Questions on interpolation, the estimation of the value of a function from given input, based on the values of the function at known points.
2
votes
0answers
12 views
Minimal surface representation from a 3D contour
I have a set of 3D points defining a 3D contour, as shown below. The points in this contour lie in their best-fit plane and I want to obtain a 3D triangular mesh representation of the surface inside ...
0
votes
0answers
17 views
Ellipse radius interpolation with different radiuses
I am writing a library for graphical LCDs and I want to incorporate a function to draw a circle on the screen. I have already succeeded in drawing simple circles, however, I want to be able to pass a ...
0
votes
1answer
29 views
Coefficients of Newton interpolation polynomial
Given distinct $y_0,...,y_m$ in $\mathbb R$, let $N_m(x)$ be the Newton interpolation polynomial of degree $m$. That is, $N_m(x) = \sum_{n=0}^{m}a_nw_n(x)$ where $w_0 = 1$, $w_n(x) = ...
0
votes
0answers
11 views
Cubic interpolation in arbitrary dimension?
Consider a $N$-dimensional space discretized with a regular cubic grid of $n^N$ cubes, each cube containing the value of a function $f$ in its center. How to correctly interpolate $f(x, y, z)$ using ...
1
vote
0answers
29 views
Curve through four points — simple algebra??
The motivation for this is Bezier curves. But, if you don't know what these are, you can skip down to the last paragraph, where the problem is described in purely algebraic terms.
Suppose I want to ...
0
votes
2answers
30 views
Forcing Bezier Interpolation
I found this very informative site that discusses forcing bezier interpolation and the site gives formulae for calculating the control points so that the curve goes through a set of four points, y0, ...
0
votes
1answer
27 views
Proof of Schur's test via Young's inequality
I am able to prove the following generalization of Schur's test using the Riesz-Thorin interpolation theorem, however I have been stuck for days now trying to prove it using Young's inequality:
Let ...
0
votes
0answers
19 views
Are high dimensional cubic interpolation and cubic spline the same?
Hi I want to implement a 2d bicubic interpolation method. I checked the official matlab implementation code of imresize and interp2, and surprisingly found that bicubic interpolation method is ...
1
vote
0answers
20 views
Multigrid Interpolation and Restriction operators
I have a question about the restriction and the interpolation operators of a Multigrid algorithm.
Let those be given:
The full weighting restriction stencil (in 2D): $\frac{1}{16} \left[ ...
5
votes
0answers
55 views
Runge's phenomen: interpolation error using Chebyshev nodes oscillates
We're trying to approximate the Runge function $f(x) = \dfrac{1}{1+25x^2}$ using Chebyshev nodes. When calculating the interpolation error, using different degrees ranging from 0 to 50, we get the ...
0
votes
0answers
27 views
Non-linear interpolation. (1D Perlin Noise)
In this document (http://webstaff.itn.liu.se/~stegu/simplexnoise/simplexnoise.pdf) about Perlin (and Simplex) Noise you can find an explanation about 1D Perlin Noise (at the top).
...
1
vote
0answers
30 views
Interpolate to 3D functions
Hy,
In math classes, I've learned that given some points in 2D space:
a(1,2),
b(7,3),
c(8,5),...
You can find an equation that goes through these points (using interpolation).
Now I was wondering if ...
2
votes
1answer
47 views
Interpolation with degree restriction
(using $f[x_1, ... , x_n]$ to denote the forward difference operator)
I have a polynomial $P(x)$ interpolating $5$ points $x_0, ... , x_4$ and $2$ derivative values $x_0, x_3$ across an evenly spaced ...
1
vote
1answer
38 views
Extension of Fourier Transform
We know that Fourier transform $ \mathcal{F} : L^1 \rightarrow C_0 $ can be extended to $ \mathcal{F} : L^2 \rightarrow L^2 $ which forms a unitary isomorphism from Plancharel Theorem. Hence as for $ ...
2
votes
1answer
39 views
Nontrivial expansion of a multivariate power series in form of a single variable series?
I am trying to interpolate a function defined over a three-dimensional real space:
$$f: R^3\rightarrow R\\(x,y,z)\rightarrow f(x,y,z)$$
Let assume I have $N_1 N_2 N_3$ points in the space which form ...
1
vote
0answers
17 views
Is there a way to estimate the range of fitting coefficients from only the data?
Considering an approximation $f$ for a set of $N$ data points $(x,y)$ using, for example, $M$ radial basis functions at arbitrary sites in the domain
$f_i = \sum_{j=1} ^M c_j\phi(||x_i-x_j||)$
where ...
1
vote
2answers
48 views
Easing function, constant velocity then decelerate to zero
I'm trying to write an interpolator for a translate animation, and I'm stuck. The animation passes a single value to the function. This value maps a value representing the elapsed fraction of an ...
2
votes
1answer
46 views
Is there a nice way to interpret this matrix equation that comes up in the context of least squares
So I am working on this problem with fitting a second degree polynomial of the form $y=a_1x^2+a_2x+a_3$ to four points using least squares. One of the parts of the problem is to write out the matrix ...
0
votes
2answers
31 views
Finding t in basic geometric interpolation
I have the following problem related to interpolation and was wondering if anyone had any ideas? In the following diagram, there exists known-points: A, B, C, D, E, unknown points F, G.
I'm trying ...
3
votes
1answer
43 views
Linear, Bi-linear or better
I have been writing some code to do some interpolation of 2D data on an irregular grid. So far what I have done is:
Triangulate the known points using Delaunay.
Find the vertices of the triangles ...
0
votes
0answers
24 views
Sinusoidal interpolation
I am new to the concept of interpolation, and know only how to interpolate with a polynomial function. What if you suspect a sinusoid will be a more accurate fit? What method should be used?
0
votes
1answer
74 views
How does one interpolate between polar coordinates?
I'm finding that when I try to use the standard methods of interpolation in polar space, the result is not what I would expect. For example, when interpolating between the following polar coordinates:
...
1
vote
1answer
20 views
What are the explicit expression for this interpolation problem
We want to fit
$f(x) = a_0 + a_1 *x + a_2 * x^2 + ... + a_n * x^n$
to the data $(x_i,f(x_i))$ for $i = 0 ... n.$
It will give rise to the following system
$ A a = b $
Here $ a = [ a_1 a_2 a_3 ...
0
votes
0answers
19 views
Coefficients of Bi-cubic interpolation using Lagrange equation
I am trying to implement Bi-cubic interpolation. For this I need to calculate 16 coefficients.
According to this article, we can calculate the coefficients using Lagrange equation.
How can Lagrange ...
0
votes
1answer
39 views
Triangle $z$-index interpolation between the vertices
I got a $2$D triangle, each vertex has a $2$D coordinate with a $z$-index value (NOT a $z$ coordinate!). The $z$-index value indicates whether a vertex lays on, in front of, or behind your screen ...
-1
votes
2answers
57 views
Prove $f(A^T)=f(A)^T$ for a matrix $A$ [closed]
As the title says, I need to prove $f(A^T)=f(A)^T$ for a matrix $A$. (where $T$ is the transpose)
I believe the proof involved the fact that an interpolation polynomial $r(A)=f(A)$ and then I must ...
0
votes
1answer
61 views
B-Spline Interpolation/Approximation
I've got a couple of probably very simple questions, yet some googling didn't bring up what I was looking for.
First what I want to do: I have a grid, and the gridpoints are function values. I want to ...
1
vote
1answer
44 views
What is the way to determin how good a sequence will interpolate?
Say I have to sequences of numbers:
$$[5, 10, 14, 21, 27, 31]$$
$$[1, 20, 21, 22, 30, 31]$$
Even though they both get to $31$ by the $6$th element, logic tells me that only the first one is a good ...
1
vote
0answers
41 views
What are the constants in the relationship between point density and median point distance?
Given $\rho$ particles uniformly distributed on a plane within a unit square ($\rho > 1$), each particle has another particle that is closest to it; the median of those nearest distances is called ...
2
votes
1answer
63 views
How to make 3D object smooth?
I want to make the below picture into an egg with smooth surface. For the implementation in Mathematica, please, see this thread here. This thread considers mathematical methods to achieve the goal ...
0
votes
1answer
25 views
nth degree interpolating polynomials
Given five ${x}$ points and for four of them I know ${f(x)}$ values.
So I have to use interpolating polynomial to estimate unknown ${f(x_i)}$
${ \displaystyle f(x) = \sum_{j=1}^{n} P_j{(x)} }$ with
...
6
votes
1answer
65 views
Cauchy Integral Formula for Matrices
How do I evaluate the Cauchy Integral Formula $f(A)=\frac{1}{2\pi i}\int\limits_Cf(z)(zI-A)^{-1}dz$ for a matrix ...
1
vote
1answer
28 views
interpolation question. From MyMaxScore free AP BC Calculus exam.
First, I disagree with the answer sheet. That is why I am posting this question here.
The question: Part B, Question 3a.
The answer given (Sorry for the picture being so small).
I know ...
2
votes
1answer
53 views
A Question About Linear Interpolation
So lets say I have two points $A=(x_1, y_1, z_1)$ and $B=(x_2, y_2, z_2)$. $A$ and $B$ are each associated with some scalar value $K_1$ and $K_2$. $K_1$ is negative and $K_2$ is positive and all the ...
1
vote
2answers
43 views
Interpolating multivariable functions
Assume I have two functions, $f_1$ and $f_2$, that both depend on $x$ and $y$, so that $f_1(x,y)$ and $f_2(x,y)$.
I don't know the exact functions, but know values of each function at some points ...
0
votes
0answers
26 views
What do you call generating a function out of a graph?
In many physical phenomena, laws an relations to their variables are somehow interpolated (example by statistical data or samples) and then an approximate set of functions are generated to work ...
1
vote
0answers
56 views
Analogue for finite sums of $$\int_{a}^{b}fg=t\bar{f}\bar{g}(b)-t\bar{f}\bar{g}(a)+\int_{a}^{b}(\bar{f}-f)(\bar{g}-g) (*)$$
Please help me to find an analogue for finite sums of
$$\int_{a}^{b}fg=t\bar{f}\bar{g}(b)-t\bar{f}\bar{g}(a)+\int_{a}^{b}(\bar{f}-f)(\bar{g}-g) (*)$$
where $ \bar{f}(t)=\frac{1}{t}\int_{a}^{t} ...
0
votes
1answer
68 views
Understanding Lagrange error.
Here is an example from my Numerical Analysis book (Burden & Faires). Trying to understand Lagrange error, but I do not understand the statements in bold.
In example 2 we found the second ...
0
votes
1answer
74 views
How to do interpolation using the newton basis?
I have these data points.
$f_i(x)= \{10, 11, 14\}$
$x_i= \{0, 1, 3\}$
then the basis functions are.
$\pi_0 = 1
\\\pi_1=(x-x_0)=x
\\\pi_2=(x-x_0)(x-x_1) =x(x-1)$
So the matrix will become.
...
1
vote
2answers
49 views
What is the equation stands for in geometry(intuitively)?
I am writing a bilinear interpolation method.
This method can be abstract by solve the equation A*x = b, A is a 4x4 matrix below:
$A=\begin{pmatrix}
1 &x_1 &y_1 &x_1y_1\\
1 ...
1
vote
2answers
61 views
Finding a simple spline-like interpolating function
I am looking for a continuous function $y=f(x,\alpha)$ for the interval $0\le x \le 1$ such that $0\le y \le 1$ and $y(0,\alpha)=0$ and $y(1,\alpha) = 1$ and $y(\alpha,\alpha) = 1-\alpha$ and ...
0
votes
1answer
62 views
finding derivative at intermediate point of known data set
I have a function $y = f(x)$, $ x \in [0,1] $ and $ y \in [0,1]$
Set of values $(x_i,y_i)$ are known for n points. I need to find derivative at point $x_{\zeta}$ such that $y(x_{\zeta}) = 0.5$
Now ...
1
vote
2answers
161 views
Cubic spline interpolation not producing an interpolant with continuous first derivative consistently
I have coded a nice cubic spline interpolator following the basic methods laid out here http://people.math.sfu.ca/~stockie/teaching/macm316/notes/splines.pdf . My program reproduces the example laid ...
0
votes
1answer
73 views
formula for an upwards-sloping convex curve with known endpoints
For a project I am working on, I need a formula that can describe a curve between two known endpoints, where the curve will always be upwards sloping and always convex (or flat). There should be many ...
2
votes
0answers
70 views
B-Spline Definition
I'm currently working on my master's project. For this, I rely on one PhD-thesis in which I found a statement I do not understand. Unfortunately, the author hasn't answered to my mails yet, so I have ...
0
votes
1answer
37 views
Closest edge to a point
I have a tricky question and can't seem to find a good solution. Let's say I have 3 convex polygons (ABCD, ADEF, BHGC) that share 2 common edges, unknown J, and a known point I:
I want to find the ...
7
votes
3answers
534 views
Is it possible to have a rule which generates: 2, 4, 6, 8, 10, 12, 14, 16, -23?
This is on Lagrange Interpolations . . .
Is it possible to have a rule which generates the sequence: 2, 4, 6, 8, 10, 12, 14, 16, -23?
The hint that he gave us is to use Summation Products, the only ...
3
votes
4answers
72 views
Profinite and p-adic interpolation of Fibonacci numbers
On the topic of profinite integers $\hat{\bf Z}$ and Fibonacci numbers $F_n$, Lenstra says (here & here)
For each profinite integer $s$, one can in a natural way define the $s$th Fibonacci ...
0
votes
2answers
44 views
Interpolating point outside quad
I'm trying to interpolate a destination point $(x,y)$ outside of a $4$ point convex polygon given a $4$ point source polygon with known values. It's sort of like inverse bilinear interpolation, except ...
-4
votes
2answers
249 views
Please help to find function for given inputs and outputs [closed]
Can you help with finding the formula for these input and output values?
When $n=1$:
$f\left(1,1\right)= 0.0000000000$
When $n=2$:
$f\left(1,2\right)= 0.0000000000$
$f\left(2,2\right)= ...
