SMS:Linear Interpolation

If the linear interpolation scheme is selected, the 2D scatter points are first triangulated to form a temporary TIN. If the surface is assumed to vary linearly across each triangle, the TIN describes a piecewise linear surface which interpolates the scatter points. The equation of the plane defined by the three vertices of a triangle is as follows:

$Ax+By+Cz+D=0\$

where $A$, $B$, and $C$, and $D$ are computed from the coordinates of the three vertices $(x_{1},y_{1},z_{1})$, $(x_{2},y_{2},z_{2})$, & $(x_{3},y_{3},z_{3})$:

$A=y_{1}(z_{2}-z_{3})+y_{2}(z_{3}-z_{1})+y_{3}(z_{1}-z_{2}^{})$

$B=z_{1}(x_{2}-x_{3})+z_{2}(x_{3}-x_{1})+z_{3}(x_{1}-x_{2}^{})$

$C=x_{1}^{}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1})+x_{3}(y_{1}-y_{2})$

$D=-Ax_{1}-By_{1}-Cz_{1}^{}$

The plane equation can also be written as:

$z=f(x,y)=-{\frac {A}{C}}x-{\frac {B}{C}}y-{\frac {D}{C}}$

which is the form of the plane equation used to compute the elevation at any point on the triangle.

Since a TIN only covers the convex hull of a scatter point set, extrapolation beyond the convex hull is not possible with the linear interpolation scheme. Any points outside the convex hull of the scatter point set are assigned the default extrapolation value entered at the bottom of the Interpolation Options dialog.

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