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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1answer
24 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 ...
1
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4answers
2k views

Smooth transition between two lines (2d)

I have function that is defined as $$ Y = \frac{1}{15} x \longrightarrow {\rm if}\qquad 0 \leq x \leq 30 $$ $$ Y = \frac{1}{70} x + \frac{11}{7} \longrightarrow {\rm if}\qquad x > 30 $$ The ...
1
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3answers
43 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

Lagrange polynomial's maximum degree

How do I prove that $$P_j(x) = \sum_{j = 0}^n \left( y_j \prod_{k = 0}^n \frac{x - x_k}{x_j - x_k} \right)$$ is a polynomial with degree at most $k$?
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0answers
11 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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0answers
10 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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0answers
18 views

Degree of Multilinear interpolation

Supposing you want to interpolate an $n$-variate polynomial on $\{0,1\}^n$, we could take the polynomial to be linear in each coordinate. What is a good interpolation procedure for this that will give ...
0
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1answer
30 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 ...
0
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1answer
19 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 ...
1
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1answer
29 views

Interpolation with a constrained range between given control points

I am trying to create an algorithm that creates smooth color gradient functions, given control points in the red, green, and blue components. Mathematically, each curve would have a domain [0, 1] ...
3
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1answer
703 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 ...
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0answers
10 views

Root finding of a Hermite interpolating polynomial

Consider a Hermite interpolation problem. I have an approach for obtaining the roots of interpolating polynomial. I would like to present an example for this approach. Can you suggest me an applicable ...
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0answers
27 views

Curve Fitting - When to use Interpolation and/or Best-Fit?

This is my first time posting on Mathematics Stack. Nice to meet you all. I have a question regarding curve fitting, interpolation, and best-fit approximation. My supervisor wants me to write a ...
1
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1answer
742 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?
1
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2answers
164 views

How to calculate a spline for points in general position?

I want to find a curve passing through (or near) $n$ points in the plane. The catch is that the curve need not be a function. That is, a vertical line might pass through the curve in more than one ...
1
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0answers
33 views

How can I interpolate the following data and get the equation?

I want to interpolate between three vectors which have 8 rows in each. For eg. X = [1 2 3 4 5 6 7 8], Y = [7 8 9 10 11 12 13 14 ], Z = [15 16 17 18 19 20 21 22] I want to find the equation Z = ...
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0answers
22 views

Computing integrals in order to find an approximation function

For a project in scientific computing I am trying to find an approximation of an unknown function $f(x)$. Given: data points $(x, f(x))$ A basis with which we can approximate $f(x)$ consists ...
0
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1answer
39 views

How to transform function values to specific interval

I'm doing a project at university about scientific computing and I'm stuck. As in: I seem to lack quite a bit of mathematical background for this project. The program has as input an array of $x$ ...
7
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3answers
58 views

Where did the idea of hermite interpolation came from?

I am given the Hermite interpolation formula directly in my text book without ANY explanations about how it was first made (obviously it was somehow constructed for the first time with some sort of ...
0
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2answers
32 views

What interpolation method makes the fewest assumptions about the function?

If I have no information about a function except a regular timeseries of samples, what is theoretically the best interpolation method to use and why? We cannot assume the function is continuous etc. I ...
0
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1answer
327 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
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1answer
25 views

Interpolating discrete points along a spline

I have 2 lines which i need to connect with a spline (or some other curve, but not the circle; it has to gradually increase its turning angle). The lines cross each other so the curve should make a ...
0
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1answer
25 views

Error of lagrange interpolation

If the original function I want to approximate using Lagrange interpolation is a polynomial the error function $(x-x_{0})...(x-x_{n})\frac{f^{(n+1)}(\xi)}{(n+1)!}$ is not working because the $n+1$ ...
1
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1answer
344 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 + ...
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0answers
10 views

What is the difference between newton interpolation polynomial and interpolation polynomial with Neville scheme?

I am trying to find the interpolation polynomial by using Neville scheme. It looks like divided difference . What is the difference between newton interpolation polynomial and interpolation polynomial ...
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1answer
2k 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? ...
3
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0answers
29 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 ...
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2answers
18 views

Interpolate/Increment Vector Rotation

For my 2D physics engine, I'm using the unit vectors of the direction an object is facing to represent its orientation; essentially, [Cos(theta),Sin(theta)] where theta is the object's rotation in ...
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0answers
12 views

“Sigmoid” function with tunable initial slope, upper asymptote and transition period

I'm looking for a function which resembles the transition between the function $f(x)=x$ for small $x$ and the function $f(x)=C$ for large $x$ ($x$ is finite and $\geq 0$). I've found the generalized ...
0
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1answer
47 views

How to find a graph's equation from its points

I have a set of data that constitutes the graph on the picture. What I want to know is how would I find the equation equivalent to that kind of graph? The X are on the interval $[1,10]$.
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1answer
22 views

Express interpolation of a polynomial in other polynomial

Let $x_0$ and $x_1$ be two distinct nodes. Let $P(x)$ be a polynomial of degree 2 or less such that : $P_0' (x_0 ) = f' (x_0 )$ , $P (x_1 ) = f (x_1 )$ , $P_0' (x_1 ) = f' (x_1 )$ , and $P (x) = f' ...
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0answers
29 views

Selecting nodes for spline interpolation

Is there a general method to determine the best sample points for spline interpolation (whether for piecewise linear or piecewise cubic Hermite) given $x$, $f(x)$, and estimating $f^\prime(x)$? Does ...
6
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3answers
9k 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, ...
1
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1answer
20 views

Bound for Lagrange interpolant

I'm having difficulty figuring out how the bound in the below example is determined. Consider the function $f(x)=e^{-x}$. For sample points $\{-1,1\}$, the Lagrange basis interpolants are ...
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0answers
33 views

$L^2$ vs $L^{\infty}$ norm for interpolation

Under what circumstances should I consider utilizing the $L^2$ norm instead of $L^{\infty}$ when interpolating a function based on sample points? Probably related: Does the answer significantly ...
0
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1answer
304 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 ...
0
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2answers
23 views

Interpolation using rate of change

I have a set of data with missing points, which I estimated using spline interpolation. I've now been given the rates of change at each data point. How will this change/improve my current ...
2
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0answers
32 views

From a set of vertices, find smallest polytope enclosing another point

Out of a set of vertices $V=\{\vec v_i\in \mathbb R^D\}$, I am constructing a piecewise linear interpolating function $f:\mathbf{conv}(V)\rightarrow R$ as follows: given a point $\vec d\in ...
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0answers
14 views

Empirical Mode Decomposition: How to make a suitable spline interpolation when the number of extrama is small?

I am doing an EMD(Empirical Mode Decomposition) project and I am getting a problem with the spline step (find the upper and lower envelopes) because there is always a big deviation in spline ...
0
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1answer
39 views

Reverse spline interpolation

Say I have a number of sets $(x, y)$ for $x \in \{0, 1, \dots, 255\}$. I want to find the least number of points to reproduce the set with a certain accuracy using linear interpolation. What is the ...
2
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4answers
131 views

Find a function with certain requirements

I'm trying to find a function $y=f(x)$ that can be described as follows: $f(x) = g(x) + c/(x-x_a)$. With $f(x)$ I want to design a function with the following properties: $f(0) = 0$; $f(x)$ has a ...
0
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2answers
29 views

Help to find the best lower bound function for a given set of data, based in the natural logarithm function

I am trying to find a lower bound function for a set of data I have, and I am struggling with it. In the following graph the blue color is the set of data and the red color is my lower bound function. ...
0
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1answer
302 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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1answer
28 views

Mathematics and Algorithms for Interpolation

I am doing some programming, where I am interpolating point a to point b, against a timer that is constantly incrementing by ...
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1answer
30 views

What to do when the chebyshev point is equal to data point in lagrange interpolation?

I am going to use Lagrange interpolation using Chebyshev nodes using the following formula $$\sum_x \prod_{k=0,k\not={j}}^n \frac {x-y_k}{y_j-y_k} f(x) $$ in which $x$ in my data points, $y_k $s ...
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0answers
28 views

How do I derive the analytical form of a discrete wavelet transform?

I guess this is more of an "applied maths" question than pure maths, and here's to hoping this is the right forum :) I am using a fast discrete wavelet transform (DWT) of a 1D vector of $2^N$ numbers ...
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0answers
18 views

Combining samples for interpolation

I'm writing a program where values at a position in a 3D field should be estimated, based on a number of existing samples. In this case it is the density of a point cloud at different positions in ...
9
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1answer
6k views

What is the difference between natural cubic spline, Hermite spline, Bézier spline and B-spline?

I am reading a book about computer graphics. It is confusing about the various splines and their algorithms. What is the difference between natural cubic spline, Hermite spline, Bézier spline and ...
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0answers
18 views

Increase number of points after interpolation

I have a set of points which would give a curved line.I want to increase the number of points on the curve. eg- In the diagram, the original points are shown in highlighted black. If I interpolate ...
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0answers
82 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 ...