I tried to explain figure below in mathematics form. As you can see I have got triangle (v1, v2, v3).
The signed shortest distance form red interface (level set value) is calculated for each vertex of the triangle. I applied linear interpolation to find a point (blue circle)on the edges with zero value of the level set.
It is clear on the below figure that for the edge v1-v3 the point has been detected correctly. In other words, the result of the linear interpolation is exactly on the interface.
although for the edge v1-v2 the result of the interpolation is not on the interface. the reason (as much as I know) is that level set values for each vertex of the edge v1-v2 is calculated from different parts of the interface (derivative of the interface is not continuous in our triangle).
My question is that, How I can show above explanation in mathematical format.