. I can calculate a finite amount of terms $\sum_{j=0}^{j^*}w_j$ and would be interested in the estimation of the sum of the ...
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2D subdivision surfaces - How to arrive at $B^1 = \large{\frac{A^0 +6B^0 +C^0}{8}}$?

In the video "Math and Movies (Animation at Pixar) - Numberphile", at around timecode 11:45, the following is said: Using the $1$, $2$, $1$ rule, you can see that $B^1 = \large{\frac{A^0 +6B^0 +C^...
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297 views

How to interpret the smoother matrix?

For example, in the smoothing spline problem, finally we get: $$\hat{y}=S_\lambda y$$ where $S_\lambda$ is the (linear) smoother matrix. But how to interpret it? Furthermore, ESL has a discussion ...
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38 views

Meaning of k-th divided differences.

The $k$-th divided difference of a function $g$ at sites $\tau_i,\ldots, \tau_{i+k}$ is the leading coefficient (that is, the coefficient of $x^k$) of the polynomial of order $k+1$ that agrees with $g$...
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226 views

NURBS: Advice for specifying the first derivative at the endpoints of a C2 cubic spline interpolation.

I am working through The NURBS Book by Piegl and Tiller. In the book they give an efficient algorithm for computing the traditional $C^2$ cubic spline for global interpolation of arbitrary data ...
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BSpline interpolation in Matlab

For a project in school we have to write a function in Matlab. [c] = spline_coeff(x,f) where • x vector of interpolation points; • f vector of function values ; • c vector of coefficienten from de ...
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373 views

Is there a formula that defines a bezier curve connecting 2 points with starting/ending directions?

Is there a way to generate a bezier curve that connects 2 points, with each point having a direction in which the bezier curve must start and stop with? For example, if point A is at (2, 2) and has a ...
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4answers
343 views

A polynomial parametric curve spanning known tangent end-points

Let $x(t)$ and $y(t)$ be unknown polynomials (of maximum order 3) defining a parametric curve $$ \mathbf{r}(t)=\begin{bmatrix}x(t)\\y(t)\end{bmatrix} $$ that fits known tangential end-points: $$ \...
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138 views

Software package for plotting 3-d splines

Given a finite point set $P \subset \mathbb{R}^2$ and a height function $h:P -> \mathbb{R}$ I want to produce a smooth surface that interpolates between the values $\left\{[p~h(p)]^\top \in \mathbb{...
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1answer
168 views

Find the natural cubic spline interpolant at the nodes?

Find the natural cubic spline interpolant to $f(x) = e^{x^2}$ at the nodes $\{x_i\}^2_{i=0} = \{−1, 0, 1\}$. Calculate the value of the interpolant at $x = 0.5$. What is the error at this point? S"(a) ...
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389 views

Line intersect spline surface

The intersection of a line with a bi-n surface is the solution of $A+Bt=Cu^nv^n+...$, where A, B, C, etc. are 3-vectors. The first step in solving this system is eliminating to get two bi-n equations ...
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77 views

Linear spline over a polynomial

The Question Suppose that $t$ is a polynomial of degree $n$ or less and satisfies $$t(x_0)=f_0 \; \; \; \; t(x_n)=f_n \; \; \; \; t(x_i)=0 \; \; , \; \; i=1,2,\dots,n-1$$ where $x_0<x_1<\dots&...
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2answers
154 views

What kind of function are cubic splines not good at approximating?

When we are given a dataset of a function sampled in many points, and we want to find approximation of the unknown function in-between these points, one of the go-to methods for practitioners is ...
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3answers
848 views

Finding a smooth path between points on a 2d map with maximum curvature

I have a set of points on a map (given by x,y coordinates) and I want to find a path between these points. The goal is to have a ship sail this path, so the path can't just be straight lines. I ...

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