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.

learn more… | top users | synonyms

1
vote
0answers
16 views

Interpolating polynomial such that it is convex in specified region

The problem I have is that I have data at two points $x_1,x_2$ and $x_2>x_1>0$. At these two points, I know that the function $f$ has values $f(x_1)$ and $f(x_2)$ respectively. It is also ...
0
votes
0answers
8 views

Basis Function Algorithm, In The NURBS book

On page 74, Peigl explained an algorithm about computing a single basis function. first lines of this algorithm are handling some special cases. ...
0
votes
0answers
15 views

Computing a percent down on a bezier curve when a control points position is moved

I have a line segment drawn as a percent down on a bezier curve, lets say which has 3 control points. I need to calculate the new percent position of the line segment when one of the control point is ...
0
votes
0answers
52 views

General issue when adding shocks on curves made of splines

Let us assume I have a "nice" curve and that I would like to introduce a small shock up/down of about 1% at a certain point along the curve. I am trying to find out what the best and most efficient ...
-1
votes
0answers
8 views

Product of B-splines

It's my first post and i'm delighted to be among you... I'm reading at the moment a paper "Some Identities for Products and Degree Raising of Splines" from Knut Morken. enter image description here He ...
0
votes
1answer
34 views

B-splines basis function

If the image domain is denoted as $\Omega_I=\{(x,y,z)| 0 \leq x<X,0\leq y<Y, 0 \leq z <Z \}$. Let $\Phi$ denote a $n_x \times n_y \times n_z$ mesh of control points points $\phi_{i,j,k}$ with ...
0
votes
1answer
51 views

Precision in Cubic spline interpolation

I am working on cubic spline interpolation with set of data points from CAD with following steps: Form piecewise spline equations between points. cubic equation : $ ax^3 + bx^2 +cx + d = P(x) $ ...
0
votes
0answers
34 views

Fitting a continuous curve over a piecewise constant data

I have some measurements that are piecewise constant over a certain variable. For example, in the following image, the vertical axis represents the measurement data and the variable is on the ...
0
votes
1answer
20 views

proving an inequality involving a linear spline / piecewise polynomial

I have $n+1$ sample points $x_i = \left(\frac{i}{n}\right)^4$ and want to approximate the function $f(x)=\sqrt{x}$ by a linear spline $f_n \in S^{1,0}(\mathcal{T_n})$ on the interval $[0,1]$. I know ...
2
votes
1answer
29 views

tangents at unit circle - parametrization leads to strange result

I'm thinking about the tangents at the unit circle in the upper right corner and I do so in terms of the parametrization $$ \begin{eqnarray} v:&[0,1]& \rightarrow \mathbb{R}^2\\ &t&...
1
vote
1answer
37 views

Optimal approximation of spline curve using linear interpolation

I have a parametric cubic spline which I need to draw in graphics. I am restricted to using a set of lines to draw this, and for performance reasons I need as few lines as possible. So I need an ...
1
vote
1answer
26 views

Support of a clamped B-spline

Let a B-spline of degree $p$ be defined by its parametric equation $$ \mathbf{r}(t) = \sum_{i=0}^n N_i^p(t)\mathbf{P}_i$$ where the $n+1$ control points are denoted by $\mathbf{P}_i$. The basis ...
0
votes
1answer
24 views

Knot sequence for a natural cubic (B-)spline interpolant

say I am given $n+1$ data points $(x_i,y_i)$ with $0\leq i \leq n$ and $x_0 < x_1 < \dots < x_n$. I want to interpolate these with a natural cubic spline $s(x)$ ($C_2$ continuous at knots ...
0
votes
0answers
15 views

Compact set of function involving BSpline functions.

Let $X = \left\{x_0,...,x_{n-1}\right\}$, $x_i-x_{i-1} = h$ for $i=1\ldots n-1$ and $$ \phi(x;X,\vec{\beta}) = \sum_{j=0}^n \beta_j \phi_{j,2}(x;X) $$ where $\phi_{j,2}$ is the $j$-th BSpline (of ...
0
votes
0answers
18 views

B spline - Partition of unity

I need the Laplace form of the second order B splines over logarithmically-spaced knot. I have read a paper in which it is mentioned that ; " Heaviside function can be represented as the sum of all ...
0
votes
1answer
37 views

Cubic spline interpolation results

I have a set of data points on which i am trying to do cubic spline interpolation. Below is the snapshot of the curve with the input data points marked in green color. And the red color marked point ...
0
votes
1answer
19 views

Estimating the curvature of a discretized curve in 3d with cubic splines

I have a computer simulation in which I'm modeling a physical curve by discretizing it and updating the locations of these points. I want to find/estimate the location of the maximum curvature of the ...
1
vote
1answer
36 views

B-Spline approximation deviates a lot while increasing the number of control points???

I'm dealing with a problem to approximate some data points with B-Spline. I follow the method and implemented the algorithm from this site: Curve Global Approximation. 1) The first step is to ...
2
votes
1answer
60 views

Cubic Spline Interpolation

My problem is to find a interpolating cubic spline to the points $$\left\{(0,0), \left(\frac{\pi}{2}, 1\right), \left(\pi,0\right), \left(\frac{3\pi}{2}, -1\right),(2\pi,0)\right\}$$ I did as ...
0
votes
0answers
20 views

Curve Conversion

I have a curve that is kind of spline. It is defined as a sequence of polynomial segments. It is defined by order, knots, and coefficients of the polynomials. How can I transform or convert this ...
0
votes
0answers
19 views

Interpolate with smoothing parameter

I need to implement in C++ interpolation with smoothing parameter. To the non-familiar with this function: The smoothing parameter gets a value from 0 to 1. 0 brings absoulte linear interpolation (...
0
votes
0answers
13 views

Example of a “abrupt function”

I need example of a simple function to show that cubic spline gives better result than Lagrange's interpolation in case of some special functions. Thank you
21
votes
11answers
5k views

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 ...
0
votes
1answer
15 views

Finding the equation of a quadratic with 2 points and a known slope. (SPLINES)

Sketch the spline of degree 2 with value 0.5 at x = 2.5 and the values 1, 1, 0, 0 at t0, . . . , t3, respectively. (t0=0, t1=2, t2=3, and t3=5) What is the value of the spline at x = 1 and 4? What I ...
0
votes
1answer
34 views

Cubic Spline for a function

I have the function $f(x)=x^3$ and I need to find the cubic spline. The given points are: $\{-1, 0, 1\}$. What is the cubic spline for this function and what would a demonstration to this be? I would ...
0
votes
1answer
19 views

n-order B-splines interpolation

I am wondering if the following statements are correct: (1) zero-order B-splines interpolation is equivalent to nearest-neighbor interpolation. $C^0$ continuity thus is not differentiable. (2) first-...
1
vote
1answer
27 views

Deriving a tridiagonal system for cubic spline interpolation

Can anyone explain how $B_{i-1} = 1/4$ and $B_{i+1} = 1/4$ were chosen in line 6 of the picture, just above the matrix? I'm trying to understand cubic splines but this result seems like it came out ...
0
votes
1answer
18 views

B-splines locally controlled

I have read that in contrast of the thin-plate splines, B-splines are locally controlled, which makes them computationally efficient even for large number of control points. I didn't understand what ...
0
votes
2answers
42 views

Interpolation by splines: how to set up the equation system for finding the coefficients of the spline (in a B-spline basis)

Problem. I want to interpolate a function $f$ in some equidistant points $x_0<x_1<x_2<x_3<x_4$ using a quadratic spline. My attempt. I assume that we can use the interpolation points as ...
0
votes
1answer
35 views

Addition of two B-spline curves

Suppose I have two B-splines, both with the same degree, $p$, and uniformly distributed knots, but with different numbers of knots and control points. Is it possible to sum the two splines to obtain ...
0
votes
0answers
13 views

Enforce Spline Section Always Greater Than/Equal Zero

I'm attempting to do some cubic spline interpolation to make incomparable discrete datasets comparable by naively making them continuous. Its largely histogram based data with different bin thresholds ...
0
votes
1answer
9 views

Normalized vs Non Normalized Bernstein

I know what Bernstein polynomials themselves are, and am intimately familiar with one of their usage cases - Bezier curves. However, I recently came across someone mentioning of "Not Normalized ...
1
vote
0answers
21 views

Gaussian process regression from predictions

Can any body provide example of a gaussian process regression being used to generate confidence intervals?
0
votes
2answers
46 views

The quadratic spline

I'd like to fit the data in table as blow x f(x) 3.0 2.5 4.5 1.0 7.0 2.5 9.0 0.5 when $x=5$, I want to find value of $f(x)$ by using ...
0
votes
1answer
26 views

Bspline matrix form?

I understand how bezier curves can be expressed in matrix form: you have a matrix multiplied by a vector containing the power series of t, and also multiply be a vector containing the control points. ...
0
votes
0answers
16 views

Does this operator produce an interpolating spline of it's argument?

I'm studying a bit of spline functions theory: In this book, chapter 6, a lot of error bounds are given when spline functions are used to approximate functions belonging to a specific function spaces. ...
1
vote
0answers
29 views

Geometric Meaning of Modulus of Smoothness

Could you explain me the "geometric meaning" of the following definition (it's taken from a book on spline functions theory)? Definition: Given $1 \leq p \leq \infty$ a positive integer $r$, and $...
2
votes
3answers
75 views

Curve-fitting using circles

I'm working for a firm, who can only use straight lines and (parts of) circles. Now I would like to do the following: imagine a square of size $5\times5$. I would like to expand it with $2$ in the $x$...
0
votes
1answer
27 views

C2 continuous interpolation on a 4-dimensional dataset

I am currently coding up a project where interpolations must be performed such that C2 continuity be preserved along the length of the whole set. The end result ought to look like a line (which will ...
1
vote
1answer
29 views

Terminology: Spline interpolation

I have read two different definitions of Splines: A differentiable piecewise polynomial. A piecewise polynomial. If I build a piecewise polynomial using cubic polynomials, it's ...
0
votes
0answers
41 views

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 ...
4
votes
1answer
53 views

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 $m$ ...
2
votes
1answer
150 views

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 ...
1
vote
1answer
49 views

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 ...
1
vote
2answers
36 views

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 ...
1
vote
1answer
34 views

Which Properties of a Natural Cubic Spline does the following function possess and not possess

I need to determine which of the properties of a natural cubic spline the following function possesses or does not possess: $$f(x) = \begin{cases} (x+1)+(x+1)^{3}, & x \in [-1,0] \\ 4+(x-1)+(x-1)^...
1
vote
0answers
52 views

Catmull-Rom: spline and filter

On this website, the author gives this definition for Catmull-Rom splines (slide 10): $$catmullRom(t) = \frac{1}{2}\left\{\begin{array}{ll} t^3 + 5t^2 + 8t + 4 & \text{if } -2 \le t \lt -1\\ -3t^...
1
vote
1answer
41 views

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 $S(-1)=5$, $S(0)=7$, ...