Interpolated Surfaces

Figure 9: Left: Transfinite interpolation along a line of constant $\alpha$. The parameter is either $\beta /(\beta + \gamma )$ or $\gamma /(\beta + \gamma )$, depending on the direction. Right: The resulting interpolated curves form a surface on the triangle.

Given the boundary curves and associated surface normals, we can create a surface over the triangle using the ``loft'' operator

$${\cal P}_0h(\alpha , \beta , \gamma ) = H_0^3(t) h(\alpha , 1 - \alpha , 0)$$

$$+ H_1^3(t) h_1(\alpha , 1 - \alpha , 0)$$

$$+ H_2^3(t) h_1(\alpha , 0, 1 - \alpha )$$

$$+ H_3^3(t) h(\alpha , 0, 1 - \alpha )$$

where

$$t = \frac{\gamma }{\beta + \gamma }.$$

The operator takes the boundary function and returns a surface (defined over the entire triangle) formed by Hermite interpolation along lines of constant $\alpha$, as shown in Figure 9. The values of $h_1$ are the directional derivatives of $h$, in the direction of constant $\alpha$, and correspond to the derivatives $g'$ computed in equations (1) and (2). The normals ${\bf n}_a$ and ${\bf n}_b$ are the edge surface normals ${\bf n}_s(\alpha , 1 - \alpha , 0)$ and ${\bf n}_S(\alpha , 0, 1 - \alpha )$, respectively.

The surface function ${\cal P}_0h$ interpolates both the boundary curve and the surface normal on the two edges ${\bf p}_0{\bf p}_1$ and ${\bf p}_2{\bf p}_0$, but only interpolates the curve on the edge ${\bf p}_1{\bf p}_2$.

Surfaces ${\cal P}_1h$ and ${\cal P}_2h$ are constructed similarly; each ${\cal P}_i$ has the proper interpolation on the two edges adjacent to vertex $i$, but not on the opposite edge.