8.2 Approximation of Space Curves
9.1.1 Main contributions and discussion
A quadrature of an integral is an approximation of this integral as a weighted sum, R f(x)dx ≈ P
iωif(xi). In this chapter, we are particularly interested in integral representations of surface curvatures. We consider the mean curvature
H = 1 2π
Z 2π 0
κ(φ)dφ, (9.2)
9.1 Introduction 93 the Gaussian curvature (the integral representation of which appeared before in
[MT98])
and the Taubin integral representation of the curvature tensor (“Taubin tensor”) M= 1
2π Z 2π
0
κ(φ)t(φ)t(φ)t dφ. (9.4) It is defined as the integral of the curvature weighted covariance matrix of the tangent vectort(φ)over the angleφbetweent(φ)andtmax, and it has the principal directions as eigenvectors with the eigenvalues equal to linear combinations of the principal curvatures [Tau95a].
We will derive novel weights that enable for the first time to obtain exact quadratures of these (and further) curvature integrals by making use of their peri-odic structure. Only minor restrictions regarding the number and positions of the samplesφi will be made.
Let Pbe a point of a smooth surface, let κi B κ(φi)be the normal curvatures at P in the tangential directions ti, and letβi B φi+1 −φi be the angle between ti and ti+1 (compare Figure 9.1 on the left hand side). First, we will present a formula to compute the mean curvature H at the surface point Pby a quadrature of integral (9.2). It is a generalization of the not-so-well-known formula
H = 1 n
X
i
κi
that holds if all anglesβi are equal. This, again, is a generalization of H= 1
2(κmax+κmin).
Our formula will be proven in Theorem 9.2; it is already stated here to give an outlook on our results.
Mean curvature is given by the weighted sum of normal curvatures:
H= X
i
ωiκi, ωi B tanβi−1+tanβi
Pj(tanβj−1+tanβj). (9.5) Similar results are obtained for the Gaussian curvature and the Taubin tensorM.
Theωi can be seen as a kind of barycentric coordinate: they determineHas a linear combination of normal curvaturesκi within the line segment[κmin, κmax]. If necessary (for example, because all theκi are smaller thanH, that is, they belong
94 9 Quadratures for Curvature Tensor Estimation
Figure 9.1: Left and Middle: Angles in the tangent plane and a mesh. Right:
Change of the normal curvature around a point.
to the grey area in Figure 9.1 (right)), then some of the weightsωibecome negative automatically.
In practice, the quadrature formula (9.5) leads to the following discrete ap-proximation ofH(given for a mesh to simplify the notation):
H ≈X
i
tanαi−1+tanαi
Pj(tanαj−1+tanαj)ki (9.6) whereαi is the angle between two consecutive outward edgesai in the one-ring aroundP (see Figure 9.4 (left) and Figure 9.1 (middle) for the relationship ofαi and βi) and ki B 2ai
a2i,n is an approximation of the normal curvature κi using the edgesai of lengthai and the normalnat P. We will discuss the influence of normal approximations fornin Section 9.3.
Our quadrature can be used to calculate integrals over all “Euler-like” formu-lae
M(φ)= M1cos2φ+M2sin2φ.
Such formulae occur repeatedly in the literature to describe surface related quan-tities. Blaschke [Bla56, §24.III] showed that the curvature radius of the osculat-ing cylindrical surface along a given tangent vector is described this way. (This statement can be considered as dual to Euler’s formula (9.1).) Pottmann et al.
[PHYK05, 7.1] derived this relationship for curvature related moments at a sur-face point. An especially important example is the Laplacian of a function f defined on the surface. After expressing it as the integral R ∂2f
∂t(φ)2 dφ, we get the
The same technique extends also to other integrals with “trigonometric” inte-grands. We applied it to the integrals (9.3) and (9.4) to obtain exact formulae and, in turn, discrete approximations of the Gaussian curvature and of the curva-ture tensor.
9.1 Introduction 95 9.1 Remark. Now, the question arises whether the previous results generalize
to higher dimensions. An analog of the Euler formula (9.1) for the directional curvature in directiontfor three-dimensional manifolds is given by
κ(t)= κmaxcos2φmax+κmedcos2φmed+κmincos2φmin
whereφmax, φmed, and φmin are the angles between tand the respective principal directionstmax, tmed, and tmin with principal curvaturesκmax ≥ κmed ≥ κmin. Cor-responding formulae hold for n-dimensional manifolds. The integral formula for the mean curvature follows easily:
H= κmax+κmed+κmin
3 = 1
4π Z
S2κ(t)dS.
Similar integral representations can be obtained for the other coefficients of the characteristic polynomial of the shape operator. Nevertheless, we were not able to derive a discrete quadrature formula similar to (9.5) to calculate the above integral.
This topic needs still further research.
Let us now summarize our contributions:
• we derive quadrature identities for trigonometric integrals; in particular, we present exact formulae for the mean curvature, the Gaussian curvature, the curvature tensor, and the Laplacian (Section 9.2),
• we show how these quadratures can be used to obtain fast and reliable ap-proximations of the curvature of a smooth surface interpolated by discrete data like a triangle mesh (Section 9.2.6),
• extending the work of Chapter 8 from curves to surfaces, we present a framework designed to enable an easy and meaningful representation of the Taylor series expansion of an analytic surface interpolated by discrete data (Appendix C),
• we use this framework to analyze the convergence speed of common estima-tions for normal vectors and our curvature approximaestima-tions, and we suggest a new normal estimation based on our analysis (Section 9.3),
• and we prove the convergence of our curvature approximations for arbitrary aspect ratios if the normal vector is known with quadratic accuracy and confirm the result experimentally (Section 9.4).
96 9 Quadratures for Curvature Tensor Estimation
Figure 9.2: The curvature function whose integral (9.2) defines the mean cur-vature. The piecewise linear function used by the trapezoidal rule shows con-siderable deviation.