By Cai M.-C., Favaron O., Li H.
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Additional resources for (2,k)-Factor-Critical Graphs and Toughness
Example text
Xσσ y˙ σ − y¨σσ x˙ σ |1/3 = 1. ¨σσ x˙ σ x which has determinant 1. Since Am (1, 0)T = y˙ σ y¨σσ ˙ σ is trivial. We have m ˙ σ , the term A−1 m m One can choose Am = A−1 m ∂σ Am = xσσ y¨σσ −¨ −y˙ σ x˙ σ (3) x ¨σσ xσσσ (3) y¨σσ yσσσ (3) (3) = (3) 0 y¨σσ xσσσ − x ¨σσ yσσσ (3) (3) 1 −y˙ σ xσσσ + x˙ σ yσσσ . (3) Since ∂σ (¨ xσσ y˙ σ − y¨σσ x˙ σ ) = y˙ σ xσσσ − x˙ σ yσσσ = 0, the only non-trivial coef(3) (3) ficient is: y¨σσ xσσσ − x ¨σσ yσσσ which can be taken (up to a sign change) as a definition of the special affine curvature: K = det(m ¨ σσ , m(3) σσσ ).
1 βm,3 Since the curve is parametrized with affine arc length, we have Q = 1 where Q is given by |β˙ 3 − 3α3 − 2β 2 /3|. This implies that αm,3 is a function of βm,3 3 and β˙ m,3 along the curve; the moving frame therefore only depends on βm,3 and its derivatives, which indicates that βm,3 is the affine curvature. 26). 4 in the projective case, because of the non-linearity of the transformations. The moving frame is still associated to a one-to-one function P0 (z0 , . . , zk ) ∈ G = P GL2 (R). 27), P0 (g z) = gP0 (z).
The computation of the expression of the curvature for an arbitrary parametrization is left to the reader. It involves the second derivative of the arc length, and therefore the seventh derivative of the curve. 1 Introduction The medial axis [24] transform associates a skeleton-like structure to a shape, which encodes its geometry. The medial axis itself (or skeleton) is the center of discs of maximal radii inscribed in the shape. The medial axis transform stores in addition the maximal radii. More precisely, represent a shape by an open connected bounded set in the plane, denoted Ω.