Page 148 - Geometric Modeling and Algebraic Geometry

8 Ridges and Umbilics of Polynomial Parametric Surfaces 149
• Real singularities of multiplicity 3 are called umbilics and they satisfy the system
S u = {p 2 =0} = {p 2 =0,P =0,P u =0,P v =0}. In addition, if δ(P 3 )
denote the discriminant of the cubic of the third derivatives of P at an umbilic,
one has:
–if δ(P 3 ) > 0, then the umbilic is called a 3-ridge umbilic and three real
branches of P are passing through the umbilic with three distinct tangents;
–if δ(P 3 ) < 0, then the umbilic is called a 1-ridge umbilic and one real branch
of P is passing through the umbilic.
As we shall see in section 8.3.5, these conditions are checked during the processing
of the algorithm.
Given this structure of singular points, the algorithm successively isolate umbil-
ics, purple points and critical points. As a system defining one set of these points
also includes the points of the previous system, we use a localization method to sim-
plify the calculations. The points reported at each stage are characterized as roots of
a zero-dimensional system — a system with a finite number of complex solutions,
together with the number of half-branches of the curve connected to each point. In
addition, points on the border of the domain of study need a special care. This setting
leads to the definition of study points:

Definition 4. Study points are points in D which are
• real singularities of P, that is S s = S u ∪ S p , with S u = S 1R ∪ S 3R and
– S 1R = {p 2 = P = P u = P v =0,δ(P 3 ) < 0}
– S 3R = {p 2 = P = P u = P v =0,δ(P 3 ) > 0}

– S p = {a = b = a = b =0,δ(P 2 ) > 0,p 2 =0}

= {a = b = a = b =0,δ(P 2 ) > 0}\ S u
• real critical points of P in the v-direction (i.e. points with a horizontal tangent
which are not singularities of P) defined by the system
S c = {P = P u =0,P v =0};
• intersections of P with the left and right sides of the box D satisfying the system
S b = {P(a, v)=0,v ∈ [c, d]}∪{P(b, v)=0,v ∈ [c, d]}. Such a point may
also be critical or singular.

8.3.3 Output specification

Definition 5. Let G be a graph whose vertices are points of D and edges are non-
intersecting straight line-segments between vertices. Let the topology on G be in-
duced by that of D. We say that G is a topological approximation of the ridge curve
P on the domain D if G is ambient isotopic to P∩ D in D.
More formally, there exists a function F : D× [0, 1] −→ D such that:

• F is continuous;
• ∀t ∈ [0, 1],F t = F(., t) is an homeomorphism of D onto itself;
• F 0 = Id D and F 1 (P∩ D)= G.