# 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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### What is a natural cubic "B-spline"?

I recently learned that natural cubic splines are strictly distinct from natural cubic B-splines while studying spline methods. It appears that natural cubic B-splines are obtained by adding ...
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### Finding Basis for specific Spline Space

Let $S = \{s \in S: s'(a) = s'(b) = 0 \}$ be the spline space that holds all cubic splines with derivate at startpoint (a) and endpoint (b) =0. I want to find a basis for this vector space. I looked ...
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### Construction of a curved grid in 2D space, based on a cubic spline interpolation

Please bear with me, this is my first question here and I hope it fits and is understandable. I want to construct a "curved" grid on the screen in a computer program. Visualization of the ...
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### Is it possible to have a quadratic spline and a cubic spline meeting and being C2 continuous at a point K?

I'm trying to interpolate some datapoints. Ideally I would like to have 3 splines: The first and third being quadratic, and the second one (in the middle) cubic. Is it possible to mix polynomials and ...
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### Frequency domain spline approximation and B-Spline inverse transforms

In short, I would like pointers to closed-form formulas, or efficient algorithms for computing the inverse of ether sine, cosine, or Hartley transforms of the B-Spline basis. The motivation, as ...
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### Intuitive understanding behind clamping of B-splines

I have been going through the implementation of B-splines, and I observe that whenever it comes to clamping, it's usually mentioned that we must repeat the end knots $p+1$ times for a spline of degree ...
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### Thin Plate Spline RBF Interpolation understanding

I have been looking at radial basis function interpolation: $f(x) = \sum w_i \phi_i(||x-x_i||)$ and examining the different kernels e.g. $\phi(r) = e^{-(\epsilon r) ^2}$ which are generally maximised ...
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### Is the Integral of a Spline multiplied with the Exponential Function of the Spline solvable?

I want to compute the integral, I have borders for $x$, however I would like a function of y as the result: $$\int_a^b \quad \frac{\partial f(y,x)}{\partial y} \exp(f(y,x)) \quad dx$$ $f(y,x)$ is a ...
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### One step beyond cubic spline interpolation, a fourth-order problem?

I am trying to fit a polynomial through three points, where I also know the derivatives at the two endpoints. I don't need a truly general solution. My specific problem is constrained as follows: <...
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### Linear fitting of values on non-uniformly parametrized B-spline surface

I have a 2d (u,v) surface in 3d space (x,y,z) that is defined as a b-spline surface. The surface is not arc-length parametrized. Additionally, I have scalar values defined on the surface at given u,v ...
1 vote
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### What is the size of the basis for cubic B - splines

I have an exam questions which says we have mortality data observed between ages 20 and 60 with knots at 20, 30, 40, 50 and 60. Then the questions say we will use a set of 7 basis splines to fit a ...
1 vote
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### Finding polynomial degree $1$ of $f(x)=\text{erf}(x-1)$ at $x=1$

Question: Find the polynomial of degree $1$ that has the highest possible order of contact with $f(x)=\text{erf}(x-1)$ at $x=1$. Plot the spline knotted at $(1,0)$ with $f(x)$ on the right and your ...
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### best high order interpolation method for set of points

what is the best way to interpolate a set of points in 2d, such that there is only one parameter to indicate position on the curve (like is the case for a Bézier curve)? one thing I know is that we ...
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### Algorithm to transform a polyline into an equidistant polyline

I have a number of points in 3D space. These points represent the positions (the toolpath) for an industrial robot or a CNC machine. The points are calculated by a software program. There can be up to ...
1 vote
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### Choosing optimal sample points for cubic spline approximation of sinusoid with polynomial argument

My problem consists as follows. Say I have a sinusoidal function whose argument is a cubic polynomial: $$f(t) = \sin(at^3+bt^2+ct+d) \quad a, b, c, d\in \mathbb{R}, \quad t\in[T_1, T_2]$$ I want to ...
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### What is the big O notation of the cubic spline algorithm?

I'm trying to do a time complexity analysis of MATLAB's cubic spline algorithm. I have a 1000 x 1000 table and I want to know what is being done when I query a point that is in between two quantities ...
1 vote
Let's say we have an arc-length parametrized curve $\mathcal{C}(s)\in\mathbb{R}^{3}$. We want to find a frame \$\mathcal{R}(s)=\left[\begin{array}{ccc} \hat{\mathbf{t}} & \hat{\mathbf{b}} & \...