extrapolate three points between line
As you interpolate a line, I assume it is a linear interpolation. The solution reduces to a linear combination of the intial starting and ending coordinates. For instance, $i_1 = \frac{1}{3}s + \frac{2}{3}e $ and $i_2 = \frac{2}{3}s + \frac{1}{3}e $ provide the coordinates of two interpolation points $i_1$ and $i_2$ between start point $s$ and end point $e$.

clear
%% Initialization
pStart =[1 3]';
pEnd = [4 2]';
nPointInter = 3;

%% Interpolation weights (including pStart and pEnd)
lInterval = linspace(0,1,nPointInter+2);

%% Interpolation coordinates
pInterp = pStart*lInterval+pEnd*(1-lInterval);

%% Display
figure(1);clf
hold  on
plot(pInterp(1,:),pInterp(2,:),'xb')
plot([pStart(1) pEnd(1)]',[pStart(2) pEnd(2)]','or')
grid on


The result will be: