Curvature line parametrization from circle patterns
We study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by respective discrete nets (circular nets and planar quadrilateral nets) with edges of order . It is shown that choosing the points of discrete nets on the smooth surface one can obtain the order approximation globally. Also a simple geometric construction for approximate determination of principal directions of smooth surfaces is given.
Department of Mathematics
Technische Universität Berlin
Discrete conjugate nets, defined as mappings with the condition that each elementary quadrangle is flat, and discrete nets of curvature lines, defined as discrete conjugate nets with additional property of circularity of the four vertices of every elementary quadrangle, play an important role in the contemporary discrete differential geometry (see e.g.  for a review) and have important applications to computer-aided geometric design .
This paper is devoted to a carefull study of the order of approximation of discrete nets to a given smooth conjugate net or the net of curvature lines on a smooth surface that can be achieved. We also present a remarkably simple approximate construction of principal directions on a given smooth surface using the standard primitive of discrete differential geometry: an elementary circular quadrangle inscribed into the surface. Roughly speaking, our results show that the previously known upper bounds () can be made one order better for conjugate nets and discrete nets of curvature lines. Moreover, one can impose one additional geometric requirement: all vertices of approximating discrete nets should lie on the original smooth surface.
The paper is organized as follows. In Section 2 we present a method of construction of principal directions in a neighborhood of a non-umbilic point on a smooth surface using only the four points of intersection of the surface with an infinitesimal circle. Our method is in fact based on a third-order approximation of the given smooth surface and gives an -approximation of principal directions if the infinitesimal circle has the radius . Different methods for approximation of principal directions have been proposed recently, see for example . Re-meshing techniques based on determination of principal directions are widely used in contemporary computer graphics ().
Section 3 is devoted to the study of the order of approximation of a single elementary quad of a discrete conjugate net (resp. discrete circular net) to the respective infinitesimal quadrangle of a the smooth net of conjugate lines (resp. curvature lines) on the original smooth surface. The results of this Section are used in Section 4 where we prove the main global results on the order of approximation of respective discrete nets to a given smooth conjugate net or the net of curvature lines on a smooth surface.
We use the following terminology and notations: is the distance from a point to another point (or a line, plane, etc.). A point is said to be -close to if , that is for . is said to lie at -distance from , if for some , we will denote this hereafter as . The same terminology and notations will be used in characterization of the asymptotic behavior for other (numeric) quantities depending on .
2 Principal curvature directions from circles
In this Section we show that one can use the elementary quadrangle of a discrete circular net inscribed into a smooth surface (see Section 3.2 for more details) for a rather precise construction of the principal directions on a smooth surface.
Euclidean construction. Take three points , , on a smooth surface without umbilic points, such that , and the angle is not -close to or .Then the circle passing through , , has radius of order as well; from elementary topological considerations one concludes that has at least one more point of intersection with . Denote it (or one of them) as . Take the straight lines , and the point of their intersection (see Fig. 1). The bisectors and of the angles formed at by and are obviously perpendicular. In fact and approximate the directions of smooth curvature lines on the surface: as we prove below, the directions of and are -close to the principal directions at any point which is -close to , , , . In fact one can find the principal directions with only -error at some specific point of which “lies over ”. Note that the order of the points , , , on the circle is important for finding the principal direction with the error of order .
Namely, let be the intersection of with the straight line passing through and perpendicular to the plane . Take the orthogonal (parallel to ) projections and of the bisectors and onto the tangent plane of at .
For a general smooth surface in a neighborhood of a non-umbilic point the Euclidean construction given above produces the directions , which are -close to the exact principal directions at .
In order to prove this Theorem we establish at first some simple but remarkable facts about planar quadrics and Dupin cyclides (Lemma 2 and Theorem 3) and give another (Möbius-invariant) construction of the principal directions (Fig. 2). In fact, the intersection curve is -close to a quadric — the Dupin indicatrix of at (see the Appendix); this explains in particular why we may assume that the number of intersection points of and is exactly four. If we take then an arbitrary circle in a plane, intersecting a quadric in points , , , (cf. Fig. 1 for the elliptic case; the order of the points , , , on the circle is not necessarily consecutive for Lemma 2 below to hold), then the directions of the angles formed by the straight lines and are parallel to axes of the quadric:
For any plane quadric with four point of intersection with a circle the bisectors and (shown on Fig. 1 for the elliptic case) are parallel to the axes of the quadric.
Proof. For simplicity we give the proof only for the case of a non-degenerate central quadric ; the other cases can be proved along the same lines. The coordinates of the intersection points , , , satisfy the system
They also lie on the one-parametric sheaf of quadrics : . Obviously the directions of the axes of all are the same. On the other hand, for some values of this quadric is degenerate. These values are the three roots of the equation
Since the bisectors of these pairs of lines define exactly the axes of the respective degenerate quadric , we have proved that their directions are the same as for .
Note that we have not used the ordering of the points , , , on the circle or on the quadric, so the statement of Lemma 2 holds true also for the bisectors of the angles formed by the lines and , as well as and . On the other hand in order to obtain the approximation of order below we fix the ordering as shown on Fig. 1.
Next we need to reformulate the Euclidean construction given above in terms of Möbius geometry:
Möbius-invariant construction. We start with a smooth surface . Take a sphere tangent to at some point , such that for its radius and the principal curvatures , () of at the inequalities hold (with non-infinitesimal differences and ). In this case intersects an -patch on along two smooth lines , (Fig. 2): , the angle between and is non-zero and non-infinitesimal (this follows from the classical lemma of Morse, cf. for example ). Take any circle of radius with inside such that the distances between and the four points of intersection of and : , are of order as well. Construct two auxiliary circles , on defined by the triples of points and respectively (not shown on Fig. 2, are very close to ). Then the bisectors , of the angles formed by , at obviously lie in the tangent plane to at . As we show below, and are -close to the principal directions of at .
First we prove that for Dupin cyclides (standard Möbius primitives, playing the role of the osculating paraboloids of the classical Euclidean differential geometry, see [6, 9]) are exactly the principal directions:
For a smooth non-umbilic point on a Dupin cyclide , the bisectors and of the Möbius-invariant construction give the principal directions of at .
Proof. Since our construction of is Möbius-invariant, we may transform into one of the normalized Dupin cyclides: a torus, a circular cone or a circular cylinder. We give below a detailed proof for the case of a torus; the other cases can be easily proved along the same lines.
Making, if necessary, another Möbius transformation, we can reduce the configuration to the following: the torus is given by the equation
the point has the coordinates and the sphere is given by the equation
After a Möbius inversion with the center :
(3) may be now reshaped to
so one can express in terms of , , , . Substituting (6), (7) and the found expression for into one gets a rational expression containing only , the constants , , and , . A straightforward calculation shows that for , this expression is a constant .
Since the images of the circles , defined above are now straight lines passing through the points of intersection of the image of the circle and the found quadric on the plane , the statement of this Theorem is now equivalent to the statement of Lemma 2.
For an arbitrary smooth surface in a neighborhood of a non-umbilic point the Möbius-invariant construction produces the directions , which are -close to the exact principal directions at .
Proof. First we perform a Möbius transformation that maps the sphere and the circle into themselves and brings into the (Euclidean) center of on , i.e. into the point such that the distance from to all the points of are equal. One can easily check that our requirement , …, guarantees that after the transformation these distances will remain of order . Inside the circle on and in its -neighborhood in where lies, the Jacobian of this transformation is limited so the distances of order , will be transformed into distances of the same order. Since the angles do not change, -close directions at remain -close at and vice versa. Introduce now a Cartesian coordinate system such that is its origin, the coordinate plane is tangent to (so to as well) and the -axes are the principal directions of at . The plane where lies will be given by the equation , the circle will be defined as and will be defined in a neighborhood of by its Taylor expansion
So the intersection points , , , of and will have respectively the coordinates with being the solutions of
so we see that
The next terms are found after substitution of into
cancellation of the second-order terms using (9) and retaining only third-order terms:
so the -shifted lines , defined by the pairs of points and are parallel to the lines , defined by the pairs and .
Thus the bisector directions between , in the plane are the same as for the unperturbed lines , ; so they are obviously parallel to the coordinate axes — the principal directions of at which is -close to both and .
Now we have two pairs of circles in : , passing through the triples of points and and the pair , passing through and . One easily concludes from (13) that , are tangent at , as well as , . Thus their bisectors are the same and parallel to the axes; the next -terms in approximations for the intersection points , , , will give a -change in the directions of , .
Remark. From the proof of the previous Theorem we see that in fact the Möbius-invariant construction gives the same (with -error) principal directions if we will substitute instead of the original surface its tangent paraboloid at (after the Möbius transformation described in the beginning of the proof). Also it should be noted that the -approximation was achieved taking into account approximating paraboloid of third order; the results obtained in this Section are therefore third order results despite the seemingly second-order construction based on an infinitesimal circle.
Proof of Theorem 1. Now we are in a position to prove the statement about -approximation of principal directions at in our Euclidean construction. For this we first perform an auxiliary construction in (Fig. 3):
take the circles , defined by the point triples and , then for the sphere where all these points and circles lie we take its tangent plane at . Next we construct another two circles , as the sections of the sphere by the “bisectoral planes” passing through and the lines , respectively. First we remark that the constructed plane is -close to the tangent plane of at : the circles , are approximating the osculating circles of the planar sections of by the planes and . As one can check by a direct computation for any planar smooth curve , if one takes the circle passing through three points lying on -distances from each other on , then the tangents to and at any of the three pins will be -close. So the tangents to the aforementioned planar sections and the tangents to the respective circles , at are -close. (Here we use the same terminology of “-closeness” for pairs of lines or planes, this means that the respective angles are .) This shows that our sphere is “very close” to the sphere (which should be tangent to at ) used in the Möbius-invariant construction. We also see that our Euclidean construction is nearly reduced to the Möbius-invariant construction, with the sphere substituted by : the orthogonal projections (along the line ) of , onto are nothing but the tangents to the circles , which are -close to the circles which exactly bisect the angles between the circles , at . This follows from the fact, that the angle between and is : as one can easily check by a direct computation, in this situation the (non-infinitesimal) angle between any two lines on one of the planes and the angle between their orthogonal projections on the other plane are -close.
Let us now estimate the angles between the tangents to the circles (lying in ) and the principal directions of at (lying in , which is -close to ). For this we use the same technology as in the proof of Theorem 4: first we perform a Möbius transformation leaving and invariant and bringing the point to the center of on , then introduce a Cartesian coordinate system with as its origin and being its -plane. Now the equation (8) of the surface will be modified by small linear terms:
with . The first terms , in the Taylor expansions of the coordinates of the points , , , will be found from the same system (9) so we have the relations (10). The next terms , are found from a modified system of the form (12):
We see that the main conclusion (13) still holds; from this point we just follow the guidelines of the proof of Theorem 4 and show that the bisectors of the angles between the circles , coincide (up to a -error) with the directions of the - and -axes. On the other hand since we know that , using the standard differential-geometric formulas for calculation of the coefficients of the second fundamental form and the principal directions we conclude that the principal directions of given by the equations (14) are -close to the - and -axes at as well (as the directions of the straight lines in ).
Remark. From the proof of this Theorem we see that in fact one can assume (with -error) in the Euclidean construction that the tangent plane at is the readily constructible plane .
3 Local results
3.1 Smooth and discrete conjugate quads
Suppose that a smooth surface parametrized locally by some curvilinear net of conjugate lines is given: , . Take some initial point and points , at -distance on the two conjugate lines of the net on and let be the fourth point of the curvilinear quad on .
Using the conjugacy condition
and its derivatives, one can easily estimate the distance from this fourth point to the plane defined by , , ; we formulate this as our next Theorem.
We remind that smooth conjugate nets are supposed to be non-degenerate, that is the directions of the curvilinear coordinate lines of such a net are non-asymptotic, so the angle between the tangent vectors and is everywhere different from zero.
For an arbitrary smooth non-degenerate conjugate net , .
Proof. Using the standard Taylor expansions one gets:
where all derivatives are taken at the point .
Taking into account (15) and its derivatives
we can compute the triple product which gives the volume of the parallelepiped spanned by the vectors , , :
Since the area of the base of this parallelepiped , we get that its height , so at least we may state that . In fact, using Taylor expansions in (16) up to order one can obtain after a lengthy computation that
which proves that for a generic conjugate net we have .
In fact we can even choose (the fourth point of an elementary planar quad ) on the plane sufficiently close to and lying on the given smooth surface :
For a given smooth non-degenerate conjugate net and the plane constructed as above one can choose a point such that .
Proof. As we prove in the Appendix, the intersection of the plane and the surface in the -neighborhood of the points , , , for sufficiently small is a curve close to a quadric – the Dupin indicatrix of , and the angle between and is in all points of . According to Theorem 5, . From this we can easily see that the distance between and the closest to point on should be .
3.2 Curvilinear quads of curvature lines and plane circular quads
For a given smooth surface parametrized by curvature lines one can prove a similar result, if we take the circle passing through the points , , defined as in Section 3.1.
For arbitrary smooth net of curvature lines in a neighborhood of a non-umbilic point on , the distance between the fourth point of intersection of the circle with and the point has order 3: .
In order to prove this theorem we need to establish a few auxiliary Lemmas.
For arbitrary smooth net of curvature lines in a neighborhood of a non-umbilic point on , .
Proof. Following an approach proposed in , we introduce for each of the points , , , purely imaginary quaternion , , , . The point will be chosen as the origin, i.e. . The basic quaternions , , are chosen to be tangent to the curvature line directions , and the normal vector to the surface in the initial point respectively. In view of future applications in the theory of triply orthogonal coordinate systems we include the given surface into such a system , being one of the coordinate surfaces . This is always possible (see e.g. ): one can take for example the one-parametric family of surfaces parallel to and the other two one-parametric families of developable surfaces defined by the curvature lines of and the normals to .
Introducing for this 3-orthogonal system the Lamé coefficients , , , , normalized vectors and the rotation coefficients , , , we have the following relations ():
where , . In the initial point we have , , . The Taylor expansions for the other points are now easily obtained after differentiation of , using (18):
Calculating now the quaternionic cross-ratio () and taking its imaginary part, one can see that . According to  we conclude that for the distance from the point to the circle of size defined by the points , , .
In the following we fix an -patch of size on the surface , such that all the points , , , belong to it. All subsequent constructions will be applied only to this -patch .
From Lemma 16 (proved in the Appendix) we know that the intersection curve is -close to the Dupin indicatrix of at some point . In fact we can take the indicatrix at the following point instead: choose the “center point” on such that is the intersection of and the normal to the plane passing through the center of the circle . In the Cartesian coordinate system (similar to the system used in the proof of Lemma 16), where the osculating paraboloid at is , the points , , , will have coordinates
so for the angle between the plane and the tangent plane to at we have . Moreover, the following is true:
The centers of the Dupin indicatrix and are -close. The angles of intersection of and are .
Proof. From (20) we easily deduce the first statement. The radius of is , while the axes of are and . Since we assume on we see that the intersection angles of and , so also of and , are .
Now using the same technique as in the proof of Lemma 16, we deduce that the Dupin indicatrix at and are -close.
4 Global results
4.1 Conjugate nets
Using the results of Section 3.1 one can try to construct inductively an approximating discrete conjugate net for sufficiently small . At the first glance the following simplest strategy may be applied: starting from the initial point