Linear interpolation code on wikipedia - I don't understand it - c++

Question

I'm reading the following code (taken from here)
void linear_interpolation_CPU(float2* result, float2* data,
float* x_out, int M, int N) {
float a;
for(int j = 0; j < N; j++) {
int k = floorf(x_out[j]);
a = x_out[j] - floorf(x_out[j]);
result[j].x = a*data[k+1].x + (-data[k].x*a + data[k].x);
result[j].y = a*data[k+1].y + (-data[k].y*a + data[k].y);
}
}
but I don't get it.
Why isn't the result[y] calculated by using the
formula?

Answer 1

It is calculated that way.
Look at the first two lines:
int k = floorf(x_out[j]);
a = x_out[j] - floorf(x_out[j]);
The first line defines x0 using the floor function. This is because the article assumes a lattice spacing of one for the sample points, as per the line:
the samples are obtained on the 0,1,...,M lattice
Now we could rewrite the second line for clarity as:
a = x_out[j] - k;
The second line is therefore x-x0.
Now, let us examine the equation:
result[j].y = a*data[k+1].y + (-data[k].y*a + data[k].y);
Rewriting this in terms of y, x, and x0 gives:
y = (x-x0)*data[k+1].y + (-data[k].y*(x-x0) + data[k].y);
Let's rename data[k+1].y as y1 and data[k].y as y0:
y = (x-x0)*y1 + (-y0*(x-x0) + y0);
Let's rearrange this by pulling out x-x0:
y = (x-x0)*(y1-y0) + y0;
And rearrange again:
y = y0 + (y1-y0)*(x-x0);
Again, the lattice spacing is important:
the samples are obtained on the 0,1,...,M lattice
Thus, x1-x0 is always 1. If we put it back in, we get
y = y0 + (y1-y0)*(x-x0)/(x1-x0);
Which is just the equation you were looking for.
Granted, it's ridiculous that the code is not written so as to make that apparent.