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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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
28 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
30 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
44 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
46 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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25 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
28 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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39 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
64 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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18 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
25 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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1answer
79 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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1answer
74 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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1answer
111 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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21 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
49 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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1answer
39 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
128 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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2answers
102 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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0answers
38 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
69 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
43 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
46 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
82 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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59 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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45 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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74 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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26 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
40 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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215 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
35 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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0answers
36 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
54 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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3answers
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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0answers
33 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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2answers
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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0answers
19 views

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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26 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 ...
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1answer
117 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}, ...
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0answers
38 views

cubic B-spline interpolation function

I read that the B-spline basis functions are the follows: $B_0(x)=(1-x)^3/6$ $B_1(x)=(3x^3-6x^2+4)/6$ $B_2(x)=(-3x^3+3x^2+3x+1)/6$ $B_3(x)=x^3/6$ The cubic b-spline interpolation function it ...
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12 views

Interpolating on the borders of differently-resolved images

I'm creating a three-dimensional model of the earth based on SRTM height data. The data set is pretty huge, so only a small fraction of the data is available at any given time. The height data is ...
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1answer
90 views

What are some alternative ways of describing n-dimensional surfaces using control points other than Bezier surfaces?

I'm interested in problems involving geometric constraints and curve subdivision. I noticed that most of these problems describe the curves/surfaces using the Bezier form. I wanted to know if there ...
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2answers
416 views

Example of cubic B-spline

I have consider a 1-D problem to figure out how B-splines work. I assume that I want to interpolate on x-values 1,2,3 and 4. I gave random values to my control points, namely w1=0, w2=-1, w3=3 and ...
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1answer
27 views

Increasing length of closed spline (scaling)

I have a 2D closed spline and I need to increase its total length by a factor k, without changing its curvature, basically scaling. If this spline was a circle, I ...
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2answers
1k 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 ...
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2answers
327 views

Relation of cubic B-splines with cubic splines

Does anyone know the relation between the cubic B-splines and cubic splines?
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1answer
73 views

Cubic B spline drawing shape

When using a cubic B spline with n+1 control points, which shape is difficult to draw / modify ? Please give an example.
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1answer
148 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 ...
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1answer
333 views

How to test if your cubic spline natural boundary interpolate correctly?

If I have constructed a cubic spline, how can you tell if your spline did the approximation correctly ? In my thinking, I will use my cubic spline to interpolate a 3rd degree polynomial function f3, ...
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2answers
140 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 ...