Part of me wants to not buy this because I find it hard to believe that for any two points x0, x1 such that d_i(x0) = d_i(x1), the areas of the triangles from this method are the same.
I'm guessing it doesn't matter which triangle we use for the ratio? Would we get the same basis if we used area(x,xi,xj)/area(xi,xj,xk)?
I guess it's because when we divide the areas, whose formula involves base and height, the base cancels out and we're only left with the ratio of the heights, which is exactly the previous formula
I agree with vtnguyen, that the lengths won't matter, it will only be based on the distance from the point x to the base length.
By dividing half of the xjxk length from both the numerator and denominator, the basis function would be the same as the function on the previous slide.
Have any works tried to do the interpolation in original 3D space for a better effect?
base*height gives the area and the base is the same so this gives the same answer as the previous one?
I just have a silly question. What is the purpose for using interpolation?
I think this intuitively makes sense. You are still using these ratios that increase and decrease proportionally to the previous method.
I think it makes sense, but I am not sure what's the use for it, is it easier to define or calculate?
Honestly I have no intuition regarding this, but does this really work? My best guess is that instead of solving for a specific ratio between heights, you solve for an equivalent ratio that is basically one for the entire triangle section.