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# 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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10 views

### How to build a cubic spline with boundaries?

I have a problem with a fit of a certain path. For a better understanding I created picture: Visualization of my problem In short: What I have is the yellow f(x), what I want is the blue g(x). ...
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### Difference between interpolation of Nd curve and multivariate interpolation

I am working on interpolating a track in 2 dimensions (x,y) as a function of time t. My question is can I simply perform two separate interpolations (x,t) and (y,t) and consider the the resulting (x,y)...
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### Spline interpolation of time-series respecting the zeros. Which technique is the best?

I'm dealing with time-series which consist of discrete measurements of a continuous signal: ...
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### What is the relationship between cubic B-splines and cubic splines?

What is the relationship between cubic B-splines and cubic splines?
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### clamped cubic spline in Matlab

The question is to use Matlab to find the clamped cubic spline $v(x)$ that interpolates a function f(x) that satisfies: $f(0)=0, f(1)=0.5, f(2)=2, f(3)=1.5, f'(0)=0.2, f'(3)=-1$ and then plot $v(x)$. ...
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### Approximation subject to derivative constraints, textbook reference?

I'm looking for a reference (preferably a textbook treatment) that can help me answer the following question. Suppose $\mathcal{F}$ is a space of functions on a subset of $\mathbb{R}^d$ with ...
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### function for which natural cubic spline and spline 'without node' are the same thing

If we're talking about cubic splines, is there a function (are there functions) for which natural spline and spline 'without node' are the same thing? If yes, please give me an example (examples), if ...
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### Difference in DOF from a cubic spline to a natural cubic spline?

enter image description here I am curious as to what the answer on this question is? My thought process is that the answer should be 4 DOF for the new natural cubic spline model. Below is how I ...
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### Find continuous curvature approximation for going from a straight line into a half circle

I have the problem that I want to steer a 4 wheel robot. It should move on a straight line, and for turning to go on a circle with a defined radius, as in this picture: The problem I have is that if ...
25 views

### Understanding Global Curve Approximation

Foreword StackOverflow referred me here, because my questions seems to be more related to math, than to algorithms. You can see the original question here: https://stackoverflow.com/questions/...
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### Does not-a-knot cubic spline interpolation always have a tridiagonal system of equations when solving for the second derivatives?

I've been looking at the question Not-a-knot cubic spline interpolation using tridiagonal solver, for much the same reasons as the original asker - to figure out how to implement not-a-knot spline ...
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### Bibliography for cubic B-spline expansion to approximate functions

I'm currently working on obtaining the deuteron wavefunction with a specific potential (OPEP) and it involves solving a system of two coupled second order differential equations. I was told to use ...
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### Minimizing cost function of curvature for quality control of weather observations

My question pertains to the implementation of the quality control of spatially distributed weather data in the following study: https://journals.ametsoc.org/doi/full/10.1175/MWR-D-10-05024.1. I will ...
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### Determining closest (smoothed) point on a spline

I have a spline defined by a list of Vector3. I have an object that roughly follows the path defined by the spline [but drifts to either side], and I have a goal that precedes the object by n points. ...
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### Evaluating basis splines at the knots

Given an order $k$ and a set of knots $\{t_m\}$, we can define the basis splines as done here, where $B_{i,k}$ denotes the $i^{th}$ basis spline of order $k$. I have written a simple script that ...
69 views

### Normal of Catmull-Rom Spline

I've implemented Catmull-Rom in python and now I want to be able to get the normal of points so I'm first finding the tangent, to look something like this. For Catmull Rom I have 4 points P0, P1, P2 ...
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### How to create a Cubic Hermite Spline interpolation equation?

I am developing a program for which I need to smoothly interpolate between some control points. Here is an example of what I need. Ie. In this gif, the knob is crossfading between no interpolation at ...
529 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 ...
1k views

### What's the best way to calculate all of the points for a curve given only a few points?

I've been reading up on curves, polynomials, splines, knots, etc., and I could definitely use some help. (I'm writing open source code, if that makes a difference.) Given two end points and any ...
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### How do you come up with the system of equations to determine the coefficients of the natural cubic splines passing through a given set of points?

Not sure if I'm doing it right, but these are the equations I came up with so far.
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### Parametric vs non-parametric transformations

I am currently concerning myself with transformations (mainly of images). However, I am not quite clear on the difference between parametric and non-parametric transformations. From my current ...
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### Natural linear spline function representation

Let $s: \mathbb{R} \to \mathbb{R}$ be a natural spline function of degree one (that is it is piecewise a polynomial of degree at most 1) and let $x_0 \leq x_1 \leq ...\leq x_n$. Show that $s$ can be ...
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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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### Confusion on the formula of Rational Bezier curve in Duncan's book

Duncan Marsh's book Applied Geometry for Computer Graphics and CAD stated I'm really confused about this formula. First of all, why do we replace $w_i\mathbf{b_i}$ with $\mathbf{b_i}$ when $w_i=0$? ...
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### Proof of equivalence between integral cost of homogenous form of NURBS curve and it's projection

The context of my problem: I'm trying to implement an optimization problem in MATLAB using YALMIP. The objective function computes the integral cost of the squared norm of the derivative of my NURBS-...
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### How to scale a catmull-rom spline

I have a catmull-rom spline represented by its end vertices plus control points. I would like to be able to scale it so that it can be resized/stretched/squeezed in a way that looks linear. However, ...
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### B-Spline basis function partition of unity

I'm struggling to prove the partition of unity property for B-spline basis functions, which states that $$\sum_{i=r-p}^rN_{i,p}(\xi)=1$$, $\xi\in[\xi_r,\xi{r+1})$, where $p$ is the degree. I know that ...
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### straight line intersection with centripetal catmull-rom: writing spline in cubic form

I'm a biologist rather than a mathematician so apologies if my definitions might be a bit off. I'm trying to find the intersection of a straight line (the normal of one of my catmull-rom curves) with ...
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### How to solve this 4x4 equation system for a cubic spline?

I am attempting to create a simple cubic spline between these lines: I have worked out the four equations as: (1) $1 = An^3 + Bn^2 + Cn + D$ (2) $g^{m-t} = At^3 + Bt^2 + Ct + D$ (3) \$0 = 3An^2 + 2Bn ...