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
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2answers
17k views

What is the relationship between cubic B-splines and cubic splines?

What is the relationship between cubic B-splines and cubic splines?
12
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1answer
10k views

Implementation of Monotone Cubic Interpolation

I'm in need to implement Monotone Cubic Interpolation for interpolate a sequence of points. The information I have about the points are x,y and timestamp. I'm much more an IT guy rather than a ...
12
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2answers
31k views

How to calculate interpolating splines in 3D space?

I'm trying to model a smooth path between several control points in three dimensions, the problem is that there doesn't appear to be an explanation on how to use splines to achieve this. Are splines a ...
13
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10answers
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.
2
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2answers
360 views

Finding a simple spline-like interpolating function

I am looking for a continuous function $y=f(x,\alpha)$ for the interval $0\le x \le 1$ such that $0\le y \le 1$ and $y(0,\alpha)=0$ and $y(1,\alpha) = 1$ and $y(\alpha,\alpha) = 1-\alpha$ and $dy/dx|_{...
2
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1answer
2k views

How to construct a parametric cubic B spline?

If I am given n+1 control point Pi(xi,yi), Po .... Pn , how do I construct a parametric relationship to draw a curve ? From what I understand , a parametric relationship is that you can express x and ...
9
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2answers
9k views

Convert a B-Spline into Bezier curves

I have a B-Spline curve. I have all the knots, and the x,y coordinates of the Control Points. I need to convert the B-Spline curve into Bezier curves. My end goal is to be able to draw the shape on ...
7
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1answer
1k views

Bezier curvature extrema

For a planar cubic Bezier curve $B (x(t),y(t))$, I would like to find the values of parameter $t$ where the curvature (or curvature radius) is greatest/smallest. The formula for curvature is: $$r = \...
9
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1answer
5k views

Natural cubic splines vs. Piecewise Hermite Splines

Recently, I was reading about a "Natural Piecewise Hermite Spline" in Game Programming Gems 5 (under the Spline-Based Time Control for Animation). This particular spline is used for generating a C2 ...
6
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3answers
3k views

Natural cubic spline interpolation error estimate

I am looking for an error estimation for natural (one with $s''(a) = s''(b) = 0$ boundary conditions) cubic spline interpolation on an evenly spaced grid. The best result I've found was $O(h^2)$ ...
3
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3answers
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 ...
2
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1answer
3k views

Is it possible to convert a B-Spline into a Bezier curve?

If so, do I lose any feature of the curve?
1
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1answer
700 views

Not-a-knot cubic spline interpolation using tridiagonal solver

I am trying to write my own cubic spline interpolant. Given the formula for the cubic spline $$S_n(x) = a_n+b_n(x-x_n)+c_n(x-x_n)^2+d_n(x-x_n)^3$$ my interpolant works perfectly for the natural ...
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0answers
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 ...
23
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11answers
6k views

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 ...
6
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2answers
9k views

Spline interpolation versus polynomial interpolation

What is the difference, if any, between spline interpolation and piecewise polynomial interpolation?
5
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1answer
1k views

What is the difference between cubic splines and cubic b-splines? [duplicate]

I am dealing with a numerical problem with cubic spline, but I am a little bit confused while using them because of terms spline and b-spline. In simple words, what is the difference between the ...
10
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0answers
1k views

What is the maximum overshoot of interpolating splines in $d$ dimensions?

Consider cubic splines $s( x, y )$ which interpolate values $y = \{ y_0, y_1, \dots,y_n \}$, on the uniform grid $\{ 0, 1,\dots, n \}$. Fix $s''(0) = s''(n) = 0$ (natural splines). How big can $$\...
4
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3answers
1k 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 $x$...
3
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2answers
4k views

How can I calculate the derivative of a Catmull-Rom spline with nonuniform parameterization?

Allow me to preface this by saying I am not a trained mathematician in any sense, so it's entirely possible I'm missing something rather fundamental. That said, I'm trying to take the derivative of a ...
2
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1answer
2k views

Is “Partition of Unity” a property of B-spline bases

Several sites about B-spline bases states that those have the "Partition of Unity"-property. Does that mean that the sum of the bases of a specific degree should be 1? If the knot vector is {0,1,2}, ...
5
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2answers
972 views

Basic B-Spline basis function question

I am studying the basic recursion formula for generating B-Spline basis functions N(i,j) of a given degree from the basis for the lower degree, and puzzling at the magic. In particular what I am ...
5
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2answers
8k views

Cubic B-Spline interpolation

The equation for B-spline with control points $(P_0, P_1,\dots,P_n)$ is \begin{equation} P(t)=\sum_{i=0}^n B_{i,k}(t)P_i \end{equation} If I have the following knots: $1,2,3,4$ and the following ...
4
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2answers
2k views

Approximating a large number of data points using (cubic) splines in l1/l2 norm.

I have a pretty large dataset ($x,y$) consisting of a few million points. There is a lot of noise in the data. I want to find a smooth but simple approximation/representation for this dataset, so that ...
4
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2answers
3k views

Is there a cubic spline interpolation with minimal curvature?

I came across the term "cubic spline with minimal curvature". However, I am not able to find any documentations/explaination on its computation method. Can anyone help me by advising how I can go ...
3
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1answer
1k views

How to find B-Spline represenation of an Akima spline?

Given points $t_i$ and values $y_i$, I'd like to use Akima interpolation to interpolate to a different set of locations $x_j$. This means I need to calculate the cubic polynomials $A_{3,t}(x)$. Given ...
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1answer
5k views

Cubic Spline Interpolation practice

Going over practice problems for our final exam. I'm stuck on a problem involving cubic splines. In fact, I don't even know where to begin. I need to find the natural cubic spline $S(t)$ at $t_0=0, ...
1
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2answers
14k views

Linear Spline Interpolation

Can someone explain to me how linear splines work and what formulas are used. I can only seem to find information on cubic splines. Which I don't really understand either Specifically, if I were ...
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2answers
1k views

Physical significance of knot vector in B-spline.

A B-spline blending curve formulation is: $P(u)=\sum_{k=0}^np_k B_{k,d}(u)$ Given $n+1$ control points, B-spline blending functions are polynomials of degree $d-1$, $(1<d<=n+1)$. ...
3
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4answers
2k views

Program to find closest function to fit arbitrary data

I've wanted this for years, but have never come across anything; a program for Windows to find the closest function to fit arbitrary data. The data I feed it is simple: A table with two columns ...
3
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1answer
876 views

How can I fit a spline curve through points in the plane with constraints on the curvature?

If I have a number of points in the plane (say I have some points $(x_n, y_n),\ n = 0,1,\ldots, N$) and I would like to fit a parametrised curve through them: $(x(t), y(t))$ where $t$ is some ...
2
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2answers
3k views

How to calculate a spline for points in general position?

I want to find a curve passing through (or near) $n$ points in the plane. The catch is that the curve need not be a function. That is, a vertical line might pass through the curve in more than one ...
2
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1answer
822 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,...
2
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1answer
337 views

Spline interpolation degrees of freedom

When using cubic spline interpolation, we have to solve $n-1$ equations with $n+2$ unknowns. What we can do is set $z_0 = z_n = 0$, which gives the natural cubic spline. But could we also set some ...
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2answers
1k views

spline derivation

Assume the following representation for cubic splines with $T$ interior knots is given. Let $g(Y)=\sum_{j=0}^3 \alpha_j Y_j+\sum_{t=1}^T \gamma_t (Y-\zeta_t)_{+}^{3}$ where $(Y-\zeta_t)_{+}:= max\{0,...
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2answers
44 views

How to create line with 6th order spline?

I am dealing with spline interpolation and what I do is basically interpolating $6$th order ($7$ control points) spline through some discrete points. Curve-based part of my algorithm is done, however, ...