# Understanding Linear Interpolation in 2D

• Bucky
In summary: This method is used to interpolate values in two dimensions, and can be expanded to three dimensions by adding another dimension and adjusting the weights accordingly. In summary, the conversation discusses the implementation of a linear interpolation method presented in a research paper. The method is used to find the value at a point (x, y) between two given points by taking a weighted average of the values at the four surrounding points. The weights are adjusted for three dimensions.
Bucky
Hi, I'm implimenting a method presented in

http://www.dgp.toronto.edu/people/stam/reality/Research/pdf/GDC03.pdf

Part of the way through it it presents a linear interpolation method, which I wanted to expand to three dimensions, but I don't understand how it works in two dimensions.

I've done linear interpolation before, but this seems very different:

Code:
void advect ( int N, int b, float * d, float * d0, float * u, float * v, float dt )
{
int i, j, i0, j0, i1, j1;
float x, y, s0, t0, s1, t1, dt0;
dt0 = dt*N;
for ( i=1 ; i<=N ; i++ ) {
for ( j=1 ; j<=N ; j++ ) {
x = i-dt0*u[IX(i,j)]; y = j-dt0*v[IX(i,j)];
if (x<0.5) x=0.5; if (x>N+0.5) x=N+ 0.5; i0=(int)x; i1=i0+1;
if (y<0.5) y=0.5; if (y>N+0.5) y=N+ 0.5; j0=(int)y; j1=j0+1;
s1 = x-i0; s0 = 1-s1; t1 = y-j0; t0 = 1-t1;

[b]

d[IX(i,j)] = s0*(t0*d0[IX(i0,j0)]+t1*d 0[IX(i0,j1)])+
s1*(t0*d0[IX(i1,j0)]+t1*d0[IX(i1,j1)]);
[/b]

}
}
set_bnd ( N, b, d );
}

(the part in bold is where the linear interpolation takes place, and I don't get what they're doing).

The basic idea of linear interpolation is to take two points (x0, y0) and (x1, y1), and then find the value at a point (x, y) that lies between them by taking a weighted average of the values at the two points. In this case, they have four points (i0, j0), (i0, j1), (i1, j1), (i1, j0). The weights are s0, s1, t0, and t1. The value at (x, y) is then given by: d[IX(i,j)] = s0*(t0*d0[IX(i0,j0)]+t1*d 0[IX(i0,j1)])+ s1*(t0*d0[IX(i1,j0)]+t1*d0[IX(i1,j1)]);Where s0 and t0 are the weights for (i0, j0) and (i0, j1) respectively, while s1 and t1 are the weights for (i1, j1) and (i1, j0) respectively. The result is the weighted average of the values at the four points.

Hi there,

Linear interpolation is a commonly used method for estimating values between two known data points. In this case, the data points are represented by the variables d0[IX(i0,j0)], d0[IX(i0,j1)], d0[IX(i1,j0)], and d0[IX(i1,j1)]. The goal of the linear interpolation is to estimate the value of d at a specific point (i,j) within the grid.

The first step in the process is to determine the coordinates of the four data points surrounding the point (i,j). This is done using the variables i0, j0, i1, and j1, which represent the integer values of the x and y coordinates of the data points.

Next, the interpolation process begins by calculating the interpolation coefficients s0, s1, t0, and t1, which represent the weights given to the four data points. These coefficients are determined by the relative distance between the point (i,j) and the surrounding data points. For example, if the point (i,j) is closer to the data point d0[IX(i0,j0)] than the other three data points, then s0 and t0 will have higher values, indicating that d0[IX(i0,j0)] will have a greater influence on the estimated value of d.

Finally, the estimated value of d at the point (i,j) is calculated by multiplying the interpolation coefficients with the corresponding data points and summing them up. This is represented in the equation in bold in your code snippet.

I hope this helps to clarify how the linear interpolation method works in this context. If you have any further questions, please feel free to ask. Best of luck with your implementation!

## 1. What is linear interpolation in 2D?

Linear interpolation in 2D is a mathematical method used to estimate values between two known data points in a two-dimensional space. It involves calculating a straight line between the two points and determining the value of any points along that line.

## 2. How is linear interpolation different from other interpolation methods?

Linear interpolation is a simple and straightforward method that assumes a linear relationship between data points. Other interpolation methods, such as polynomial interpolation, use more complex mathematical functions to estimate values between data points.

## 3. What is the purpose of using linear interpolation in 2D?

The main purpose of using linear interpolation in 2D is to estimate values between known data points in a smooth and continuous manner. This can be useful in creating visualizations, filling in missing data, or making predictions based on existing data.

## 4. How is linear interpolation calculated in 2D?

Linear interpolation in 2D involves using the slope-intercept formula (y = mx + b) to calculate the value of a point along a straight line between two known points. The slope (m) is calculated using the two known points, and the y-intercept (b) is determined by plugging in one of the known points.

## 5. Are there any limitations to using linear interpolation in 2D?

Yes, linear interpolation in 2D assumes a linear relationship between data points and may not accurately estimate values in cases where data points do not follow a linear pattern. It is also not suitable for extrapolation, as it can only estimate values within the range of the known data points.

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