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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Cubic Spline with Exponential Term

I have the following function that I want to use for a spline instead of the standard $a + bt + ct^2 + dt^3$ cubic spline: $$a[F - \sum_{m=1}^{100}Ge^{-Y^2t}] + b[Ft - \sum_{m=1}^{100}\frac G{Y^2} + \...
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Bandwidth in Gaussian kernel smoothing

I am trying to implement a Gaussian kernel smoother. The equation is $$\begin{split}f(x)&=\sum_{i=1}^N w_i(x)y_i\\ w_i(x)&=\frac{\kappa_h(x-x_i)}{\sum_{i'=1}^N \kappa_h(x-x_{i'})}\end{split}$$ ...
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What is $B_{i,n}(x)$ in the Wiki page for B-splines

A spline is a piecewise polynomial function with k knots or places where the polynomial function can change between the knots. It has order n if its knots are denoted $t_0,t_1,...,t_n$. For a given ...
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Find the coefficients of a cubic spline

Let $S_3 : [x_0,x_n] \to \mathbb{R}$ be a cubic spline on $I_i=[x_i, x_{i+1}]$ such that $S_3(x_i)=y_i$ and $S_3'(x_i)= z_i, i=0,...,n$ we consider $S_{3,i}$ the restriction of $S_3$ on each interval ...
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Square normalized Splines

The M-Spline basis functions have the neat property that they integrate to one $$ \int M_i(x, k, t) dx = 1 $$ and therefore any M-Spline $$ \int \sum_i a_i M_i(x,k,t) dx = 1 $$ if the coefficients $...
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quadratic splines

Consider $f \in C[0,1]$ and $\{x_i \}_{i=0}^n$ is an increasing equally space sequence where $x_i \triangleq i h$ and $h\triangleq 1/n$. It is known that if $s_f$ is the linear spline (piecewise ...
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is there a NURBS but have weights on different axis?

We know that NURBS has the form $p(u)=\frac{\sum_{i=0}^n\ w_i\ N_{i,k}\ (u)\bf d_i}{\sum_{i=0}^n\ w_i\ N_{i,k}\ (u)}$ which adds weights on different control point $\bf d_i$. I am looking for a form ...
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Summation Expression with Range and Sequence expression

I've been trying to search for the meaning for this summation for a couple of days now. I've used https://approach0.xyz/ to search on the stack, but I cannot seem to find an explanation that helps me. ...
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$C^3$ monotone spline interpolation

I am looking for a spline interpolation that is 3 times differentiable, strictly monotone and doesn't involve the inclusion of extra control nodes. I found this paper that describes the construction ...
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Hermite Curve to Approximate Sin(X) X and Y values incorrect

So I'm trying to approximate sin(x) using Hermite curves. I have four separate curves that when graphed are supposed to look similar to sin(x) on the period 0 through 2PI. I am doing the calculations ...
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Solving fractional differential equations using B-Spline Collocation

I am working on implementing the method shown in this paper to solve a particular fractional differential equation using the method of collocation (where the basis function used is fractional b-spline)...
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Equation to find the Principal Unit Normal of a Bezier curve at $t$ ($0\leq t\leq1$)?

I have recently been working on Bezier curves and have come across a very interesting snippet of code (don't worry its almost entirely mathematical) relating to finding the principal normal of a ...
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Are cubic splines with the not-a-knot condition undefined when using 3 data nodes?

The following problem was posed: find the cubic not-a-knot spline interpolant of the given function and node vector: $$ \cos(\pi^2x^2), \overrightarrow{t}=[-1,1,4]^T $$ However, isn't it impossible to ...
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Is "torsion continuity" a misnomer for G³ geometric continuity?

I've recently been researching parametric vs geometric continuity of splines (piecewise polynomials) in 2D space. The most common terms for each level of geometric continuity are: $G^0$ is positional ...
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Catmull Rom with slope for end points

I have a number of data points that I want to connect with a Catmull Rom spline. Normally you can't calculate the spline from the first and second data points or from the next to last data point to ...
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Partial derivative of divided difference w.r.p to knots (with multiplicity)

I'm self-studying Spline Functions: Basic Theory by Larry Schumaker. In Theorem 2.55 on page 52, we consider the partial derivative of divided difference w.r.p to knots when an entire block of knots ...
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How to sum up two Bernstein polynoms

The sum/difference of two parametric surfaces descripted in monmomial bases is straight forward as the following example shows: $G_{1,1}(u_1,v_1) = 3*v_1(v_1-1)^2(u_1-1)^3 + 3u_1$ $G_{2,1}(u_2,v_2) = ...
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Intersection of two parametric surfaces (governing equation)

in order to solve the interaction between two parametric surfaces (represented as Bezier oder B-Splines) i need to "solve" the non linear equation system. As both surfaces are depended in ...
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Is there any Bezier-like spline that is explicit function?

What I actually need is to allow users to generate 1D data array (actually, MIDI CC values) through some control points on GUI. The Bezier curves has a very good property that the tangent on end ...
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How to quantify the error of explicit methods of merging Bezier curves

I came across this paper here describing a method of creating an n_th order Bezier curve that approximates multiple other Bezier curves that are connected. The method is based on minimzing the error ...
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When to choose Bezier curve over B-Spline curve?

I am reading about Bézier curves and B-Spline curves. I have understood both mathematically and intuitively that the big difference between these two kind of curves is that when dealing with Bézier ...
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Monotonic surface interpolation with non-uniform data points

I have a problem where I have a set of non-uniformly spaced data points in 3D space and I wish to generate a monotonic surface across these points. In practice I'd have to interpolate within a ...
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Joining a straight line to a Bezier curve

Suppose that we have a Bezier curve of order n represented in the matrix notation as $P(t) = T(t) \Lambda P_c$ (where P(t) is a point on the Bezier curve, $\Lambda$ is a lower triangular matrix ...
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What energy do even degree splines minimize?

For odd degree splines, with degree $k$ I know they minimize an energy: $$E[u] = \int_{a}^{b} \left\|\frac{d^{(k+1)/2}u}{dx^{(k+1)/2}}(x)\right\|^2 \, dx, \, u(x_i) = f(x_i), \, \frac{d^l u}{dx^l}(c) =...
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Uniform interpolation on a Cubic Hermite Spline

I have a 3D spline with points $p_0,p_1,...,p_n$ and tangents $m_0,m_1,...,m_n$. I'm using the formula described in this page. $p(t) = (2t^3-3t^2+1)p_0+(t^3-2t^2+t)m_0+(-2t^3+3t^2)p_1+(t^3-t^2)m_1$, ...
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Piecewise quadratic interpolation in a symmetric manner

I am trying to derive a reasonable symmetric interpolant for quadratic $C^0$ interpolation. For odd degrees I have no trouble since things are symmetric. For example for a piecewise cubic interpolant ...
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Calculate length of a cubic spline, given it's vertices in 3d space?

Given a cubic spline defined by $n$ number of vertices in a 3d space, how would one calculate the length of this spline? (Attached picture is just for illustration, to explain what I mean by vertices, ...
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Constructing $C^{k-2}$ and $C^{k-1}$ splines on a regular grid through convolution

I am currently trying to figure out how spline interpolation works on a regular grid. The simplest spline interpolation basis function I can think of is the box function: $f(x) = 1, \, |x|<1$ $$ \...
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Understanding this Inclusive Notation

I am learning about splines in my computational analysis class, and this question came up: I have not seen this notation before (that is "$(-1 \leq x \leq 0.5)$"), and it is confusing me ...
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Determine spline coefficients by linear interpolation between 2 other splines

I am looking for a way to calculate a spline function from 2 previously calculated spline functions by linear interpolation. I have these functions (590.33, 911.4, 1192.51 -> linear interpolation ...
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Recursively determine cubic spline coefficients

I am looking for a way to fit a cubic spline on previously recorded points without using matrices. The software is running on an embedded microcontroller which is low on RAM, so calculating my ...
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Upper bound of multi-dimensional integral

It seems clear to me that for a function $f: [a,b] \rightarrow \mathbb{R}$, the following upper bound holds, $$ \left| \int_u^v f(x) dx \right| \leq ||f||_{\infty} h $$ where $h = b - a$ and $u,v \in [...
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How do you fill a composite Bézier curve composed of a list of cubic Bézier curves?

According to Wikipedia: "A composite Bézier curve is a series of Bézier curves joined end to end where the last point of one curve coincides with the starting point of the next curve." I'...
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Smooth function passing through a countable number of points

Given a sequence $(y_n)_n$ of real numbers, can we find a smooth ($C^1$, $C^2$ or even $C^\infty$) real function $\phi$ such that $\phi(2^n)=y_n$ for all $n\in\mathbb{N}$ ? It's clear if we want a ...
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What are exponential-polynomial splines?

I am a bit confused on the topic of exponential-polynomial splines. Would this be an example of such a spline? aex + b(x + 1 + ex) + c(x2 + x + 1 + ex) + d(x3 +x2 + x + 1 +ex) I ask because I want to ...
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Are knots in spline models basically the change-points in a change-point analysis?

I'm just trying to get my head around using Bayesian spline model to detect changepoints. Are the knots in the spline point what I'm basically trying to find to detect the changepoint?
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how many control points are needed if n+1 data points are interpolated by p degree(or p+1 order) B-Spline?

Exactly, I suppose that both the number of control points and knot points are decided by the number of data points to be interpolated and the degree of B-Spline basis. However, I found two different ...
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Find a basis for $S(1,0)h[a,b]=\{p∈C0[a,b],p|Ii∈P1(Ii)\}$

On the intervall $[a,b]$ let $a = x_o< x_1 < x_2 < ... < x_n = b$ be a decomposition. Consider the Vectorspace of the piecewise linear functions $$S_h^{(1,0)}[a,b] = \{p \in C^0[a,b], p|_{{...
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Determine whether a spline curve describes an elliptical arc

I'm currently writing software which analyses STEP files to hunt for engineered parts with particular shapes. The software uses Open Cascade which presents me with topological and geometrical objects ...
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Exponential Term in Spline Polynomial [closed]

Need some help here. A cubic spline takes the form of: $a + b^t + ct^2 + dt^3$ Is there such a spline that incorporates an exponential term to take the form of: $a + bt + ct^2 + dt^3 + fe^{Xt}$ where $...
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Error in first derivative of cubic spline interpolant

Let $f: [a,b] \rightarrow \mathbb{R}$ be a $C^{\infty}$ function, and let $a = x_0 < x_1 < \cdots < x_n = b$ be a partition of the interval $[a,b]$. Let $s(x)$ be a piecewise polynomial ...
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Will smoothing splines always lead to continuous $\hat{f}$?

I found these notes referencing smoothing splines in elements of mathematical learning. I'm just confused why (3) is False because I really can't think of a counter example λ can be chosen by cross-...
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Is there a convex function satisfying certain growth and smothness conditions?

I would like find (if it exists) a function $F:\mathbb R^2\to\mathbb R$ such that the following conditions hold true: $F\in C^\infty(\mathbb R^2)$ $F$ is radial, that is $F(x,y)=f(\sqrt{x^2+y^2}\,)$ $...
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B-Spline Basis Function Discrepancy in Definition

Introduction The standard recursive definition of the B-Spline basis, given a knot vector $U = u_0 , ... ,u_n$ is: $$B_{i,0}(u)= \begin{cases} 1, & u_i \le u< u_{i+1} \\ 0, & \text{else} \...
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why it is not possible to use quadratic spline

The following four points are known to lie on a closed curve in the (x, y)-plane: (−1,0), (−1/2,3), (1/2,−3), (1,0) and the goal of this question is to fit a piecewise-polynomial approximation of the ...
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How do I create a bezier spline section out of many points?

I am building a program where I need to simplify N number of points into a single section of a bezier spline, ie describe them using just 2 end points and 2 control points. Naturally this will lead to ...
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Find the parameters for a cubic spline

I have the cubic spline $$f(x) = \begin{cases} f_{0} = ax^{3}+bx^{2}+ c(x-1), \quad \text{if} \quad -1 \le x \le 0 \\ f_{1} = d(x-1)^{2} +cx, \quad \text{if} \quad 0 \le x \le 1 \end{cases}$$ ...
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Proving the integration of piecewise cubic functions as 0

Given that $g(x)$ is a piecewise cubic function, and $h(x) = \bar g(x)-g(x)$. I am struggling to show that $\sum_{i=1}^{N}\int_{x_{n-1}}^{x_n}g'''(x)h'(x)dx = 0$ I understand that $g'''(x)$ will be a ...
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How do you move multiple points with a set distance (linear) between them on a non-linear spline?

I'm doing some things related to spline code, and I've run into a problem. Please take a quick look at the picture below. Black = spline (bezier), red circles = points on spline Imagine that the ...
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Can an Elastica curve have a discontinuity in the curvature?

This question is related to this one: What are the conditions for the union of two Elastica curves to be an Elastica curve as well? An Elastica curve is defined as one that minimises the bending ...
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