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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non parametric bsplines

As you know bspline functions are parametric. I used chord length method to approximate 2 dimentional (x,y) data. I have a problem, as bspline functions are parametric (t) in nature, they need a ...
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+50

Compute Fourier coefficients of spline fit to data

Suppose you have data $$\{(x_i, -1^{i+1})\}_{1\dots N}, \quad x_1=0<x_i<x_{i+1}<x_N=2\pi \ \forall i \in\{2,\dots N-1\} $$ In other words, we have a sequence of $y=\pm1$ values at distinct ...
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Interpolating data points using a closed B-Spline with nonuniform knot vector

I want to create B-Splines which interpolate a set of given data points (knot points). The data points either describe a closed curve or an open & clamped curve. My main source is this website. ...
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2answers
106 views

How do I calculate this loop spline given the length, angle and horizontal offset?

I'm developing a formula to calculate a loop spline from a length, angle and horizontal offset. I can successfully calculate the loop from the first two parameters, but taking the horizontal offset ...
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11 views

B-spline for function with expanding support

In a nutshell my problem is as follows: I have some sampled data for a function $F$ and I want to estimate $F$ by fitting a tensor product B-spline. Additionally, I have a partial derivative ...
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Derivation of B-Spline basis function recursion formula

Can anyone explain the logic behind the derivation of the seemingly magical b-spline basis function recursion formula (deBoor-Cox) $N_{i,0}(u)=1 $ if $u_i\leq u < u_{i+1}$ otherwise, $=0$ $...
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1answer
16 views

How does using l1 norm or l2 norm for parametric spline affect the shape

When constructing a parametric cubic spline in three dimensions, I get three splines x(h), y(h) and z(h). When calculating the parameter h I would intuitively use the l2 norm between each successive ...
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23 views

B-Spline knot insertion - boundary conditions

I try to understand how to insert a knot in a given B-Spline curve (while keeping the same degree). The B-Spline curve is defined by the following data: degree $n$ control points $d_0, ..., d_m$ ...
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1answer
15 views

From Bezier Curve basis to B Spline basis functions

Bezier basis functions can be determined using recursion: $B_{i,p} = (1-t)B_{i,p-1}+tB_{i-1,p}$ So for a quadratic bezier basis, we get: $1-2t+t^2$ $2t-2t^2$ $t^2$ So for a quadratic bezier ...
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How to do Knot Selection For B-Spline Approximation

As you may know, for B-Spline approximation, we need to determine a knot sequence. what are the ways of finding a reasonable sequence to approximate points?
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24 views

Interpolating polynomial such that it is convex in specified region

The problem I have is that I have data at two points $x_1,x_2$ and $x_2>x_1>0$. At these two points, I know that the function $f$ has values $f(x_1)$ and $f(x_2)$ respectively. It is also ...
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21 views

Basis Function Algorithm, In The NURBS book

On page 74, Peigl explained an algorithm about computing a single basis function. first lines of this algorithm are handling some special cases. ...
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15 views

Computing a percent down on a bezier curve when a control points position is moved

I have a line segment drawn as a percent down on a bezier curve, lets say which has 3 control points. I need to calculate the new percent position of the line segment when one of the control point is ...
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55 views

General issue when adding shocks on curves made of splines

Let us assume I have a "nice" curve and that I would like to introduce a small shock up/down of about 1% at a certain point along the curve. I am trying to find out what the best and most efficient ...
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1answer
39 views

B-splines basis function

If the image domain is denoted as $\Omega_I=\{(x,y,z)| 0 \leq x<X,0\leq y<Y, 0 \leq z <Z \}$. Let $\Phi$ denote a $n_x \times n_y \times n_z$ mesh of control points points $\phi_{i,j,k}$ with ...
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1answer
56 views

Precision in Cubic spline interpolation

I am working on cubic spline interpolation with set of data points from CAD with following steps: Form piecewise spline equations between points. cubic equation : $ ax^3 + bx^2 +cx + d = P(x) $ ...
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41 views

Fitting a continuous curve over a piecewise constant data

I have some measurements that are piecewise constant over a certain variable. For example, in the following image, the vertical axis represents the measurement data and the variable is on the ...
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1answer
20 views

proving an inequality involving a linear spline / piecewise polynomial

I have $n+1$ sample points $x_i = \left(\frac{i}{n}\right)^4$ and want to approximate the function $f(x)=\sqrt{x}$ by a linear spline $f_n \in S^{1,0}(\mathcal{T_n})$ on the interval $[0,1]$. I know ...
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1answer
29 views

tangents at unit circle - parametrization leads to strange result

I'm thinking about the tangents at the unit circle in the upper right corner and I do so in terms of the parametrization $$ \begin{eqnarray} v:&[0,1]& \rightarrow \mathbb{R}^2\\ &t&...
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1answer
46 views

Optimal approximation of spline curve using linear interpolation

I have a parametric cubic spline which I need to draw in graphics. I am restricted to using a set of lines to draw this, and for performance reasons I need as few lines as possible. So I need an ...
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28 views

Support of a clamped B-spline

Let a B-spline of degree $p$ be defined by its parametric equation $$ \mathbf{r}(t) = \sum_{i=0}^n N_i^p(t)\mathbf{P}_i$$ where the $n+1$ control points are denoted by $\mathbf{P}_i$. The basis ...
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1answer
28 views

Knot sequence for a natural cubic (B-)spline interpolant

say I am given $n+1$ data points $(x_i,y_i)$ with $0\leq i \leq n$ and $x_0 < x_1 < \dots < x_n$. I want to interpolate these with a natural cubic spline $s(x)$ ($C_2$ continuous at knots ...
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17 views

Compact set of function involving BSpline functions.

Let $X = \left\{x_0,...,x_{n-1}\right\}$, $x_i-x_{i-1} = h$ for $i=1\ldots n-1$ and $$ \phi(x;X,\vec{\beta}) = \sum_{j=0}^n \beta_j \phi_{j,2}(x;X) $$ where $\phi_{j,2}$ is the $j$-th BSpline (of ...
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21 views

B spline - Partition of unity

I need the Laplace form of the second order B splines over logarithmically-spaced knot. I have read a paper in which it is mentioned that ; " Heaviside function can be represented as the sum of all ...
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1answer
40 views

Cubic spline interpolation results

I have a set of data points on which i am trying to do cubic spline interpolation. Below is the snapshot of the curve with the input data points marked in green color. And the red color marked point ...
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1answer
24 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 ...
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1answer
53 views

B-Spline approximation deviates a lot while increasing the number of control points???

I'm dealing with a problem to approximate some data points with B-Spline. I follow the method and implemented the algorithm from this site: Curve Global Approximation. 1) The first step is to ...
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1answer
66 views

Cubic Spline Interpolation

My problem is to find a interpolating cubic spline to the points $$\left\{(0,0), \left(\frac{\pi}{2}, 1\right), \left(\pi,0\right), \left(\frac{3\pi}{2}, -1\right),(2\pi,0)\right\}$$ I did as ...
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20 views

Curve Conversion

I have a curve that is kind of spline. It is defined as a sequence of polynomial segments. It is defined by order, knots, and coefficients of the polynomials. How can I transform or convert this ...
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Interpolate with smoothing parameter

I need to implement in C++ interpolation with smoothing parameter. To the non-familiar with this function: The smoothing parameter gets a value from 0 to 1. 0 brings absoulte linear interpolation (...
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Example of a “abrupt function”

I need example of a simple function to show that cubic spline gives better result than Lagrange's interpolation in case of some special functions. Thank you
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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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1answer
15 views

Finding the equation of a quadratic with 2 points and a known slope. (SPLINES)

Sketch the spline of degree 2 with value 0.5 at x = 2.5 and the values 1, 1, 0, 0 at t0, . . . , t3, respectively. (t0=0, t1=2, t2=3, and t3=5) What is the value of the spline at x = 1 and 4? What I ...
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1answer
35 views

Cubic Spline for a function

I have the function $f(x)=x^3$ and I need to find the cubic spline. The given points are: $\{-1, 0, 1\}$. What is the cubic spline for this function and what would a demonstration to this be? I would ...
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1answer
23 views

n-order B-splines interpolation

I am wondering if the following statements are correct: (1) zero-order B-splines interpolation is equivalent to nearest-neighbor interpolation. $C^0$ continuity thus is not differentiable. (2) first-...
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1answer
29 views

Deriving a tridiagonal system for cubic spline interpolation

Can anyone explain how $B_{i-1} = 1/4$ and $B_{i+1} = 1/4$ were chosen in line 6 of the picture, just above the matrix? I'm trying to understand cubic splines but this result seems like it came out ...
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1answer
19 views

B-splines locally controlled

I have read that in contrast of the thin-plate splines, B-splines are locally controlled, which makes them computationally efficient even for large number of control points. I didn't understand what ...
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43 views

Interpolation by splines: how to set up the equation system for finding the coefficients of the spline (in a B-spline basis)

Problem. I want to interpolate a function $f$ in some equidistant points $x_0<x_1<x_2<x_3<x_4$ using a quadratic spline. My attempt. I assume that we can use the interpolation points as ...
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1answer
38 views

Addition of two B-spline curves

Suppose I have two B-splines, both with the same degree, $p$, and uniformly distributed knots, but with different numbers of knots and control points. Is it possible to sum the two splines to obtain ...
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13 views

Enforce Spline Section Always Greater Than/Equal Zero

I'm attempting to do some cubic spline interpolation to make incomparable discrete datasets comparable by naively making them continuous. Its largely histogram based data with different bin thresholds ...
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1answer
9 views

Normalized vs Non Normalized Bernstein

I know what Bernstein polynomials themselves are, and am intimately familiar with one of their usage cases - Bezier curves. However, I recently came across someone mentioning of "Not Normalized ...
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25 views

Gaussian process regression from predictions

Can any body provide example of a gaussian process regression being used to generate confidence intervals?
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2answers
48 views

The quadratic spline

I'd like to fit the data in table as blow x f(x) 3.0 2.5 4.5 1.0 7.0 2.5 9.0 0.5 when $x=5$, I want to find value of $f(x)$ by using ...
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1answer
30 views

Bspline matrix form?

I understand how bezier curves can be expressed in matrix form: you have a matrix multiplied by a vector containing the power series of t, and also multiply be a vector containing the control points. ...
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Does this operator produce an interpolating spline of it's argument?

I'm studying a bit of spline functions theory: In this book, chapter 6, a lot of error bounds are given when spline functions are used to approximate functions belonging to a specific function spaces. ...
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Geometric Meaning of Modulus of Smoothness

Could you explain me the "geometric meaning" of the following definition (it's taken from a book on spline functions theory)? Definition: Given $1 \leq p \leq \infty$ a positive integer $r$, and $...
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3answers
77 views

Curve-fitting using circles

I'm working for a firm, who can only use straight lines and (parts of) circles. Now I would like to do the following: imagine a square of size $5\times5$. I would like to expand it with $2$ in the $x$...
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28 views

C2 continuous interpolation on a 4-dimensional dataset

I am currently coding up a project where interpolations must be performed such that C2 continuity be preserved along the length of the whole set. The end result ought to look like a line (which will ...
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1answer
30 views

Terminology: Spline interpolation

I have read two different definitions of Splines: A differentiable piecewise polynomial. A piecewise polynomial. If I build a piecewise polynomial using cubic polynomials, it's ...