4.3.1 Interpolation and extrapolation
The most direct application of mean value coordinates is their use for interpola-tion and extrapolainterpola-tion using (2.3). In Figure 4.2, color values are specified at the eight vertices of the cube. In (a) and (c), the values are interpolated on the faces.
36 4 Barycentric Coordinates for Arbitrary Polyhedra
(a) (b) (c) (d)
Figure 4.2: An example of interpolation of color values using 3D mean value coordinates. The color values are specified at the vertices of the cube. They are interpolated on the faces ((a) and (c)) and on a plane passing through the cube ((b) and (d)). If the cube is triangulated beforehand ((a) and (b)), the interpolation is less smooth than with our method ((c) and (d)).
(a) (b) (c) (d)
Figure 4.3: An example of a space deformation using 3D mean value coor-dinates with respect to the polygonal control mesh. If the control mesh is triangulated beforehand, strong artifacts may be introduced ((a) and (b)).
No triangulation is necessary with our method ((c) and (d)).
In (b) and (d), the color values are interpolated and extrapolated on a plane that passes through the cube. In (a) and (b), the cube was triangulated before the in-terpolation. The piecewise linear structure of the interpolation on the triangles is clearly visible. With our 3D mean value coordinates, a triangulation is no longer necessary, and the resulting interpolation is much smoother.
4.3.2 Space deformations with 3D mean value coordi-nates
Figure 4.3 shows an example how mean value coordinates can be used for space deformations. We determine the mean value coordinates of the vertices of the
4.3 Applications 37
Figure 4.4: The quadratic mean value Bernstein polynomialsB22000,B21100, and B21010+B20101.
tube with respect to the black control mesh with verticesvi. Then we deform the control mesh by moving the vertices to pointswiand calculate the new location of the tube byx = P
iλ3Di x; (vj)j
wi. Note that we can compute these coordinates for non-convex (control) meshes with non-convex faces ((c) and (d)). If all faces are triangulated, the number of faces is nearly tripled, the result depends on the chosen triangulation, and large artifacts may be introduced ((a) and (b)).
Although this approach is very simple, it is possible to obtain pleasing results.
These can be considerably improved by modifying the barycentric coordinates such that derivatives are taken into account. This is discussed in detail in Chap-ter 5.
4.3.3 Bernstein polynomials on polygons and polyhedra
In this section, we introduce Bernstein polynomials in mean value coordinates on polygons. They can be used to define generalized Bézier surfaces. The full theory, including examples of Bernstein polynomials on polyhedra, is developed in Part II.
Bézier surfaces are defined by a linear combination of Bernstein polynomi-als which are polynomipolynomi-als in barycentric coordinates. Using classical barycentric coordinates, this was only possible for triangles. Using tensor product polynomi-als, Bernstein polynomials can be defined on quadrangular domains as well, but this leads to a higher degree of the polynomial. The only approaches for general polygons that we are aware of are restricted to convex polygons [LD89, Gol02].
We can define mean value Bernstein polynomials for arbitrary polygons and polyhedra. For a polygon or polyhedron withkvertices, the general form for the Bernstein polynomials in the coordinatesλ= (λ1, . . . λk)is
Bnα(x)= n!
α!λα(x)
38 4 Barycentric Coordinates for Arbitrary Polyhedra where we use multi-indices α = (α1, . . . αk) ∈ k with the notation α! B α1!· · ·αk! and λα B λα11· · ·λαkk. In Figure 4.4, we show some quadratic Bern-stein polynomials on a square using mean value coordinates [Flo03].
Important properties of classical Bézier surfaces like the convex hull property and the de Casteljau algorithm still hold in this extended setup.
4.4 Summary
We have shown that spherical mean value coordinates can be used to construct 3D mean value coordinates for polyhedra with arbitrary polygonal faces while before only 3D mean value coordinates for triangular polyhedra were known. The same method can be used to construct mean value coordinates for arbitrary polytopes of successively higher dimensions. We showed that then-D mean value coordinates are well defined. This concludes the generalization of mean value coordinates from two tondimensions.
In the future, it would be interesting to find a general theory for barycentric coordinates for arbitrary polytopes similar to the one given in [FHK06, JLW07].
It should shed light on the relationship between “Euclidean” and spherical coor-dinates. To construct the general 3D mean value coordinates, we had to make two choices. First, we chose to use the “mean value” face vector as in [FKR05, JSW05], then we chose spherical mean value coordinates as coefficientsµf,i. How-ever, we do not yet know which choices have to be made to obtain other types of coordinates like the Warren-Wachspress coordinates [War96]. While it seems ob-vious that the face vector of the respective type should be chosen by integrating over the respective generating surface (see [JLW07] for details) to ensure consis-tency with existing definitions, it is less obvious which choice is the “right” one for the coefficientsµf,iand which effects would result from different choices.
Chapter 5
Higher Order Barycentric Coordinates
In recent years, a wide range of generalized barycentric coordinates has been sug-gested. However, they usually lack control over derivatives. We show how the notion of barycentric coordinates can be extended to specify derivatives at control points. This is also known as Hermite interpolation. We introduce a method to modify existing barycentric coordinates to higher order barycentric coordinates and demonstrate, using higher order mean value coordinates, that our method, al-though conceptually simple and easy to implement, can be used to give easy and intuitive control at interactive frame rates over local space deformations such as rotations.