Articles of spline

Approximating a large number of data points using (cubic) splines in l1/l2 norm.

I have a pretty large dataset ($x,y$) consisting of a few million points. There is a lot of noise in the data. I want to find a smooth but simple approximation/representation for this dataset, so that for a given value of $x$ I have a decent estimate of $y$. I expect the data to have […]

How to find B-Spline represenation of an Akima spline?

Given points $t_i$ and values $y_i$, I’d like to use Akima interpolation to interpolate to a different set of locations $x_j$. This means I need to calculate the cubic polynomials $A_{3,t}(x)$. Given that these $A_{3,t}(x)$ are actually splines, it should be possible to find their B-Spline coefficients, i.e., $\alpha_l$ such that $A_{3,t}(x) = \sum_l \alpha_l\cdot […]

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 about finding more information (maybe there is another more common name)? Thanks a lot! Ryou

Bezier curvature extrema

For a planar cubic Bezier curve $B (x(t),y(t))$, I would like to find the values of parameter $t$ where the curvature (or curvature radius) is greatest/smallest. The formula for curvature is: $$r = \dfrac{(x’^2+y’^2)^{(3/2)}}{x’ (t) y”(t) – y'(t) x”(t)}$$ The problem is that there is that square root in it so I was wondering whether […]

Spline interpolation versus polynomial interpolation

What is the difference, if any, between spline interpolation and piecewise polynomial interpolation?

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}, there is 1 basis of degree 1, namely: $$N_0^1(t) = \left\{ \begin{array}{ll} t & \quad 0\leq t< 1 \\ 2-t […]

Natural cubic splines vs. Piecewise Hermite Splines

Recently, I was reading about a “Natural Piecewise Hermite Spline” in Game Programming Gems 5 (under the Spline-Based Time Control for Animation). This particular spline is used for generating a C2 Hermite spline to fit some given data. I kinda understand how natural cubic spline interpolation works (ie: setup a tridiagonal matrix, solve Ax=b where […]

What is the maximum overshoot of interpolating splines in $d$ dimensions?

Consider cubic splines $s( x, y )$ which interpolate values $y = \{ y_0, y_1, \dots,y_n \}$, on the uniform grid $\{ 0, 1,\dots, n \}$. Fix $s”(0) = s”(n) = 0$ (natural splines). How big can $$\operatorname{overshoot}( s; y ) \equiv \max_{0\le x\le n} s( x; y ) $$ be, over all splines $s( […]

What equation produces this curve?

I’m working on an engineering project, and I’d like to be able to input an equation into my CAD software, rather than drawing a spline. The spline is pretty simple – a gentle curve which begins and ends horizontal. Is there a simple equation for this curve? Or perhaps two equations, one for each half? […]

Natural cubic spline interpolation error estimate

I am looking for an error estimation for natural (one with $s”(a) = s”(b) = 0$ boundary conditions) cubic spline interpolation on an evenly spaced grid. The best result I’ve found was $O(h^2)$ without any clarification what the actual constant in $O(\cdot)$ is. Intuitively, the error should have the form of $$\max_{x\in[a,b]}|f(x) – s(x)| < […]