I don't think you have misunderstood anything fundamental. The description is unclear. Here are examples to illustrate how it works. We will check each how extrapolation method "boundary" works with each of the three interpolation methods: "nearest", "linear", and "natural".
Example 1. Method=nearest, extrapolation method=boundary. In this case, it appears that the extrapolated value is the value of the nearest of the original points.
figure; plot(X,Y,'ro','MarkerFaceColor','r')
Now examine the behavior around the top right corner of the plot above (x=3 to 5, y=2 to 4), since that is where the point set is concave.
[Xq,Yq]=meshgrid(3:.05:5,2:.05:4);
XL=X([15,16,19,20]); YL=Y([15,16,19,20]); VL=V([15,16,19,20]);
F1=scatteredInterpolant(X(:),Y(:),V(:),'nearest','boundary');
figure; surf(Xq,Yq,Vq1); colorbar; hold on; view(0,90)
plot3(XL,YL,VL,'ro','MarkerFaceColor','r')
xlabel('X'); ylabel('Y'); zlabel('Z'); title('nearest, boundary');
Careful examination of the points above suggests that the extrapolated value equals the value of the nearest original point, not the value of the nearest point on the boundary of the convex hull. For example, at Xq,Yq=4.05,3, extrapolated Vq1=4.47, which is the value of the closest original point (at X,Y=4,3). The closest point on the convex hull is at about X=3.525,Y=2.475, and this point has extrapolated value V=4.02 (by nearest neighbor). So, if Vq were using the extrapolated value at the closest point on the convex hull, Vq would be 4.02 when Xq,Yq=4.05,3. But Vq1 is not that.
Example 2: Method=linear, extrap.method=boundary.
F2=scatteredInterpolant(X(:),Y(:),V(:),'linear','boundary');
figure; surf(Xq,Yq,Vq2); colorbar; hold on
plot3(XL,YL,VL,'ro','MarkerFaceColor','r')
xlabel('X'); ylabel('Y'); zlabel('Z'); title('linear, boundary');
Adjust the view to further appreciate and understand the plot.
In the case above, points along the convex hull boundary are intepolated linearly between the original points that are on the convex hull, ignoring original points interior to the convex hull. Points outside the convex hull are extrapolated from the convex hull boundary, and are not affected by points interior to the convex hull.
Example 3: Method=linear, extrap.method=boundary.
F3=scatteredInterpolant(X(:),Y(:),V(:),'natural','boundary');
figure; surf(Xq,Yq,Vq3); colorbar; hold on
plot3(XL,YL,VL,'ro','MarkerFaceColor','r')
xlabel('X'); ylabel('Y'); zlabel('Z'); title('natural, boundary');
Adjust the view to understand the surface better. It looks the same as the plot when method='linear'.
This case is like the linear case: points on the convex hull boundary are intepolated linearly between the original points that are on the convex hull, ignoring original points interior to the convex hull. Points outside the convex hull are esitmated from the points on the convex hull boundary. Inside the convex hull, the esitmated values are slightly different with natural method than with the linear method, consistent with natural versus linear interpolation.
If you vary the height of the point inside the convex hull, you will see that the results with method="nearest" still work the same way. With method="linear" or "natural", the interpolated points on the convex hull boundary, and the extrapolated points outside the convex hull, are not altered by the adjustment of the nearby point inside the convex hull. Their height is determined only by the original and interpolated points along the boundary of the convex hull.
