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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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 ...
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1answer
33 views

Derivates of periodic parametric cubic splines

My Problem is sort of solved, I overlooked, that paameters $B$ to $D$ are dependent on $x$ and $y$ one question remains, see bottom of question. I implemented a periodic parametric cubic spline, and ...
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13 views

Interpolation with a constrained range between given control points

I am trying to create an algorithm that creates smooth color gradient functions, given control points in the red, green, and blue components. Mathematically, each curve would have a domain [0, 1] ...
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1answer
26 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)_{+}:= ...
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24 views

Reverse spline interpolation

Say I have a number of sets $(x, y)$ for $x \in \{0, 1, \dots, 255\}$. I want to find the least number of points to reproduce the set with a certain accuracy using linear interpolation. What is the ...
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1answer
16 views

minimum order of bspline curve for C2 continuity

Given a control polygon with five pairwise different points $d_0,...,d_4$ what is the minimum order of B-Spline curve for this polygon such that it is $C^2$ continuous ?
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1answer
24 views

Smooth curve between 2 points with given gradient at first point?

I'm trying to create a smooth curve between 2 given points with a given gradient/tangent at the first point and any gradient at the last. The idea being to be able to join these to create a smooth ...
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0answers
47 views

Polynomial parametrization of a quadric with two given points

Let $X^1, X^2 \in \mathbb{R}^3$ be two distinct points of the quadric surface defined by the implicit function $$ \phi(X)= X^T\cdot A\cdot X + b^T \cdot X+c=0, $$ where and A, b and c are unknowns. ...
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43 views

Question about continuity of a polynomial curve (Spline)

I'm getting a little bit confused trying to write my own algorithm for calculating a Spline. Let's start saying that for my application I need that the curve, interpolating between more points, must ...
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25 views

Tangent at middle vertex of 3 vertices

One of implementations that I found of Centripetal Catmull-Rom spline was using a simple, yet hard for me to grasp, equation to calculate a tangent at vertex (?). Below I state my understanding of the ...
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1answer
64 views

Looking for help for building a Spline's algorithm 10th order

I'm trying to code the following algorithm in C++ and need help to understand the build of Splines from a mathematical point of view (found on page 129 on this paper). $$ f(t) = \boldsymbol{t} \cdot ...
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1answer
35 views

Understanding normal and binormal of a vector or of a spline

I found a paper where it computes the 3D trajectory of a quadrotor and defines an error position as the difference between 2 vectors (here the source, under 3D trajectory control): $$ e_{p} = ...
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2answers
34 views

Spline approximation for $g(t) = \frac{t e^{-t}}{(x+t^2)^2}$

Is there any nice way to do a spline approximation for $$ g(t) = \frac{t e^{-t}}{(x+t^2)^2}\,, $$ where $x$ is some constant? I tried finding nice interpolation points, however this proved very ...
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2answers
38 views

how to find a spline function from given control points

consider having an n-amount of control points in 2D space, what's the best way to find the function passing through the start and end points while approximating the path according to the other given ...
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1answer
45 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 $O(h^2)$ ...
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1answer
60 views

Simpson's rule over cubic splines

I'm helping a friend of mine to do her homework, but i need help understanding some results (sorry but i took numeric methods class a looooong time ago) So, the task is to fit a cubic spline over ...
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1answer
61 views

Application for interpolating periodic B-spline

I need to draw a cubic C^2 continous, closed (periodic boundary conditions) B-spline which should interpolate a set of control points. If possible it would be great if I could specify the knot vector. ...
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1answer
51 views

Interpolation between two points

I am looking for an interpolation between two points $P$ and $Q$. I need the curve to have derivative (direction) $\vec{v_1}$ at point P and $\vec{v_2}$ at point Q. In addition, there is a maximum ...
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1answer
31 views

proving C2 continuity given closed spline constraints

Given the closed spline's constraints as below $$P(0) = P_k $$ $$P(1) = P_{k+1}$$ $$P''(0) = P_{k-1} - 2P_{k}+P_{k+1}$$ $$P''(1) = P_{k} - 2P_{k+1}+P_{k+2}$$ How do I prove that this spline ...
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48 views

Equation of smooth spline curve

This is a homework question a)Assume an equilateral triangle ABC of a side AB = a = 10.The coordinate of A is (5, 3).The slope of the segment AB is 2.This triangle controls a curve.This smooth ...
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3answers
76 views

Why do we choose cubic polynomials when we make a spline?

Good morning, I want to learn more about cubic splines but unfortunately my class goes pretty quickly and we really only get the high level overview of why they're important and why they work. To me ...
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24 views

MATLAB implementation Spline Fitting

Check the attached problem please. I am a beginner in spline fitting and have a few questions: 1) How to find the coefficients c[n]. Is it by DTFT? 2) I understand how to find the derivative but ...
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1answer
28 views

Determining unknown coefficients of cubic splines

The problem : Find $c$ in the following cubic spline. $S \scriptstyle{1}$$(x)$ = $\large4 - \large\frac{11}{4}x + \large\frac{3}{4}x^3$, on $[0,1]$ $S \scriptstyle{2}$$(x)$ = $\large2 - ...
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103 views

B-spline: compute control points given equations and knots?

Assuming a cubic or higher-order 2-D B-spline: if all piecewise polynomial equations for the final spline (and thus the knot vector as well) are already known, is there a relatively "streamlined" ...
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101 views

Normal of a Point on a Catmull Rom Spline

Im working on a program that makes use of catmull rom splines. I start with a list of points (ex. (50, 50), (75, 125), (200, 50) and (225, 300)) and then later calculate additional points for the ...
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192 views

One strange way to generate b-spline base

I am trying to use curve fitting under Matlab. There are two kinds of spline in Matlab: piecewise polynomials and b-spline. For b-spline, we know that the basic functions can be derived by means of a ...
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23 views

Is there any interactive spline fitting software?

I'd like to know if there's any software (freeware) for interactive data interpolation. What I want is to be able to visualize my data on an XY plot and drag the points to see how it affects the ...
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1answer
54 views

Bounding volume of catmull-rom splines

I need to compute a 2D "spherical" bounding volume for the part of a catmull-rom spline $S(t)$ with four control points $P1$, $P2$, $P3$ and $P4$ in the domain $0 \le t \le1$. The purpose is to reduce ...
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40 views

How can missing data be organised or classified (Interpolation vs Approximation)?

I'm looking for a way to distinguish between the various types of missing data techniques? Can someone help to clarify or organize these categories in sub-sections or indicate similarities or ...
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1answer
174 views

derivative B-spline with own knot set

Define the spline function of degree $q$ on the interval $[\xi_0,\xi_K]$ $$f(t)=\sum_{j=1}^{K+q}b_j B_j(t)$$ where $B_j$ are degree $q$ B-spline basis functions determined by the knots ...
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121 views

Deformable circle from a cubic Bezier approximation

I plan to draw approximate circles using a piecewise cubic Bezier representation. The representation should use four Beziers and be defined by four interpolating control points (let us call them ...
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39 views

Coordinate frames along the bounday of a minimal area (soap-film) surface

I would like to calculate coordinate frames along a closed Bezier (Or Catmull-Rom) spline. One axis should be tangential to the curve, and another axis normal to the minimal-area surface (soap-film ...
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1answer
73 views

Calculating a cubic spline goes wrong

I am trying to solve a old exam and really stuck at the cubic splines. We have the function $f(x) = \cos^2(\frac{x}{2})$ and the points $x_0 = \frac{\pi}{2}$, $x_1=0$ and $x_2 = \frac{\pi}{2}$. ...
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1answer
45 views

How to calculate intermediate points in spline?

I have two point $p_1$ and $p_2$. The velocity vectors are $v_1$ and $v_2$ respectively. The length of the velocity vectors are constant. I want to draw a path from $p_1$ to $p_2$ that enters $p_1$ ...
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1answer
49 views

Cubic spline interpolation realization

dear mathematicians! I was trying to code cubic spline interpolation algorythm. So I found one here. But I was confused. Let's see why. So let say I got 2 vectors - vector $X$ and $Y$ (with ...
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1answer
99 views

How to calculate the length of a cubic hermite spline between two points

I am using the following equation to create a cubic hermite spline: $$p_n(t) = a_nt^3+b_nt^2+c_nt+d_n$$ $$1\geq t\geq 0$$ $p_n(t)$ is the unit interval interpolation equation for dimension n. $t$ is ...
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72 views

How to calculate a bezier curve given derivative of endpoints, location of endpoints, and points on the curve?

I know how to calculate a hermite spline, which has known derivatives and locations for each point, and I know how to calculate bezier curves which go through certain points, but I need to be able to ...
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48 views

How can I generate a spline with a maximum second derivative without specifying first derivative for mid points?

I've done interpolation before with bezier splines and cubic splines, but I need to find a way to limit the second derivative throughout the curve so that there is a limit to how sharp the corner can ...
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79 views

NURBS surface fitting for a closed region on mesh

I'm developing a tool that allows users to select a closed boundary (a polygon) on the triangle mesh and then from this boundary, generate a NURBS surface fitting the original mesh surface. My idea ...
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27 views

Knot sequences in different basis

Given vector space $$ P_{3,1}^5[0,1,2,3,4,5] $$ I get the standard basis of V as: $$ \{ 1, t, t^2, t^3, (t-1)^2_+, (t-1)^3_+, (t-2)^2_+, (t-2)^3_+, (t-3)^2_+, (t-3)^3_+, (t-4)^2_+, (t-4)^3_+\} $$ ...
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1answer
42 views

Where did I go wrong: B-Spline recursion and B-Spline using determinants

For $ B^2_1(t) $ with knot values $ t_1 = 1, ..., t_4 = 4 $ Using the determinant method $ B^d_i(t) = (-1)^{d+1} (t_{i+d+1}-t_i) \frac1D A $ where D is the determinant of $\begin{bmatrix} 1 \ t_i \ ...
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1answer
279 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 ...
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1answer
36 views

Can you interpolate a function $f: \mathbb{R} \rightarrow \mathbb{R}^2$ piecewise (by two interpolations)?

I am currently trying to improve on-line handwriting recognition. On-line means in this case that I have the information how the symbols are written as a list of $n$ tuples of coordinates $(x(t_i), ...
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37 views

Approximate function algorithm using a polynomial and Boor splines

I have a defined function and a set of points with equal distance between them. The problem is that I have to approximate the graphic of that function using a polynomial function of 3rd degree and a ...
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1answer
60 views

Defined matrix in Catmull Spline Curve

I am trying to use Catmull spline curve in my program , I am trying to understand it but why we only use below given Matrix , because the examples I saw I only found the below one In Catmull spline ...
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58 views

What is a “control point”?

I'm trying to figure out a good definition of control point for use in wikipedia (see https://en.wikipedia.org/wiki/Control_point_(mathematics) ) There seems to be a bias towards ascribing a ...
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36 views

Thin-plate spline energy function value normalisation

I have an energy function as follows which produces a scalar denoting the thin-plate spline energy between point-sets $V$ and $X$. $E_{TPS}(d,w)=||Y-Vd-\Phi w||^2 + \lambda_1\text{trace}(w^T \Phi w) ...
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65 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 $d-1$, $(1<d<=n+1)$. ...
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Concept of knots in B-splines [duplicate]

Given $n+1$ control points, B-spline blending functions are polynomials of degree $d-1$, $(1<d\leq n+1)$. This much is easy to comprehend. Now comes the part I am not able to make any sense ...
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27 views

Schwarz–Christoffel-like mapping on differentiable simple cubic spline boundary

For a concept of a computer game I have in mind I came to need that. I have a 2D pond, which has a boundary that is a simple differentiable cubic spline. There are ducks floating around, looking at ...