Questions on interpolation, the estimation of the value of a function from given input, based on the values of the function at known points.

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3
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2answers
58 views

Shamir's secret sharing interpolation problem

I try to understand this protocol - Shamir's secret sharing - threshold scheme. I got my data and I made interpolation basing on examples published on Wikipedia. You can see them below (sorry, I am ...
0
votes
1answer
28 views

linear or bilinear interpolation

I want to know how to use linear and bilinear interpolation in 2D. Specifically the pairs $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$, and $(x_4,y_4)$ are given in a quadrilateral. In this case how to ...
1
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3answers
24 views

Constructing Polynomial Function from Set of Points and Slopes

I only have a basic knowledge of calculus but I would like to know if it's possible to, given a set of points each with their own slopes, construct the simplest (or any) polynomial function that ...
0
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0answers
13 views

How to use bilinear interpolation on 2d axis?

In my application i find 4 nearest points on the grid $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$, $(x_4,y_4)$ for a signal that detected with unknown location using knn. Each of these points have a ...
0
votes
1answer
19 views

How to use bilinear interpolation?

I need an explanation about Bilinear Interpolation. I use KNN and find $4$ points which I need to use bilinear interpolation to find unknown position. I was unable to understand explanations in other ...
0
votes
1answer
386 views

Spline with varing tension, selection of tension factor

I need to perform a special interpolation, using that kind of basis : $$\varphi_{i,j}(x) = a_i + b_ix + c_i(\cosh(\tau\ x) - 1) + d_i(\sinh(\tau\ x) - \tau\ x)$$ where the $a_i$, $b_i$, $c_i$ and ...
0
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2answers
13 views

Formula for $N$-Dimensional linear interpolation

Linear interpolation between values $A$ and $B$ can be defined as: $f(x) = A(1-x)+Bx$ Bilinear interpolation between values $A,B,C,D$ is defined as: $f(x,y) = g(x)(1-y) + h(x)y$ where $g(x) = ...
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0answers
22 views

How to select the number of nodes in a spline interpolation?

I am writing a program to test the precision of different methods for imputing missing data in a time series. One of the methods I am going to test is a natural cubic spline interpolation. I'll be ...
0
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0answers
34 views

Number of points for local spline interpolation

I have a large scattered data set of (x,y = f(x)) points and I want to interpolate them to regularly spaced grid points. To do this I have chosen to use cubic splines as my interpolation method. The ...
2
votes
1answer
16 views

Interpolate between 3D plane and 3D hemisphere

I have a simple 3D plane whose points (different $x, y$ values, but all $z = 0$) need to be mapped to 3D Cartesian coordinates in order to form a hemisphere. However, I also would like to be able to ...
3
votes
1answer
759 views

Akima spline interpolation

I want to use Akima interpolation on series of points. I have those points in 3D [x, y, z]. But in all resources, I found, there is only f(x) and x (so [x,y]). In Natrual Cubic Spline I am using this ...
2
votes
1answer
34 views

Does $\Vert f-s_n \Vert_\infty \to 0$ still hold for $f\in C^0[a,b]$?

If $f\in C^2[a,b]$ and $s_n$ its piecewise linear interpolation at points $x_0, \ldots, x_n$ with $h_n = \max_{j=0,\ldots,n-1} (x_{j+1}-x_j)$ then one can show that $$\Vert f-s_n \Vert_\infty \leq ...
0
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0answers
12 views

When would a Fourier Product (made up term) exist for a finite sequence of the form $C_{\text{Max}}\prod _{i=1}^k A_i \cos \left( B_i n\right)$

Let us say that we are given a finite list of points of the form C = {i,$x_i$} where i goes from 0 to the card(C) that when plotted in the Euclidean plane has some vertical axis that splits the graph ...
0
votes
1answer
465 views

Plane fitting using svd

I am trying to get a best fit plane in a 3d space of points. I am using an svd as described in http://stackoverflow.com/questions/10900141/fast-plane-fitting-to-many-points. If I use the data provided ...
0
votes
1answer
24 views

Cubic Uniform BSpline surface interpolation

I want to understand cubic BSpline surface( very hard for me to figure out). I prefer matrix form which presented here. Equation 4.12 in page 33, describes how data point should be presented ...
8
votes
3answers
389 views

A Curious Binomial Sum Identity without Calculus of Finite Differences

Let $f$ be a polynomial of degree $m$ in $t$. The following curious identity holds for $n \geq m$, \begin{align} \binom{t}{n+1} \sum_{j = 0}^{n} (-1)^{j} \binom{n}{j} \frac{f(j)}{t - j} = (-1)^{n} ...
0
votes
4answers
46 views

Interpolation between 2 points on the perimeter of a circle?

I'm trying to produce movement on a unit circle from one point to another in equal increments, but I'm having trouble doing this without the use of angles (which isn't an option). Given 2 points on a ...
1
vote
1answer
796 views

How to evaluate Newton's Divided Difference Polynomial in MatLab with an unknown degree?

I already have the code that finds the coefficients for the polynomial, but how do you find a value for the polynomial if given an x coordinate in MatLab code?
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0answers
35 views

How to proceed with this simple proof?

If $$\alpha_k = \sum_l a_l \ \ g((k-l)T-l\Delta T)$$ $$s_k = \sum_l \alpha_l \ \ q((k-l)T+k\Delta T)$$ where $a_l \in \pm1$ and $g(t) = \frac {\sin(\pi t/T)}{\pi t/T}$ and $q(t) = \frac {\sin(\pi ...
2
votes
1answer
37 views

What is the best way to interpolate over the 25th and 75th percentile of SAT scores?

The problem: I know the 25th and 75th percentiles of SAT scores for students admitted to a given university, and I want to interpolate over those two points in order to estimate all the percentiles ...
1
vote
3answers
128 views

Get equation for a curve which intersects x at seemingly randomly distributed points?

Is there any type of function that when graphed would show a curve which intersects the x axis multiple times, with each point being an arbitrary distance from the last? I mean, not like a trig ...
1
vote
4answers
227 views

How to find the 4th degree polynomial with given values at $0,1,2,3,4$?

Determine a fourth degree polynomial p that has $p(0), p(1), p(2), p(3), p(4)$ equal to $7, 1, 3, 1, 7$, respectively. Using my ideas, I first write out the points on the polynomial as $(0,7), (1, ...
2
votes
0answers
69 views

C++: Library to interpolate polynomial and find a polynomial roots [migrated]

I need to know what library, compatible with C++, can be used to interpolate a polynomial. So given $n$ point-value pairs it can recover the polynomial. The library must support big integer ...
0
votes
1answer
317 views

Reconstruct Control points in a Bézier Curve?

I have a curve that I know is a (non-periodic) Cubic Bézier Curve (because I constructed it as such). I stored each ordered pair in the curve, but not the control points. Is it mathematically ...
1
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1answer
374 views

Newton backward interpolation in Mathematica

I have the following task: Create a function (in Wolfram Mathematica), called $\mathrm{NewtonBackward}$[n_,x0_,h_,f_] which interpolates backwards the function $f(x)$ with nodes {x_i = x_0 + ...
0
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1answer
19 views

Knowing two Vectors, and the distance to a 3rd, how to get the 3rd

If i know the two Vectors v1 and v2, which discripe points in a 2D space, and i also know that a vector v3 is on the line segment between v1 and v2, how can i get the x and y coodinates of v3 if the ...
0
votes
0answers
28 views

Interpolating a set of GPS points on ellipsoid earth model.

I have a set of GPS (latitude, longitude) co-ordinates along with the time at which the coordinates were collected. Additionally, I have the speed and the heading of the vehicle at those coordinates. ...
0
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0answers
19 views

periodic radial basis function

A have a point cloud ,described in spherical coordinates, which I need to fit with a smooth surface. I'm trying to do this with a bivariate radial basis function network, which operates on a spherical ...
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0answers
13 views

Interpolation of jacobians for point wise defined transformation

Here is my question: let's say that I have a transformation function T from the image A to the image B which is pointwise defined. That is, T(x) = J, where J is the jacobian representing the ...
1
vote
2answers
58 views

Positive linear combinations of intervals

Given two intervals at $i\in\{0,1\}$ $I_i=[-a_i,a_i]$ where $0<a_0<a_1=1-a_0<1$ and a third interval $I=[-a,a]$ where $0<a<\frac{1}2$, when is there an $\alpha,\beta\in\Bbb R$ such that ...
1
vote
3answers
19 views

How can I characterise the error of an interpolated surface?

I am writing a program in which I can interpolate and display a surface by kernel interpolation. Lets say I interpolate a function $f(x)$ by the function $f^*(x)$. Clearly the error at any given point ...
0
votes
2answers
438 views

How to find Chebyshev nodes

I want to use Chebyshev interpolation. But I am a little confused for finding Chebyshev nodes. I use the following figure to illustrate my problem. Consider I have a vector of numbers I depicted as a ...
0
votes
1answer
47 views

Need a formula / method to get a value between 0 and 1 if a point lies in an area between two rectangles

I'm trying to figure out a way to construct a formula or method for the following process: If Point E is inside the smaller rectangle (red area), the value should be 1. If Point E is outside the ...
0
votes
2answers
373 views

2D cubic B-splines

I have been looking at B-splines to interpolate points. Having 1-D B-splines makes perfect sense to me, but haven't been able to find something that explains 2-D B-splines well for me nor provide me ...
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0answers
6 views

Finding a point using weighted points - creating a weights (camera) system

I'm making a game and trying to create a very specific camera system. I want to have Camera Weights (CWs) around the 2D level, and a circle-cast around my main character. The circle cast will find ...
0
votes
1answer
33 views

Continuously differentiable interpolation

I have real values $y_i$ given on uniform grid. I want to build interpolating function $f(x)$ such that: $f(x) = y_i$, when $x=i$, $f$ is continuously differentiable. Instead of using famous ...
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0answers
11 views

Error estimation - spline interpolation

I got a question regarding error estimation and spline interpolation. I got a parabola shaped graph that I've used spline interpolation on to get more accurate data. I've used a much smaller step on ...
0
votes
1answer
55 views

B-Spline: How to generate a closed curve using Uniform B-Spline curve?

Given $n+1$ control points; $P_0,P_1,...,P_n$ (where all are 2-dimensional points), and $k$ (which defines the order of the polynomial, and hence its degree; $k-1$) the B-Spline curve is defined by: ...
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vote
1answer
20 views

Explicit Bezier Curves: Lerping between curves same as lerping control points?

Let's say that you have two explicit (one dimensional) quadratic Bezier curves: $f(t) = A(1-t)^2+B(1-t)t+Ct^2$ $g(t) = D(1-t)^2+E(1-t)t+Ft^2$ Where $A, B, C, D, E, F$ are scalar constants. Then, ...
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0answers
111 views

Centripetal Catmull–Rom spline

What is "t" in this short and simple example below? There are 4 points Pn[xn,yn] in 2D space: A[1,6] B[3,1] ...
0
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1answer
22 views

How to spline together Bezier curves to form a smoth closed curve?

Given $k=m\cdot n$ points: $P_1,P_2,...,P_k$ (all points are two dimensional points), how can I spline together $m$ Bezier curves of $n$ degree to form a smooth closed curve? Denote $B_{i,j}(t)$ to ...
2
votes
1answer
24 views

Can Runge's approximating rat. fns. be required to take certain prescribed values?

Suppose $f$ is analytic on an open set $U$ containing the compact set $K$, and $\{r_n\}$ is a sequence of rational functions provided by Runge's theorem (having poles in some prescribed set $A$). For ...
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2answers
24 views

How to interpolate values when x-values are ranges, not numbers?

I am given this information about the incidence rate of stroke (per 1,000) for males for these age groups: 18-44: 6 45-54: 19 55-64: 35 65-74: 64 However, I need the incidence rates in these age ...
0
votes
1answer
25 views

Partition of function into pieces for interpolation needs

I've got some experimental data obtained from my mate's research. There are two sets of (x,y) points for each curve. He asked me to interpolate function values between this points, so for each curve I ...
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0answers
62 views

proving linear interpolation of Level Set

I tried to explain figure below in mathematics form. As you can see I have got triangle (v1, v2, v3). The signed shortest distance form red interface (level set value) is calculated for each vertex ...
1
vote
1answer
20 views

Explicit formula for interpolating polynomial

$a\in(0,1)$ is fixed. $M\in\Bbb Z_{>1}$ is fixed. What is $f(x)$ given that $$f(0)=0\mbox{, }f(M)=1+a\mbox{, }f(1)=1-a$$$$\mbox{ }f(x)\in(1-a,1+a)\mbox{, }\forall x\in(1,M)?$$ What is $g(x)$ ...
2
votes
1answer
3k views

Interpolation of a logarithmic function

I have a logarithmic function $$m \ln(x) + b$$ And three points $$(x_0, y_0), (x_1, y_1), (x_2, y_2)$$ The task is to find $m$ and $b$. Do I understand right that the third point is redundant? ...
8
votes
3answers
11k views

Newton's Interpolation Formula: Difference between the forward and the backward formula

I was taught that the forward formula should be used when calculating the value of a point near $x_0$ and the backward one when calculating near $x_n$. However, the interpolation polynomial is unique, ...
0
votes
0answers
16 views

Minimum of a cubic fitted to two points and their derivatives

I'm trying to understand a line search method used to find a step length in a minimsation algorithm. There is an interval $[a, b]$ containing desirable step lengths and there are two previous ...
2
votes
0answers
108 views

Chebyshev Interpolation and Expansion

I am seeking connections between pointwise Lagrange interpolation (using Chebyshev-Gauss nodes) and generalized series approximation approach using Chebyshev polynomials. Pointwise Lagrange ...