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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28 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
57 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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17 views

Maple Cubic Spline Interpolation [closed]

Need help on this one... Using the strategy of the second derivative, solve the interpolation problem with cubic spline for a problem of $n+1$ points. In order to do this, two Maple procedures will ...
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1answer
31 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
28 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
19 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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16 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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24 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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24 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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19 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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32 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
34 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
34 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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23 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
36 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
55 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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29 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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50 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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18 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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21 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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58 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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30 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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10 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
86 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
236 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
23 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
250 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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205 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
67 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
87 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
173 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
69 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 ...
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1answer
24 views

What does the vertical bar means in this B-Spline evaluation formula

I have a homework with the problem set like the following: When a B-spline is evaluated at one of its knots it can be simplified according to the formula $B(t_i | t_j,...,t_{j+1+p}) = B(t_i | ...
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25 views

Possibility of 3D interpolation without decouple axis

I am wondering if it is possible to do 3d spline interpolation without decoupling the axis. Such as creating a spline function on x then a different one on y and another on z. Then for any given ...
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1answer
44 views

Trajectory spline 3D illusion

While playing around with 2D quadratic splines of a trajectory, I sometimes perceive the resulting curve rotating "in 3D" when changing the parameters. Here is a crude GIF example: And another ...
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2answers
102 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
194 views

Interpolating between points in 3D

I want to interpolate a spline of some sort between several points in three dimensions, but I have some very specific requirements. I must know the length of each spline segment between two points I ...
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1answer
17 views

What's the detailed expression

Thank you in advance. I am not sure if B-spline/NURBS can express as basic function in matrix, as, $$ x(t)=B(t)c $$ $$ B(t)=[b_1(t)...b_M(t)] $$ in which x(t) is a Dx1 states, B(t) is known temporal ...
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1answer
24 views

Using the TPS to interpolate between points

I have implemented the Thin Plate Spline interpolation. I successfully plotted surfaces which go through all my 3D data points. What I want to do is calculate a few points which lie between my known ...
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1answer
55 views

Determining the surface with given polynomial borders

Let's say we'd like to guess the shape (I'm not sure the word 'approximate' is appropriate here) of some surface when we are given its borders via third order polynomials (i.e. we are given their ...
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1answer
50 views

Showing that a thin-plate spline RBF approximation is real analytic

I am finishing my Ph.D. dissertation in engineering and I would like to show a simple proof. I am having troubles formalizing my ideas into a proof though. I think in a mathematics paper this concept ...
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2answers
253 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 ...
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1answer
454 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, ...
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1answer
107 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 ...
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2answers
2k 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 ...
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1answer
189 views

How do I interpolate the derivative of a catmull-rom spline?

I am creating an implementation of a cubic hermite spline in Python. One feature I would like to add is a method to compute the slope (IE the derivative) for a given T value. Currently, I can do it ...
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38 views

Spline interpolation problem akin to Bezier spline

Given three pairwise distinct points $p_1, p_2, p_3 \in \mathbb{R}^2$, I'd like to find a function $f: \mathbb{R} \to \mathbb{R}^2$ with at least $f \in C^1$ such that $f(0) = p_1, f(1) = p_3, f'(1) ...
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1answer
27 views

Non-zero terms in a B-Spline surface

Questions For question 4 I know that in the u direction there is at most 5, non-zero basis functions N_(i,4) to N_(i-4,4) and in the v direction there is at most 4 non-zero basis functions N_(j,3) to ...
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273 views

Natural Cubic Spline 3 points

I am trying to do a natural cubic spline but I'm having trouble. f(-.0247500)=-.5, f(.3349375)=-.25, f(1.101000)=0 I tried doing the matrix, Ax=b where, h0=h1=.25 an a0=-.0247500, a1=.3349375, ...
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1answer
67 views

B-spline parameterization and derivatives

I have a question regarding the re-parameterisation of a B-spline. Some info: The B-spline is of order 4 (degree 5), hence $C^3$ continuity There is no knot multiplicity The end conditions are not ...