# 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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### Minimization problem with latent function and splines

I have a dataset consisting of pairs $(x_i, y_i)$. I want to determine the function $f$, so $$f(x)f(y) = 1$$ with the constraint that $f(x) \leq x$, $f'(x) \geq 0$ and $f''(x) \geq 0$. I was ...
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### B-spline surfaces fitting references

I am looking for reference papers / publications regarding b-spline / NURBS surfaces fitting (I think I have noticed NURBS fitting is not very widespread so I guess most references will be related to ...
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### How to interpolate figure with splines?

Lets say I have a set of coordinates which when correctly (directly - a straight line) connected form a shape (a simple curve if you will). Think of star-shape, important is that a xalue x might have ...
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### Random non-intersecting cubic bezier curves between prescribed anchor points

I am given 4 pairs of red-green points in 2D. each pair corresponds to end points of a cubic bezier curve. My objective is to generate random control points for 4 curves (one going through each pair) ...
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### How to generate variable pitch helix in nurbs form

I would like to define a helix with a start pitch end pitch start radius end radius start angle The pitch and radius parameters should be interpolated linearly from each end along the length of the ...
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### Fitting an Akima Spline curve

I'm trying to fit an Akima Spline curve using the same method as this tool: https://www.mycurvefit.com/share/4ab90a5f-af5e-435e-9ce4-652c95c3d9a7 This curve gives me the exact shape I'm after for my ...
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### B-Spline Definition

I'm currently working on my master's project. For this, I rely on one PhD-thesis in which I found a statement I do not understand. Unfortunately, the author hasn't answered to my mails yet, so I have ...
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### Slope matching in spline interpolation

When using a Curve parameterization, I build segment functions as follows: $X(u) = A_x+B_x \times u+C_x \times u^2+D_x\times u^3$ $Y(u) = A_y+B_y \times u+C_y \times u^2+D_y\times u^3$ $uļ¼[0,1]$ In ...
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### Roots/zero crossing of additive piecewise linear splines

Suppose we have a function $f$ in p-variables $x_1,...,x_p$ which has the form $f(x_1,...,x_p) = f_1(x_1)+...+f_p(x_p)$. Assume further that each function $f_p$ is a piecewise linear spline. What I ...
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### Is CatmullāClark subdivision surface a refinement scheme?

To do the knot refinement, one important thing is to not change the geometry. Also, it is done in the parametric domain. However, for CatmullāClark subdivision surface, it is done on the physical ...
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### Should a natural cubic spline ensure 3rd derivative on the boundary to be 0?

Say we have K interior knots (t_1 to t_K). We can form K + 1 piecewice polynomials. If we want to fit a natural cubic spline s(x) In regression, we would enforce ...
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### Flexible grid update algorithm

I have a 2D square grid where the edges are line segments and the vertices have moved from their initial positions. Based on external constraints, the vertices are submitted to further motion, under ...
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### Relation between a Bezier curve and B-Spline curve

While the ideas behind Bezier curves are rather straight forward, I'm really struggling trying to understand B-Splines. I really researched quite a lot about it and still can't figure it out. I ...
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### Approximate a ($y²-x²=1$) hyperbola with line segments and elliptic ($a(x-x_0)²+b(y-y_0)²=1$) arcs

On some IT graphic systems, you have tools which draw line segments and circle or elipse arcs, but which do not draw parabolas or hyperbolas, and in many cases, those system keep track of graphical ...
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### “fastest” curve through n points

I'm programming an AI for a race game, where my car has to drive through some checkpoints. If I let it drive straight in direction of the checkpoints, it has to slow down and make a huge turn after ...
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### Monotonic and smooth interpolation between three points

The problem I have is the following: Given three $x,y$ points, $(-1, -y_{0})$, $(0,0)$ and $(1,y_{2})$, where $y_{0} \geq 0$ and $y_2 \geq 0$, I want to interpolate smoothly between them with a ...
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### How to calculate smoothing spline coefficients

I am attempting to calculate smoothing spline coefficients based on the description in Reinsch's 1967 paper, but I'm having some trouble. The first derivative is not continuous. Here are the ...
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### Convert multiple Bézier segments to a nurbs curve

I have multiple cubic BĆ©zier curve segments which are contiguous and G1 (they are the result of the fitting of many curve samples). Now I would like to transform these BĆ©zier segments into a single ...
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### Iterated backward difference quotient from splines

I am working on a certain problem: say we have a function $f$ defined on an interval $[a,b]$ and we want the spline $Sf$ of order $k$ (and knots needed for the B-splines are in $[a,b]$) to agree to $f$...
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### The advantage of B-spline compared to Bézier if the number of control points is very small

If the number of control points is n+1, and the degree of the basis function is p If n = p, B-spline is as same as BĆ©zier curve. Suppose I have a chance to increase the number of control points say ...
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### What is the purpose of having repeated knots in a B spline?

A primer on the cpr package in R (page 2 of https://arxiv.org/pdf/1705.04756.pdf) writes the following about B-splines. A B-spline basis matrix is defined by a polynomial order $k$ and knot ...
The $k$-th divided difference of a function $g$ at sites $\tau_i,\ldots, \tau_{i+k}$ is the leading coefficient (that is, the coefficient of $x^k$) of the polynomial of order $k+1$ that agrees with $g$...
### How to compute the quadric form matrix of the B-spline norm $\int_{k_1}^{k_n}f(t)^2dt$?
Let $f:[k_1,k_n]\to\mathbb{R}$ be a degree $d$ B-spline function with real knots $\{k_1,\ \ldots\ ,\ k_n\}$ and real coefficients $\{c_1,\ \ldots\ ,c_{n-d-1}\}$. The norm \$\ \int_{k_1}^{k_n}f(t)^2dt\ ...