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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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 ...
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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 ...
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1answer
26 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 ...
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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 ...
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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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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 ...
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33 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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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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 ...
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3answers
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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 ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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1answer
54 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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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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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 ...
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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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110 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] ...
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2answers
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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 ...
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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 $g(x)$ ...
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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 ...
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23 views

Find the clamped spline with the initial and final derivative are equal, and the derivative at $x = 3$ is set to $0$.

The set of $x$ and $y$ are $x=\{0,1,2,3,4,5\}, y=\{0,1,2,3,4,5\}$ I want to do a clamped spline with the two end derivatives equal and the derivative at $x=3$ is $0$. I tried splitting the data into ...
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1answer
34 views

$\left( 1 - \frac{1}{n} \right)\left( 1 - \frac{2}{n} \right) \cdot … \cdot \left( 1 - \frac{k-1}{n} \right) = \frac{n!}{n^k r! (n-k-r)!}$

I'm trying to understand a proof in "Interpolation and Approximation by Polynomials" by Phillips. Let me quote (page 253): "For $k\geq 1$ we begin with $$B_{n+k}^{(k)}(f;x)=\frac{(n+k)!}{n!} ...
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1answer
45 views

Show that S is a cubic spline (natural or clamped)

Please see question. I believe the answer should be: $S_0(2)=\frac12(x^3-3x+2)=2$ $S'_0(2)=\frac12(3x^2-3)=\frac{9}{2}$ $S''_0(2)=\frac12(6x)=6$ $S_1(2)=\frac12(x^3-12x^2+45x-46)=2$ ...
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39 views

prove this using lagrange and newton divided difference error!

suppose f(x) is polynomial with degree of three.prove $f[{x}_{0},{x}_{1},{x}_{2}] = \frac{1}{2}{f}^{(2)}(\frac{{x}_{0}+{x}_{1}+{x}_{2}}{3})$ and ${x}_{0},{x}_{1},{x}_{2}$ are distinct point. I ...
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37 views

Proof of Hunt's Interpolation

I'm new to weak $L^p$ spaces and I'm doing a book exercise. Can someone enlighten me on the proof of the Hunt's interpolation theorem, which goes as follows: Theorem Let $\langle \,M, \mu \, ...
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Does the condition $S \in C^2[x0, x2]$ lead to a meaningful solution, when conctructing quadratic spline

we have three data $x_0 , x_1 , x_2$ we want to find the quadratic interpolation . $$S_0(x) = a_0 + b_0(x − x_0) + c_0(x − x_0)^2 on [x_0, x_1]$$ $$S_1(x) = a_1 + b_1(x − x_1) + c_1(x − x_1)^2 on ...
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cubic interpolation and data on a straigh line

Suppose the data we want to interpolate lie on a straight line. What can be said about the natural cubic spline ? i think i should show that the cubic spline in fact becomes a line . so if the spline ...
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1answer
18 views

Given an interpolating polynomial, how do I find another polynomial that interpolates at 1 less point?

The polynomial $$p(x) = x^5-2x^4-5x^3+15x^2+4x-12$$ interpolates x = -2,-1,1,2,3,0 with p(x) = f1, f2, f3, f4, f5, -12 respectively. In general, how do I find another polynomial of a lower degree ...
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25 views

how to learn the fast method of finding the cubic splines

in finding cubic splines if we have n points we get system of equations of magnitude 4(n-1) in the naive approach . in more sophisticated approach one only solves a system of equations with (n-1) ...
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solving $c_j$ system of equations for cubic splines?

the problem is like this : There are $N$ points $(x_0,y_0),(x_1,y_1),\dots,(x_{N-1},y_{N-1}) \in \mathbb{R}^2$ where $x_0 < x_1 < \cdots < x_{N-1}$. Cubic spline interpolation should give ...
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1answer
75 views

What is the difference between nodes and knots in interpolation?

I have been reading literature about polynomial interpolation (Lagrange) where the principles are described around nodes. The literature I have read about spline interpolation, however, talks only ...
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1answer
59 views

Calculating the x, y coordinate a set distance between two points

I'm trying to calculate the x and y coordinates that are a set distance between the coordinates of two pixels in an image. For example, if I travel from my original location (x1=4, y1=3) to a new ...
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3answers
48 views

Parameterizing cliffs

I am looking for a function $f(x; \alpha, X_1, X_2, Y_1, Y_2)$ that has the following property: For $\alpha=0$ it behaves linearly between $(X_1, Y_1)$ and $(X_2, Y_2)$, and as $\alpha$ gets closer to ...
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0answers
15 views

$(x,y)$ points extrapolation

My process is generating $x,y$ points. The points are result of tracking movement of an object so its a bit inertial. The tracked object can't change direction rapidly. I need to extrapolate data ...
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16 views

Re-initialize a 3D spline surface using different control points

I have a 3d spline surface is that is modeled with 65 points where the x,y,z position of each point is known. I want to keep the surface shape, but have different x,y positions for my control ...
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1answer
63 views

The Gherkin (an egg shaped building) - equation for the curve in order to calculate the surface area of revolution

I am trying to calculate the surface area of revolution for The Gherkin, an egg-shaped building in London, UK. Not sure about how to obtain the equation of the curve but I have the data points that ...
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1answer
22 views

What is the physical meaning of 2 nodes being same while fitting an interpolating polynomial?

When we are trying to find out constants for Newton's interpolating polynomial, we use divided difference method to find the constants. Then we have Hermite-Genocchi formula to find those constants ...