Advances in Discrete Differential Geometry pp 267286  Cite as
Vertex Normals and Face Curvatures of Triangle Meshes
Abstract
This study contributes to the discrete differential geometry of triangle meshes, in combination with discrete line congruences associated with such meshes. In particular we discuss when a congruence defined by linear interpolation of vertex normals deserves to be called a ‘normal’ congruence. Our main results are a discussion of various definitions of normality, a detailed study of the geometry of such congruences, and a concept of curvatures and shape operators associated with the faces of a triangle mesh. These curvatures are compatible with both normal congruences and the Steiner formula.
Keywords
Principal Curvature Shape Operator Triangle Mesh Quadratic Cone Focal Surface1 Introduction
The system of lines orthogonal to a surface (called the normal congruence of that surface) has close relations to the surface’s curvatures and is a well studied object of classical differential geometry, see e.g. [14]. It is quite surprising that this natural correspondence has not been extensively exploited in discrete differential geometry: most notions of discrete curvature are constructed in a way not involving normals, or involving normals only implicitly. There are however applications such as support structures and shading/lighting systems in architectural geometry where line congruences, and in particular normal congruences, come into play [21]. We continue this study, elaborate on discrete normal congruences in more depth and present a novel discrete curvature theory for triangle meshes which is based on discrete line congruences.
Contributions and overview. We organize our presentation as as follows. Section 2 summarizes properties of smooth congruences and elaborates on an important example arising in the context of linear interpolation of surface normals.
Section 3 first recalls discrete congruences following the work of Wang et al. [21] and then focuses on the interesting geometry of a new version of discrete normal congruences (defined over triangle meshes). We shed new light onto the behavior of linearly interpolated surface normals and discuss the problem of choosing vertex normals.
In Sect. 4, discrete normal congruences lead to a curvature theory for triangle meshes which has many analogies to the classical smooth setting. Unlike most other concepts of discrete curvature, it assigns values of the curvatures (principal, mean, Gaussian) to the faces of a triangle mesh. We discuss internal consistency of this theory and show by examples (Sect. 5), that it is well suited for curvature estimation and other applications.
Previous work. Smooth line congruences represent a classical subject. An introduction may be found in the monograph by Pottmann and Wallner [16]. Discrete congruences have appeared both in discrete differential geometry and geometry processing. Let us first mention contributions which study congruences based on triangle meshes: A computational framework for normal congruences and for estimating focal surfaces of meshes with known or estimated normals has been presented by Yu et al. [22]. The paper by Wang et al. [21] is described in more detail below.
Congruences associated with quad meshes are discrete versions of parametrized congruences associated with parametrized surfaces. In particular, the socalled torsal parametrizations are discussed from the integrable systems perspective by Bobenko and Suris [3]. An earlier contribution in this direction is due to Doliwa et al. [6]. These special parametrizations also occur as node axes in torsionfree support structures in architectural geometry [12, 15, 17].
Curvatures of triangle meshes are a well studied subject. One may distinguish between numerical approximation schemes (such as the jet fitting approach [4] or integral invariants [18]) on the one hand, and extensive studies from the discrete differential geometry perspective on the other hand. Without going into any detail we mention that these include discrete exterior calculus [5], the geometry of offsetlike sets and distance functions [13], or various ways of defining shape operators [8, 9]. Naturally, also Yu et al. [22] address this topic when studying discrete normal congruences and focal surfaces. We present here yet another definition of curvatures for triangle meshes which is based on discrete normal congruences, and which is at the same time motivated by the Steiner formula (which also plays an important role in [2, 13, 15]).
2 Smooth Line Congruences
Normal Congruences.
The normals of a surface constitute the normal congruence of that surface. For such congruences the analogy between torsal directions and principal directions mentioned above is actually an equality: The surface normals along a curve form a developable surface if and only if that curve is a principal curvature line [14].
Focal surfaces and focal planes.
For normal congruences, the focal points are precisely the principal centers of curvature; they exist always unless one of the principal curvatures is zero. In each point of the surface, the focal plane (i.e., torsal plane) is spanned by the surface normal and a principal tangent.
Example: Congruences defined by linear interpolation.
 (i)
Each intersection line \(L = P_{\alpha }\cap P_{\beta }\) of two planes in the family \(P_{\lambda }\) is contained in the congruence Open image in new window . This follows from the fact that L is spanned by the points \(X=\phi _{\alpha \beta }^{1}(L)\cap L\) and \(\phi _{\alpha \beta }(X)=L\cap \phi _{\alpha \beta }(L)\).
 (ii)
The lines \(P_\alpha \cap P_\beta \) with \(\alpha \) fixed, constitute a developable surface Open image in new window which is planar and contained in \(P_\alpha \) (in general, it is the tangent surface of a parabola \(r_\alpha \)).
 (iii)
For properties of the focal surface, see Fig. 2.
3 Discrete Normal Congruences
Discrete normal congruences—Version 1.
It is not straightforward to define which correspondence between triangle meshes defines a normal congruence. Firstly this is because congruences of the form (4) are never normal except for degenerate cases. Secondly such a normal congruence would automatically lead to a good definition of constantdistance offset mesh of a triangle mesh which is lacking so far.
Proposition 3.1
Discrete normal congruences—Version 2.
Proposition 3.2
Consider two combinatorially equivalent triangle meshes and the line congruence Open image in new window defined by the piecewiseaffine correspondence of faces, and consider in particular one such pair \({\mathbf a}_1{\mathbf a}_2{\mathbf a}_3\), \({\mathbf b}_1{\mathbf b}_2{\mathbf b}_3\) of corresponding faces. In the generic case, the normality condition (6*) implies the following property:
For each plane \(P_{\lambda }\) spanned by the vertices \((1\lambda ){\mathbf a}_i + \lambda {\mathbf b}_i\) there is a congruence line \(N_\lambda = L(u_\lambda ,v_\lambda )\) such that the two focal planes of that line together with \(P_\lambda \) form a trihedron of mutually orthogonal planes.
The meaning of “generic” is discussed in the proof.
Proof
Generically, vectors \({\mathbf e}_i={\mathbf b}_i{\mathbf a}_i\) are linearly independent, so we can express a normal vector \({\mathbf n}\) of the triangle \({\mathbf a}_1{\mathbf a}_2{\mathbf a}_3\) (which spans \(P_0\)) as a linear combination \({\mathbf n}=\sum _{i=1}^3\alpha _i{\mathbf e}_i\). Generically, \(\sum \alpha _i\ne 0\), so by multiplying \({\mathbf n}\) with a factor we can achieve \(\sum \alpha _i=1\) and by relabeling the coefficients \(\alpha _i\) we get \({\mathbf n}= (1uv){\mathbf e}_1 + u{\mathbf e}_2 + v{\mathbf e}_3\). Then Equation (5) shows that the line L(u, v) is orthogonal to \(P_0\).
As illustrated by Fig. 2, congruences defined by the affine correspondence of triangles have counterintuitive properties: The planes \(P_\lambda \) generated by linear interpolation of the defining triangles at the same time are the focal planes of Open image in new window (and vice versa) since any \(P_\lambda \) carries the developable surface generated by the lines \(\{P_\lambda \cap P_\alpha \}_{\alpha \in \mathbb R}\). The torsal planes \(P_\lambda \) are tangent to the focal surface F of Open image in new window . It is known that F is the tangent surface of a cubic polynomial curve, cf. [16, Ex. 7.1.2]. Proposition 3.2 now tells us that this curve has infinitely many triples of mutually orthogonal tangent planes. Translating these planes (the principal trihedra) through the origin, they become tangent planes of the directing cone of F, which is a quadratic cone. This cone is quadratic and must likewise have infinitely many orthogonal circumscribed trihedra. It is therefore a socalled Monge cone, see Fig. 6.
There is a phenomenon in geometry, called porism, cf. [7]. It refers to situations where existence of one object of a certain kind implies existence of an entire 1parameter family of such objects. Monge cones are an instance of a porism: If a quadratic cone has one circumscribed orthogonal trihedron, then one can move this trihedron around the cone while it remains tangential. This fact is classical knowledge in projective geometry, see e.e. [1, pp. 33–34].
The same porism is hidden in the proof of Proposition 3.2: The normality condition (6*) was equivalent to existence of the principal trihedron associated with \(P_0\), but it also implied existence of the trihedron for all \(P_\lambda \).
Details on principal trihedra in discretenormal congruences.
Proposition 3.3
If a congruence is defined by the affine correspondence between two triangles \({\mathbf a}_1{\mathbf a}_2{\mathbf a}_3\) and \({\mathbf b}_1{\mathbf b}_2{\mathbf b}_3\) and satisfies the normality condition (6*), then its focal surface has a 1parameter family of circumscribed ‘principal’ orthogonal trihedra whose apex moves on a straight line and whose edges form an algebraic surface of degree 4 which contains that line as a triple line.
The complicated geometry of these congruences reflects the difficulties in defining offset pairs of triangle meshes.
Discrete normal congruences — Version 3.
Comparison of definitions.
The various definitions of discrete normal congruences have different advantages. When one wants to design a normal congruence (as in Wang et al. [21]), version 1 may be better because it ensures orthogonality of focal planes in the part of the line congruence which is actually realized. Using version 2, orthogonal focal planes may occur outside the realized part. On the other hand, when using the normal congruence of a given surface, version 2 has the advantage that one plane of a principal frame contains the base mesh triangle; moreover discrete principal directions are orthogonal and lie in the plane of the triangle. Version 3 normality is not used here except for Fig. 8 where we show that imposing version 3 normality leads to results comparable to version 2. Since the weaker condition of version 2 is sufficient to achieve the same results, it is not necessary to impose version 3 normality.
4 Curvatures of Faces of Triangle Meshes

If Open image in new window is normal in the sense of Eqs. (6) and (7), then we apply the projection mentioned in Proposition 3.1, resulting in vertices \(\bar{\mathbf a}_1\bar{\mathbf a}_2\bar{\mathbf a}_3\), \(\bar{\mathbf b}_1\bar{\mathbf b}_2\bar{\mathbf b}_3\). The projection is in the direction of a certain unit vector \({\mathbf n}\).

As an alternative, the congruence may be normal in the sense of Eqs. (6*), (7*). Here we consider orthogonal projection onto the plane \(P_0\) which contains \({\mathbf a}_1{\mathbf a}_2{\mathbf a}_3\). This projection results in vertices \(\bar{\mathbf a}_i={\mathbf a}_i\) and \(\bar{\mathbf b}_i\). The projection is in direction of the unit normal vector \({\mathbf n}={\mathbf n}_0\) of the plane \(P_0\).
Proposition 4.1
The eigenvalues of the shape operator \(\varLambda \) are the principal curvatures \(\kappa _1,\kappa _2\), and its trace and determinant are given by 2H and K, respectively. Eigenvectors of \(\varLambda \) indicate the principal directions.
Proof
We first show the statement for ‘version 2’ normality. Recall the linear mapping \(\alpha \) in the proof of Proposition 3.2 which maps \(\bar{\mathbf a}_i\bar{\mathbf a}_j\mathop {\longmapsto }\limits ^{\alpha }(\bar{\mathbf a}_i+\bar{\mathbf e}_i)  (\bar{\mathbf a}_j+\bar{\mathbf e}_j)\). Since by construction, \(\varLambda =\mathord {\mathrm{{id}}}\alpha \), \(\varLambda \) has the same eigenvectors as \(\alpha \), i.e., the torsal directions. The statement about \(\mathop {\mathrm{{tr}}}\varLambda \) and \(\det \varLambda \) follows from the relations \( \det \varLambda = \frac{\det (\varLambda ({\mathbf x}),\varLambda ({\mathbf y}))}{\det ({\mathbf x},{\mathbf y})}\) and \(\mathop {\mathrm{{tr}}}\varLambda =\frac{\det (\varLambda ({\mathbf x}),{\mathbf y})+\det ({\mathbf x},\varLambda ({\mathbf y}))}{\det ({\mathbf x},{\mathbf y})}\) which generally hold for linear mappings of \(\mathbb R^2\). The statement about eigenvalues follows immediately.
For version 1 normality the proof is the same, only the bars have a different meaning. The mapping \(\alpha \) is also referred to in the proof of Proposition 3.1 in [21].
Since we have defined principal curvatures \(\kappa _1,\kappa _2\) implicitly via mean curvature H and Gauss curvature K, their relation to focal geometry is still unclear. In the smooth case, points at distance \(1/\kappa _i\) from the surface are focal points of the normal congruence. This property holds in the discrete case too, if we use version 2 normality:
Proposition 4.2
Consider a congruence with parametric representation \({\mathbf x}(u,v,\lambda )\) which is defined by the correspondence of two triangles \({\mathbf a}_1{\mathbf a}_2{\mathbf a}_3\) and \({\mathbf b}_1{\mathbf b}_2{\mathbf b}_3\). Assume that it is normal in the sense of Eq. (6*), and consider (in the notation of Proposition 3.2) the plane \(P_0\) which contains \({\mathbf a}_1{\mathbf a}_2{\mathbf a}_3\) and the corresponding normal \(L(u_0,v_0)\). Then the focal points of that line lie at distance \(1/\kappa _1\), \(1/\kappa _2\) from the plane \(P_0\), with \(\kappa _i\) as the principal curvatures, i.e., the focal points are precisely the points \({\mathbf x}(u_0,v_0,1/\kappa _i)\).
Proof
We consider the parametrization (8) which is with respect to an adapted coordinate system, so that \(u_0=0\) and \(v_0=0\). It is easy to see that the values \(\kappa _1,\kappa _2\) occurring there are indeed the principal curvatures. A simple computation shows that for the special case \(u=v=0\), the determinant of partial derivatives of \({\mathbf x}(u,v,\lambda )\) specializes to \( \mathopen [{\mathbf x}_u,{\mathbf x}_v,{\mathbf x}_\lambda \mathclose ] = (1\lambda \kappa _1)(1\lambda \kappa _2). \) Thus we have a singularity if \(\lambda =1/\kappa _i\).
Special cases.
An umbilic point is characterized by equality of principal curvatures, i.e., \(\kappa _1=\kappa _2=\kappa \). In this case some of the geometric objects discussed above simplify. E.g. the abovementioned cubic family of planes becomes the set of tangent planes of a quadratic cone with vertex \((0,0,1/\kappa )\). Such an umbilic occurs every time two corresponding triangles \( {\mathbf a}_1 {\mathbf a}_2 {\mathbf a}_3\) and \({\mathbf b}_1 {\mathbf b}_2 {\mathbf b}_3\) are in homothetic position, but the converse is not true.
A parabolic point is characterized by one principal curvature, say \(\kappa _1\), being zero. In this case, Eq. (8) immediately shows that the congruence vectors \({\mathbf e}_1, {\mathbf e}_2, {\mathbf e}_3\) associated with vertices \({\mathbf a}_1, {\mathbf a}_2, {\mathbf a}_3\) are not linearly independent, so Proposition 3.2 does not apply. Along the x axis, the lines of the congruence are parallel to each other, which is in accordance with the fact that the focal point \((0,0,1/\kappa _1)\) has moved to infinity. The abovementioned cubic family of planes is quadratic (in fact, it is the family of tangent planes of a parabolic cylinder).
Remark 4.3
We should mention that the approach to curvatures presented here carries over to relative differential geometry where the image of the Gauss map is not a sphere but a general convex body [19]. Another straightforward extension is to curvatures at vertices, which however does not lead to a shape operator in such a natural manner.
5 Results and Discussion
Numerical examples. Vertex normals of a mesh can be estimated (e.g. as areaweighted averages of face normals). Any such collection of sensible normals is not far away from being a “normal” congruence in our sense. By applying optimization, we can make it as normal as possible, meaning that (6) is fulfilled in the leastsquares sense. Numerical experiments show that this improves the quality of the normal field (even if there are not enough d.o.f. to satisfy (6) fully if the vertices of the mesh are kept fixed). Since curvatures and the distribution of normals are inseparable, it makes sense to study curvatures not only as quantities derived from a mesh, but as quantities which arise naturally from the the result of the optimization procedure just mentioned. In this way the natural sensitivity of curvatures with respect to noise is moderated.
Comparison of residuals regarding normalcy of the congruence (“c”) and unit vectors being normalized (“n”) when optimizing congruences
Sphere  Torus  Disk w/holes, see Fig. 10  

Fixed vertices  Fixed vertices  Moving vertices  Fixed vertices  Moving vertices  
c  n  c  n  c  n  c  n  c  n  
v. 1  \(7.8\times 10^{3}\)  0  \(7.7\times 10^{0}\)  0  –  –  \(1.5\times 10^{~0}\)  0  –  – 
v. 2  \(9.7 \times 10^{5}\)  0  \(9.6\times 10^{1}\)  0  \(6.9\times 10^{5}\)  \(8.1\times 10^{7}\)  \(1.9\times 10^{2}\)  0  \(4.0\times 10^{5}\)  \(2.2\times 10^{9}\) 
v. 3  \(9.0\times 10^{2}\)  0  \(1.3 \times 10^{1}\)  0  \(6.9\times 10^{4}\)  \(6.3\times 10^{4}\)  \(2.4\times 10^{1}\)  0  \(9.6\times 10^{5}\)  \(9.5\times 10^{10}\) 
Computing Curvatures. Once a normal congruence is available, we can compute curvatures (see Fig. 9) and we can integrate the field of principal curvature directions as well as the field of asymptotic directions (see Fig. 10 for an example). It must be said, however, that we do not want to compete with the many other methods for computing curvatures, and we do not regard the ability to compute curvatures a main result of this study.
Relevance for discrete differential geometry.
The idea of employing the Steiner formula for defining curvatures has proved very helpful in bringing together various different notions of curvature, and indeed, various different notions of discrete surfaces (like discrete minimal surfaces and discrete cmc surfaces) which were defined in a way not involving curvature directly but by other means like Christoffel duality. We refer to [2, 3] for more details. The theory presented in [2] is restricted to offsetlike pairs of polyhedral surfaces where corresponding edges and faces are parallel. There are ongoing efforts to extend this theory to more general situations (we point to recent work on quad meshes [10] and on isothermic triangle meshes of constant mean curvature [11]). It is therefore remarkable that at least for the situation described here, triangle meshes allow an approach to curvatures and even a shape operator which is likewise guided by the Steiner formula, but without the rather restrictive property of parallelity (which for triangle meshes would be even more restrictive).
Future Research.
As to discrete differential geometry, it is still unclear how known constructions of special discrete surfaces relate to the curvatures defined here: For instance, it seems difficult to gain nice geometric properties from the condition vanishing mean curvature. Nevertheless one of the known constructions of discrete minimal surfaces might be equipped with a canonical normal congruence such that, when our theory is applied, mean curvature vanishes.
Further applications of line congruences have been discussed by Wang et al. [21], but there might be other examples of geometry processing tasks where the notion of line congruence, or even normal congruence, becomes relevant.
Footnotes
 1.
The vector of coefficients \((n_1,n_2,n_3)\) of the equation of a plane is a normal vector of that plane. This shows that the orthogonal polar cone of the Monge cone fulfills the equation \((\kappa _1\kappa _2)x_1x_2ax_2x_3+bx_1x_3=0\). Since the Monge cone had many circumscribed orthogonal trihedra, its polar cone has many inscribed orthogonal frames. These frames are generated by translating the frames seen in Fig. 7b through the origin.
Notes
Acknowledgments
The authors are grateful to Alexander Bobenko for fruitful discussions and to the anonymous reviewers for their suggestions. This research was supported by the DFG Collaborative Research Center, TRR 109, “Discretization in Geometry and Dynamics” through grants I705 and I706 of the Austrian Science Fund (FWF).
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