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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Linear splines approximation error
i have some problems with the following task:
Let $f\in C^2([a,b])$ and s the interpolating linear spline of f with the grid points $x_i=x_0+ih, i=0,...,n$ and $h=\frac{1}{n}(b-a)$. Proof for every ...
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write cubic spline as a sum of the local splines times interpolation values
Not sure if this is the right place to ask. If so, please let me know.
I am performing cubic spline interpolation to produce a globally $C^2([a,b])$ function $S(x)$ on the one-dimensional interval $[a,...
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Construction of a curved grid in 2D space, based on a cubic spline interpolation
Please bear with me, this is my first question here and I hope it fits and is understandable.
I want to construct a "curved" grid on the screen in a computer program.
Visualization of the ...
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Is it possible to have a quadratic spline and a cubic spline meeting and being C2 continuous at a point K?
I'm trying to interpolate some datapoints. Ideally I would like to have 3 splines: The first and third being quadratic, and the second one (in the middle) cubic. Is it possible to mix polynomials and ...
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Frequency domain spline approximation and B-Spline inverse transforms
In short, I would like pointers to closed-form formulas, or efficient algorithms for computing the inverse of ether sine, cosine, or Hartley transforms of the B-Spline basis. The motivation, as ...
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Intuitive understanding behind clamping of B-splines
I have been going through the implementation of B-splines, and I observe that whenever it comes to clamping, it's usually mentioned that we must repeat the end knots $p+1$ times for a spline of degree ...
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Thin Plate Spline RBF Interpolation understanding
I have been looking at radial basis function interpolation: $f(x) = \sum w_i \phi_i(||x-x_i||)$ and examining the different kernels e.g. $\phi(r) = e^{-(\epsilon r) ^2}$ which are generally maximised ...
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Is the Integral of a Spline multiplied with the Exponential Function of the Spline solvable?
I want to compute the integral, I have borders for $x$, however I would like a function of y as the result:
$$
\int_a^b \quad \frac{\partial f(y,x)}{\partial y} \exp(f(y,x)) \quad dx
$$
$f(y,x)$ is a ...
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Is it possible to rewrite the B-Spline Basis functions nonrecusively
B-Splines have a recursively-defined basis function $N_{i,j}(t)$ as shown here:
$$
\newcommand{\defeq}{:=}
N_{i,j}(t)\defeq\frac{t-t_i}{t_{i+j}-t_i}N_{i,j-1}(t)\frac{t_{i+j+1}-t}{t_{i+j+1}-t_{i+1}}N_{...
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Define position of an object as a function of time such that object travels in fancy ways to a given point within a given duration
For context, I am trying to write animation software and, in doing so, have encountered this problem. I ask that you not be too critical of the setup I have going on because I have no background on ...
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Is it possible to construct a SQUAD quaternion spline in a piecewise fashion?
If I have a cubic Bezier spline that interpolates the points $p_1 ... p_n$, I can extend it to another point $p_{n+1}$ by just adding a new segment with the first two control points determined by the ...
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How to match an ellipse with a b-spline curve with C2 continuity?
I want to describe a Turbine Airfoil using B-Splines. For the leading edge (blue) of the airfoil I'm using an ellipse arc which is then followed by b-spline curve which describes the airfoil body (...
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To find a interpolating cubic spline to the points
My problem is to find a interpolating cubic spline to the points
$$\left\{(-2022,8043), (-4, 1989), (-2,1983), (0, 1977), (1, 1974), (3, 1968),(2022,-4089)\right\}$$
I know I can use a formula here or ...
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choosing between Multivariate, Univariate and Spline interpolation
I have dataset of points with coordinates and temperature measured at each point.
I would like to interpolate the points to generate a continuous image.
I have checked Scipy and I have seen that there ...
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Best approximation with unknown basis functions
Given an analytic real known function such as $f(x)=\exp(-x^2)$ defined on a finite interval such as $x \in [0,1]$, find the solution of the following problem
$$\min_{c_k,\phi_k(x)} \int_a^b|e(x)|^2 ...
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One step beyond cubic spline interpolation, a fourth-order problem?
I am trying to fit a polynomial through three points, where I also know the derivatives at the two endpoints. I don't need a truly general solution. My specific problem is constrained as follows:
<...
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Linear fitting of values on non-uniformly parametrized B-spline surface
I have a 2d (u,v) surface in 3d space (x,y,z) that is defined as a b-spline surface. The surface is not arc-length parametrized.
Additionally, I have scalar values defined on the surface at given u,v ...
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Whittaker smoother 2D, $C^0$ continuity
Suppose the 2D version of the Whittaker smoothing spline
https://eigenvector.com/wp-content/uploads/2020/01/WhittakerSmoother.pdf
minimizing
$$
\phi=(X-X_{s})^{T}(X-X_{s})^{T}+(Y-Y_{s})^{T}(Y-Y_{s})+...
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Are B-spline basis functions also B-splines?
When you write a spline curve as a linear combination of b-spline basis functions, it's called a "b-spline".
The basis functions are generated recursively by the deBoor-Cox algorithm, ...
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What exactly is the basis Function for B-splines?
I am using splines for 1D model in a research project. I found a reference for how to write the basis function that I put into my code but can no longer find it. Most guides and references on splines ...
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integrate hat function
Let $\{0=x_0<\dots <x_n=1\}$ a decomposition of the intervall $[0,1]$ and $h_i:=x_i-x_{i-1}$. Furthermore, define
$\phi_i (x) = \left\{
\begin{array}{ll}
\frac{x-x_{i-1}}{h_i} & x \in [x_{i-...
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Creating higher order cardinal B-Spline basis functions
I am currently trying to grasp how B-Spline functions work and seem to have hit an issue with its definition.
One type of B-Spline basis function seems to be a cardinal B-Spline function, which I ...
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What is a general form cubic-spline approximation of an arc?
Say I have an arc from $P_0$ to $P_3$, that has unit tangents $t_0$ and $t_3$, respectively. Let $L$ denote the distance between $A$ and $B$.
If $P_0 = (0, 1)$, $P_3 = (1, 0)$, $t_0 = (1, 0)$ and $t_3 ...
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Interpolation: cubic splines
Cubic splines
Given $n+1$ data points $\left(x_i, f_i\right)$ such that: $x_i<x_{i+1}$
We want a function $y(x)$ such that this function $f(x)$ interpolates continuously (up to and including the ...
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Proof clamped nurbs curve is tangent to first and last legs of its control points
I want to prove that $\mathbf{C}$ a quadratic NURBS curve is tangent to the first and last legs of its control polygon.
Given three control points $\mathbf{P}_0$, $\mathbf{P}_1$, $\mathbf{P}_2$ with ...
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Where are the control points for a non-parametric b-spline?
There are many wonderful illustrations of parametric curves with b-splines and control points, but I mostly use splines for uni-variate functions. So I have a vector of control values, but not really ...
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An elegant way to verify the convexity of a spline with certain boundary conditions
It is known that, given constants $a, b$ with $a \geq 1$ and $-1 < b < 0$, one can construct on the interval $[0, 1]$ a cubic spline
$$ g(x) = c_{0} + c_{1}(x-1) + c_{2}(x-1)^{2} + c_{3}(x-1)^{3}...
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What is the size of the basis for cubic B - splines
I have an exam questions which says we have mortality data observed between ages 20 and 60 with knots at 20, 30, 40, 50 and 60. Then the questions say we will use a set of 7 basis splines to fit a ...
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Finding polynomial degree $1$ of $f(x)=\text{erf}(x-1)$ at $x=1$
Question: Find the polynomial of degree $1$ that has the highest possible order of contact with $f(x)=\text{erf}(x-1)$ at $x=1$. Plot the spline knotted at $(1,0)$ with $f(x)$ on the right and your ...
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Non-constant cubic spline interval
I have the following Numerical Methods homework question that I am working on.
Question: The velocity profile of a rocket against time is give as below
\begin{array}{c|ccccc}
\bf t & 0& 10&...
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Convert B-Spline into Bezier surfaces
There is one topic regarding converting B-Spline curves into Bezier curves. (Given knots and control points),
Convert a B-Spline into Bezier curves
I wonder if there is any way to convert(or partition)...
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How to avoid Cusps and Corners when generating spline?
I already asked this in robotics.stackexchange.com, but no reply. So I've decided to ask here
Here I wrote Qubic spline trajectory generation. The following are the result.
Spline that generated
...
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Matrix spline simplification
On the APIC paper page 6 under equation 11 the paper states that the spline weights for quadratic and cubic splines simplify for their formulation of $D$ such that:
$$D = \frac{1}{4}\Delta x I$$
$$D = ...
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best high order interpolation method for set of points
what is the best way to interpolate a set of points in 2d, such that there is only one parameter to indicate position on the curve (like is the case for a Bézier curve)?
one thing I know is that we ...
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Algorithm to transform a polyline into an equidistant polyline
I have a number of points in 3D space. These points represent the positions (the toolpath) for an industrial robot or a CNC machine. The points are calculated by a software program. There can be up to ...
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Choosing optimal sample points for cubic spline approximation of sinusoid with polynomial argument
My problem consists as follows. Say I have a sinusoidal function whose argument is a cubic polynomial:
$$f(t) = \sin(at^3+bt^2+ct+d) \quad a, b, c, d\in \mathbb{R}, \quad t\in[T_1, T_2]$$
I want to ...
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Showing that that a curve of degree $d$ by an affine combinations can also be expressed in terms of Lagrange polynomials of degree d
Im working with splines and interpolation and bit stuck on showing the following:
Given $d+1$ points $(c_i)^d_{i=0}$ and parameters $(t_i)^d_{i=0}$ I have worked with and constructed the curve $q_{0,d}...
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What is the big O notation of the cubic spline algorithm?
I'm trying to do a time complexity analysis of MATLAB's cubic spline algorithm. I have a 1000 x 1000 table and I want to know what is being done when I query a point that is in between two quantities ...
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RKHS of B-spline kernel
I am doing a homework on Kernel methods. In the first question, B-spline functions are introduced as iterated convolutions of the rectangle function $I$:
\begin{align*}
I(x) = 1& \text{ if } x \in ...
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uniform approximation of lower order B spline with higher order B spline
Suppose I have a compact domain $\Omega$ and a given function $f(x) = \sum_{k=1}^Kw_iB_{k,p}(x)$ where $\{B_{k,p}(x)\}_{k=1}^K$ is a set of $p$th degree B-spline basis ($p$ is some general non-...
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Rotation interpolation knowing values at specific points
Let's say we have an arc-length parametrized curve $\mathcal{C}(s)\in\mathbb{R}^{3}$.
We want to find a frame $\mathcal{R}(s)=\left[\begin{array}{ccc}
\hat{\mathbf{t}} & \hat{\mathbf{b}} & \...
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Approximating an arc with NURBS
I am trying to find a way to approximate an arbitrary arc (given by an angle phi and a radius) with a degree 3 NURBS (ie at least 4 control points).
This website: https://www.ibiblio.org/e-notes/...
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Smoothing a series of points in 3D space where each point has a high and low threshold
Say I have a series of points in 3D space where the dimensions are strike, time, and price. I'm describing option chain data. At each point there is a bid and an ask. The bid is the lower price and ...
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Bounding the error of Ellipse approximation by a cubic Spline
So situation is like this. I have an Ellipse $(a cos(t), bsin(t))$. My final goal was to approximate it by a cubic spline it such a way that modulus of velocity along a curve is approximately a ...
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How do I multiply two BSPlines curves?
Question: How do I compute the knot-vector and control points of $C_3$ such $C_3(u) = C_1(u) \cdot C_2(u) \ \ \ \forall \ u$?
Description:
Let $C_1(u)$ and $C_2(u)$ be two BSpline scalar from $\left[...
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Find the fixed points of a pair of related equations like $c_{n,x+1} = \frac{d_{n-1, x}}{4} + P_{n} -\frac{c_{n+1, x}}{4}$
Problem:
$P$, $c$, and $d$ are points (A vertex and its two tangents for a bezier spline). $n$ is the index of the point in the spline, and is infinite. $x$ is the iteration number. What happens when ...
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Closed formula for the integral of BSpline Base Function?
Let $N_{ij}(u)$ be a BSpline function of degree $j$:
$$
N_{i0}(u) = \begin{cases}1 \ \ \ \ \text{if} \ u \in \left[u_i, \ u_{i+1} \right) \\ 0 \ \ \ \ \text{else}\end{cases}
$$
$$
N_{ij}(u) = \dfrac{u-...
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Proving that Bernstein polynomials are basic B-splines
I want to prove that if we use the knot vector $t_0=\dots=t_n=0, t_{n+1}=\dots =t_{2n+1}=1$ then $N^n_i=B^n_i$ on $[0,1)$. I have the following definitions:
$N^0_i=1$ on $[t_i,t_{i+1})$, $0$ ...
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General continuous piecewise polynomials solving linear recurrences
While reading The spline interpolation of sequences satisfying a linear recurrence relation, it is proved a necessary and sufficient condition for the existence of piecewise polynomial which satisfies ...
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What is the Quadratic trigonometric Nu Spline basis functions?
can someone help me identify what is exactly the quadratic trigonometric Nu spline basis function based on an article "A Quadratic Trigonometric Nu Spline with Shape Control"
here is the doi ...
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