riangle
irface using
d geometric
. The faces
smoothness
ree, in most
the vertices
normals at
1e boundary
: interpolate
rves of the
5 must have
ns allowed)
1. Our con-
ine the field
[his field is
ves at vari-
| of surface
elimiters of
nterpolated
garding the
stized fields
Ids, two ad-
)sSsess some
ly small and
ing the sur-
es, etc. To
bined with
ire.
), some pre-
gles, which
faces, some
on. In sec-
grammetry
| of surface
along with
o insert the
ing remarks
h.
photogram-
details, see
ézier patch)
represents a polynomial surface of degree n in R* with help
of a control net Pj, with i, j, k 2 0 andi+j+k=n (see
Figure 1). The points P;; are called control points, as the
shape of the surface can be controlled by their position. The
parameter domain is a triangle A(R, S, T). A Point U in this
triangle can be described by its barycentric coordinates, the
triple (r, s, t) with r4-s--t = 1 and U(r, 5,1) — rR-F sS-FtT.
Poos 2
R S
Po
P 030
Figure 1: Control net and parameter triangle
The point P(r,s,t) on the surface to the point U in the
parameter triangle is computed with the following recursive
de Casteljau algorithm (see Figure 2):
1: P?, = pt
l rt play =) ~1 ;
o Pop. zn P oaaaad pm ith
Op ug dl]. p is therefore the image
of U under the affine map from A(R,S,' T) onto
AP P^! P!-!
0-F17p,q" ^ 0,p3-1,9? Pot»
3. Pooo := P(r, s, t) is the desired point on the surface.
Bézier triangles have the following properties:
e The surface has a polynomial parametric representation
of degree n, expressible with Bernstein polynomials as
P(rie 4 = S P jk Bij(r, s, t) (1)
i+j+k=n
ni-— gon
AR S
with. Bl, (r, 5,4) =
e End point interpolation: Poo is the point on the
surface corresponding to the point R in parameter
space. The tangent plane at Poo contains the points
P._1,1,0 and P,,_1,0,1. Analogy applies for Po,0 and
Poor
e The boundary curves of a Bézier triangle are Bézier
curves. Their control polygons are the boundary poly-
gons of the control net. The other control points of the
surface are called ‘inner points’.
n=2 Poo»
R S
P020
Figure 2: Constructing a point, subdivision and derivatives
639
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
e Thetangent plane at Pg, is defined by the three points
Pico > Poro » Poor
e The directional derivative D, to the direction r in para-
meter space is given by the affine image of r in the
triangle that is obtained in the last but one step of the
de Casteljau algorithm.
e The de Casteljau algorithm has the following subdivi-
sion property: During the computation of P(r, s,t) the
control nets of the three subpatches to the parameter
triangles A(R,S,U), A(U,S,T) and A(R,U, T)
are obtained. The control points are Pl with 7 +
J +1 = n and analogously for the other two control
nets. As an example, in Figure 2 one of the new nets is
P200, P110, P020, Pls, Phos Plo: The new control
nets lie closer to the surface than the original one.
The subdivision algorithm mentioned above can be performed
for more points in parameter space simultaneously. Choosing
three points in the parameter space at the midpoints of each
edge of the parameter triangle yields four new control nets
describing the surface. Each of these control nets can be sub-
divided with the same points in parameter space again. This
leads to a sequence of control nets that are fastly converging
towards the surface (see [Hoschek,1993]). If subdivision is
mentioned in the following, it has always to be understood
as subdivision based on the edge midpoints of the parameter
triangle.
2.2 Variational surface design
Pleasing surface shapes or surfaces with specific physical
properties are often obtained as solutions of variational prob-
lems.
A frequently used fairness measure for a surface S(u, v) is
the linearized thin plate energy
E= fe. 4- 282, 4- S2,)dudv. (2)
Although it is dependent on the parametrization, it is of-
ten sufficient to minimize this quadratic functional in a linear
space of surface candidates. This clearly amounts to the solu-
tion of a linear system. Other functionals, partially paramet-
rization invariant, are discussed in [Greiner, 1994], along with
a technique to solve the nonlinear optimization iteratively by
linear problems. These are obtained when the solution sur-
face in step N is used as parameter domain for the improved
solution in step N +1.
In our approach, variational design is basically used for regu-
larization. Therefore, it is sufficient to use even simpler meas-
ures that may be applied directly to a piecewise linear mesh,
such as the control net of a Bézier triangle. The net shall con-
sist of points P;,$ — 1,. ..,n, and'of edges e;j,j — 1,... m.
Then, an energy for the network may be formulated as the
energy of a configuration of springs with one spring placed
along each edge e;,
m
Es Yale (3)
zl
The spring constants &; introduce additional flexibility in
choosing the functional. We often set all constants equal to 1.
Another choice discussed in [Eck,1995] yields approximations
of harmonic maps.
