A Smoothing Displacement Function

Since we have a mechanism to create images with displacements, it is useful to have a displacement that creates a smooth mesh. This would allow rendering smoothed versions of tessellated models with or without additional displacements. To make the problem as local as possible, we would like the smoothing displacement to only have knowledge of a given triangle's vertices and vertex normals. Knowledge about neighboring triangles could allow a smoother surface, but create additional complexities in the representation of the triangle mesh.

Although examining how to smooth triangle meshes has been examined by many researchers (e.g., [7]), this problem is different in that the function must have the algebraic form of a height function in barycentric coordinates with respect to barycentric interpolated normals.

Figure 8: An icosahedron with a smoothing displacement that only uses the vertices and vertex normals for the triangle being displaced for $N=1,4,100$.

We have created a simple smoothing displacement as a proof of concept. This displacement interpolates the triangle vertices, and has a smooth tangent plane on the transition between two adjacent triangles. This implies a number of constraints:

• the surface must depend only on the vertices and vertex normals,
• the surface must be smooth over the triangle,
• the surface must interpolate the vertices of the triangle,
• the surface normal at each vertex must match the prescribed vertex normals,
• the tangent plane along each edge of the surface must match that constructed on an adjacent triangle, so that joined patches meet with $G^1$ continuity.

The final requirement listed above is the one which is the most difficult to satisfy, because Hermite (derivative) interpolation is more difficult to enforce over a line than at single points.

We use the Coons patch approach to construct the surface. First, boundary curves and prescribed tangent planes are constructed using ordinary Hermite interpolation. Then three surfaces are constructed which interpolate the boundary curves and tangents along two of the edges. These three surfaces are blended in such a way as to preserve the derivatives and remove the ``bad'' edges from the final surface. The surface will be constructed in terms of barycentric coordinates. The approach applied to an icosahedron is shown in Figure 8. This displacement function, although useful for eliminating artifacts in certain models, creates objectionable artifacts for other models. More details of the smoothing function are available in a technical report [14].