Curve and Surface Intersection Problems
In many applications, we need to find the points (curves) where two curves
(surfaces) intersect, for example in constructing a countour map to graphically
represent a prescribed surface o hypersurface, in computing silhouettes, in
performing Boolean operations on solid bodies, in constructing smooth blending
curves and surfaces, in finding offset curves and surfaces, etc.
A good (surface) intersection algorithm should have the following properties:
Although presently there is no single algorithm which is optimal with respect
to all four of these characteristics, there are many available algorithms.
accuracy, in the usual numeric sense;
robustness, in the sense that all intersections are correctly identified,
even where they decompose into pieces;
speed, in particular in interactive setting;
self control, in the sense that it does not require help from a user.
Tipically, a numerical algorithm is efficient, but is not fully robust and
so may fail in certain cases. Furthermore, the accuracy a numerical method
can deliver varies with the surface degree, with the local surface geometry
at the intersection curve, and with the angle at which the local surface
intersect. Algorithms based on exact arithmetic, on the other hand,
are fully robust and accurate, but are normally slow.
The difficulty of the intersection problem, and the type of algorithm which
best suits it, depends also on the general form of the curves or surfaces
we are dealing with (implicit, explicit, parametrized), along with their
particular properties (special or free-form).
A CCI (Curve/Curve Intersection) or SSI (Surface/Surface Intersection) problem
can be approached with a variety of different techniques, including numerical,
analytic, geometric, and algebraic methods, and using various methods such
subdivision, tracing, and discretization. While the classic algorithms of the first
generation are mostly based on just one method, more recently the tendency is to
combine two or more methods into a multi-step method or so called hybrid-
algorithm. The idea is to preserve the advantages of each one, but to eliminate
their disadvantages if possible.
In our research activity we dealed with the NURBS SSI problem; we experimented some
known approaches in literature and finally, as our best result for NURBS, we proposed a
modified version of the marching method given in Barnhill, Farin, Jordan, Piper
"Surface/surface intersection" CAGD (1987), specialized and optimized for NURBS
surfaces. The balance between robustness and efficiency, an open question in SSI
algorithms, was the main contribution of that work.
This proposal is the SSI algorithm used in the xcbool system to compose
solid bodies in the xcmodel system.
Follows some graphic representations of the output of our SSI algorithm.
First Glass domain
Second Glass domain
In the following figures we consider the bottle/plane intersection computed by
our SSI algorithm.
They compare the intersection between the two well-parametrized surfaces
(fig.1) with the intersection between the same surfaces, in
which the bottle surface is badly parametrized; in fact it is built as a revolving
surface with a badly parametrized circular trajectory curve (fig.2).
In the second case our algorithm presents an increased computational cost due to
the fact that the solution for the parametric domain shown in fig.2 is given by
a less smooth curve than in the first case (see fig.1). In this situation the
marching phase encounters problems in the spans
of higher curvature making it impossible to find all the intersection points to the
This our experience where working with badly parametrized curves or surfaces can be
disadvantageous was the motivation to deal with the problem of parametric NURBS
curve (surface) reparametrization. This consists in changing the current parameter
of a curve with another parameter using a reparametrization function. It should be
noted that the shape of the curve remains unchanged during this process; only the
way the curve is described is altered. Follows an example.
NURBS periodic curve of 3 degree with its control points.
drawing in dots relating to the initial parametrization.
drawing in dots relating to a reparametrization.
As well known, the mathematical nature of the intersection problem is very
simple when one plane curve o surface is given implicitly while the other is
given in parametric form. In view of this, if both curves or surfaces are
given in parametric form that is the more useful rapresentation in CAGD, it
is natural to try to convert one of them to implicit form. In the literature,
this idea, which comes from algebraic geometry is called implicitization.
Because of its algebraic complexity, implicitization for curves or surfaces
via parametric elimination can usually be done only with the help of formula
manipulation such as Mathematica.
In addition, for high polynomial degree, the algebraic method leads to
significant numerical problems.
For example to intersect curves of degree n
we need the expansion of a determinant leading to an equation of degree n^2,
whose zero are to be found. Thus the available software for finding zeros of
polynomials has an essential influence on the robustness, accuracy and
efficiency of the algorithm. Because of the associated numerical problems,
at the moment of aour research, results using known algorithms for n>=5 have not yet been obtained.
Recently we have proposed a suitable realization of the fast fraction free
algorithm proposed in Bini, Gemignani, "Fast fraction-free triangularization
of Bezoutians with applications to sub-resultant chain computation", LAA (1998),
for the triangulation of a Bezout matrix with
integer entries. This algorithm has been applyed successfully to the
rational Bezier CCI problem working with great accuracy and avoiding
the need for symbolic computation.
As already noted, using double precision
arithmetic, there is an upper limit to the degree of the curves to be intersected.
So our proposal, for planar curves of degree >5,
uses the GNU Multiple Precision arithmetic library, which provides optimized
tools to perform floating point arithmetic with arbitrarily high precision.
CCI example; the two curves have respectively 5 and 8 degrees.
The next steps are to extend our approach for planar curve of any degree and
to other intersection problems
in CAGD, from ray/surface intersection, that is the main computational problem
in a ray-tracing algorithm, to surfaces/surfaces intersection.
G.Casciola, S.De Santis, R.Quadalti, Algoritmi paralleli per il problema
NURBS Surface/Surface Intersection, Dipartimento di
Matematica, Universita di Bologna, (1992)
G.Casciola, S.Morigi, Il problema SSI nella modellazione solida con
superfici NURBS, Atti dell'Accademia delle Scienze dell'Istituto di
Bologna, Serie V, n.6 (1995)
G.Casciola, S.Morigi, Reparametrization of NURBS curves,
International Journal of Shape Modeling, vol.2 n.2\&3 (1996)
G.Casciola, F.Fabbri, L.B.Montefusco,
An Application of Fast Factorization Algorithms in Computer Aided
submitted to Linear Algebra and its Applications, (2001)