# Tagged Questions

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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### Calculating B-Splines and dimension of spline space

I've got the following assignment: Let $S$ be the space of piecewise polynomials of degree $3$ on the intervall $[-1;1]$ with knots $x_i = -1+\frac{i}{2}, 0 \leq i \leq 4$. (a) Calculate a basis of ...
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### The dimension of the space of continuous functions that are piecewise polynomials of degree $k$

I am trying to calculate $$\dim( \{ f\in C^0 ([a,b]) : f_{|[x_{j-1},x_j]} \in \mathcal{P}_k, j = 1,...,m \}) \text{ with }m,k \in \mathbb{N}$$ Is $m(k+1)$ correct? My thoughts: I have $m$ ...
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### Error Bounded Cubic B-Splines with fewest segments

I have some odd constraints in my project. Suppose we want to use Cubic BSplines to approximate a set of Points. There is two Constraints: error value should be an input to the algorithm (error ...
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### Spline terminology

I am reading up on splines and as a beginner I have a basic question - Does it make sense to say - "I will fit a cubic b-spline to the data". As b-spline is just a representation of spline in terms ...
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### How sensitive is a natural cubic spline?

I am interested in using natural cubic splines to generate possible replacement values in the quality control of data. I would like to do this as close to real-time as I can. That is, I would like ...
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### Natural Cubic Spline Problem - what do I do next?

Related to this similar problem. I am attempting to solve the following: Find a natural cubic spline function whose knots are $-1$, $0$, and $1$ and that takes on the values $S(-1)=5$, $S(0)=7$, ...
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### The Elements of Statistical Learning: How does this nature cubic spline have K basis functions with K knots with the given solution? See Details.

I'm referring to this found in chap 5: Picture from ELS If K=2 (as in Sec 5.2), then we have N1, N2, N3 and N4 basis functions. So 4 basis functions and 2 knots. I know that a natural cubic spline ...
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### Which interpolation method for complicated, smooth curves?

Which interpolation method should I use for complicated "smooth" curves such as $\frac{sin(x)}{x}$ for $x>0$.
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### Evaluating a Hermite Quaternion Curve

I have a set of fixed poses (position and orientation) and want to interpolate C1 continuous between the orientations. I tried to follow A General Construction Scheme for Unit Quaternion Curves with ...
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### Find real constants $c$ and $k$ such that $y=cx^k$ passes through point $(a, b)$ with slope $m$

In the Cartesian plane, can a power function of the form $y=cx^k$ (where $c>0$ and $k>1$, not necessarily an integer) be found such that its graph passes through any arbitrary point $(a, b)$ ...
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### Cubic spline. What is symmetrical form and why?

I'm trying to understand the algorithm for cubic spline from Wikipedia. It says the polynomial can be written in symmetrical form: A third order polynomial $q(x)$ for which $q(x_1)=y_1$, $(x_2)=y_2$, ...
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### Find the natural cubic spline function whose knots are $-1$, $0$, and $1$ and that takes the values $S(-1)=13$, $S(0)=7$, and $S(1)=9$.

Find the natural cubic spline function whose knots are $-1$, $0$, and $1$ and that takes the values $S(-1)=13$, $S(0)=7$, and $S(1)=9$. I'm not sure how to go about this. Any solutions/hints are ...
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### Forecasting electricity load based on various parameters using neural networks.

I have a task of predicting the electricity load for 2 months of a region based on various parameters, like avg. rainfall, avg. solar radiation, winter or summer. etc. There are two sets of data ...
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### Quadratic spline?$f(x)=x$ when $x\in (-\infty, 1]$, $f(x)={-1\over 2}(2-x)^2+{3\over 2}$ when $x\in [1,2]$, and

Determine whether this is a quadratic spline $f(x)=x$ when $x\in (-\infty, 1]$, $f(x)={-1\over 2}(2-x)^2+{3\over 2}$ when $x\in [1,2]$, and $f(x)={3\over 2}$ when $x\in [2,\infty)$. I think this is a ...
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### Computing splines using Hermite interpolants

The form for a cubic Hermite interpolant has the form $p_i(x)=a_i+b_i(x-x_i)+c_i(x-x_i)^2+d_i(x-x_i)^3$ according to the following conditions: $p_i(x_i) = y_i$ $p_i(x_{i+1}) = y_{i+1}$ ...
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### Spline interpolation explanation

I'm trying to learn about spline interpolation, and I'm struggling to understand what h_i^3 is in the second and third equation. I don't understand how they derived that equation. I'm trying to ...
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### Can I make this numerical integration continuously differentiable?

Suppose I have the discrete values $f(x_i)$ for every $x_i$ greater than some $\varepsilon$, and I want to numerically calculate the following integral: n = \int_\varepsilon^\infty ...
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### How to set control points for spline curves

I've written a program that calculates points on spline curves (including Hermite, Bezier, and B-splines) and plot the curve on the screen (the curve is plotted on an html canvas using javascript). ...
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### Tangent of Cubic Hermite curve

I have created cubic curve using CatmullRom Spline or Akima spline. From those, I obtain $a, b, c, d$ parameters. To get point on the curve, I do this $f(t) = a + bt + ct^2 + dt^3$ How to get ...
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### Why do bsplines of order 3 go through the endpoints only when the endpoints have multiplicity 3

Hi I'm learning about different types of splines and interpolating curves. I wrote a script to generate cubic bsplines. I noticed that the bspline does not go through the endpoints if I give a knot ...
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### Can monotone cubic interpolation be implemented explicitly in B-spline form?

I have been interpolating cubic splines to some data, but it is now clear that I need my curves to be monotonic. Wikipedia and StackExchange sources describe how to impose the monotonicity condition ...
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### Does cubic spline interpolation preserve both monotony and convexity?

I have a question. Let's say i have a function $f(\cdot)$ such that $Dom(f) = [a,b]$. The function is at least of class $C^2$ and it is both strictly monotone and convex. My question is, does a ...
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### Linear, Bilinear and B-spline interpolation

I've read that the linear interpolation isn't differentiable everywhere and it would be better to model a continuous-space image using quadratic or cubic B-spline interpolation because is ...
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### Spline interpolation degrees of freedom

When using cubic spline interpolation, we have to solve $n-1$ equations with $n+2$ unknowns. What we can do is set $z_0 = z_n = 0$, which gives the natural cubic spline. But could we also set some ...
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### How to get normal of Catmull Rom spline

I have a piece wise track made out of Catmull Rom splines. I originally crossed the tangent vector with (0,0,1), then the result of that crossed with the tangent to get the normal vector. However, ...
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### Cubic spline with clamped boundaries

I have a cubic spline interpolation problem to work through. I think I understand what is required of the question, but my biggest concern is the nature of clamped versus natural boundaries. All the ...
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### Is cardinal $B$-spline of order $n$ really piecewise Bezier order $n$ curve?

Is cardinal $B$-spline of order $n$ really piecewise Bezier curve $n$? I think I saw this in some lecture notes, but I can't recall where.
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### Numerical integration with limited support

This is somewhat of a crossover between mathematics and programming, but I suspect the core idea I'm missing is mathematical. Starting with the following integral: \tilde{F}(k) = ...
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### Derivative of multivariate splines using tensor products

I am trying to compute the derivative of a multivariate spline, in fact bi-variate I use a b-spline univariate to create a basis, for the first $x_1$ and second variable $x_2$, then I use the tensor ...
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### Subdividing a Bézier curve into N curves

NOTE: I am only concerned with quadratic Bézier curves. So, dividing a Bézier curve into two is remarkably easy; just interpolate between start and control points by $t$, and get the end point for ...
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### Extrapolation and Splines

If you have a smooth curve, and at a certain point in time you want to predict the next turning point, and you assume it is a non-periodic, stationary, smooth process, then what would be the best way ...
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### B-spline weighted least squares fit

Can someone please point to an easy to read source for Bspline curve fitting with weighted least squares. Basically I want to fit a function, and I have some points which are more important then ...
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### How is the B-Spline definition constructed?

I'm trying to understand how the B-Spline definition is constructed. That is, where did the knot vector and the basis functions and their recursive definition come from. The definition can be seen ...
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### How to Find the Error for Spline Interpolation Without the Original Function?

Most of the literature (e.g.: http://pages.cs.wisc.edu/~amos/412/lecture-notes/lecture11.pdf) I have consulted thus far indicates how one would determine the error of a cubic when the original ...
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### Alternatives / Extensions to the Thin Plate Splines method

Thin Plate Splines are a great method to find a smooth interpolating surface given scattered data. Essentially, the method involves calculating weights for a radial basis function centred around each ...
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### a smooth bandlimited and short-lived B-spline function?

Is there exist a smooth Spline function which lives only between $[-1, 1]$ and is also bandlimited? I have not found any! I will need to replace the $\text{sinc}$ function/interpolator which this ...
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### How to apply a time shift to a pulse-shape, spanned with spline functions?

I have a sampled pulse shape: $h = [1, 0.5]$ and I do not know what is its real underlying continuous-time pulse. I want to compute the samples of $h(t-\Delta t)$. If I write the continuous pulse ...
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### Definitions of BSpline

I can think of 4 ways of defining BSpline with separate, equally spaced knots: recursion convolution sum divided difference. Methods 1 & 2 start with the indicator function of the unit ...
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### Nurbs parametric coordinate span

I am using the Nurbs definition of Wikipedia. I might have missed something in the definition but I cannot understand how to know on which interval does the parametric coordinate span. Particularily ...
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### Deriving uniform cubic B-spline matrix

The page 6 of this paper says: Condition 1: $p(0) = q(1)$ – Symmetry: $p(0) = q(1) = 1/6(\pi-2 + 4 \pi-1 + \pi)$ Condition 2: $p’(0) = q’(1)$ – Geometry: \$p’(0) = q’(1) = 1/2 ((\pi – \pi-1) + ...
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### Measure of curve smoothness

Could someone please give me the intuition behind using integral of squared second derivative as a measure of curve smoothness? I was thinking that since curvature measures how fast a curve changes, ...