Rational blending surfaces between quadrics

Johannes Wallner, Helmut Pottmann

Abstract Using tools from classical line geometry and the theory of kinematic mappings, it is possible to dene an intrinsic control structure for NURBS curves and surfaces on the sphere, the cylinder and on any projectively equivalent quadratic surface. These methods are further used to construct exact C 1 blends between these surfaces, such that interactive design of trim lines and surface tension is possible. The lowest possible degree of a blend that can be achieved with this method is (4, 3). Keywords: kinematic mapping; line geometry; blending surface; NURBS surface Blending surfaces are surfaces smoothly joining two other surfaces. There are a variety of dierent types of blends: implicit blends, which are given by a relation of the type F (x, y, z) = 0 or parametric blends, which are given by an equation of the form F = (x(u, v), y(u, v), z(u, v)). Depending on the original surfaces we have to join, we can use approximate or exact blends. For a survey of dierent blending methods using parametric surfaces see e.g. (Varady, 1994) and the extensive bibliography given therein, or, for implicit blending, (Homann, 1987) and (Warren, 1989). It is the aim of this paper to present a method to construct exact C 1 rational blending surfaces between quadrics, which are frequently used primitives in solid modeling systems. For this purpose we use quadratic projections which have their origin in the theory of kinematic mappings to dene intrinsic control structures for NURBS curves and surfaces on quadrics.

The spherical kinematic mapping

In this section we describe well known connections between the group SO 3 of rotations in three-space and the space of quaternions. In euclidean four-space R 4 we have coordinates (x0 , . . . , x3 ). We identify (a0 , . . . , a3 ) R4 with the quaternion a = a0 + ia1 + ja2 + ka3 . The conjugate quaternion is denoted by a = a0 ia1 ja2 ka3 . The canonical scalar product in R4 gives rise to a norm N (a) = a2 + a2 + a2 + a2 = a, which is multiplicative, that is N (ab) = N (a)N (b). a 0 1 2 3 We embed R3 into R4 by letting (x, y, z) ix + jy + kz. Then the unit sphere S 3 = {a R4 |N (a) = 1} acts on R3 by multiplication: If N (a) = 1 and x R3 , then ax R3 . The mapping x ax is linear and because of N (ax) = N (x) it is a a a isometric. It can also be shown that it is orientation preserving, which means that it is a rotation. Now it is well known that for all rotations L SO3 there exists an a S 3 such that precisely a and a describe L. This allows us to dene a 1-1 mapping from the group of rotations onto projective three space P 3 , which will be called the spherical kinematic mapping. In P 3 we will use homogeneous coordinates, a point Ra P 3 consists of all scalar

multiples of a homogeneous coordinate vector a = (a0 , . . . , a3 ). Choose L SO3 and let a S 3 correspond to L. Then (L) is Ra. Note that is well dened. The scalar product in R4 induces a distance d(., .) in P 3 by cos d(Ra, Rb) = (ab)/ N (a)N (b). Thus P 3 becomes an elliptic space according to the CayleyKlein classication. For details on , see for instance (Mller, 1962). The following u list of properties can be easily veried: 1. One-parameter subgroups of SO3 and their cosets are mapped to straight lines. 2. The group-invariant distance in SO3 which equals the angle between two rotations, is mapped to twice the distance in P 3 . 3. Fix a unit vector n S 2 R3 . For all unit vectors v S 2 the set of all rotations which map n to v has a straight line as image under . These lines form a bration of P 3 , which is known to geometers as an elliptic net. The induced bration in S 3 is known as the Hopf bration.

A spherical control structure

We choose a unit vector n S 2 and dene a mapping : P 3 S 2 by letting

n can be chosen such that has the following representation in homogeneous coordinates: : R4 R4 , x y = (x), y0 y1 y2 y3 = x2 + x2 + x2 + x2 , 3 2 1 0 = 2(x0 x1 x2 x3 ), = 2(x1 x3 x0 x2 ), = x2 + x2 x2 x2 . 1 2 0 3 (1)

Sometimes we will look at as at a mapping P 3 P 3 , sometimes as at a mapping R4 R4 . When restricted to the unit sphere S 3 , is called the Hopf map S 3 S 2 . It has been proved in (Dietz et al., 1993) that a rational curve or surface of degree 2m contained in the unit sphere has a representation of the form y = (x), where the xi are univariate or bivariate polynomials of degree less or equal to m (see also Dietz et al., 1995). also is the composition of an elliptic net projection and the inverse of a stereographic projection, and therefore is also called generalized stereographic projection. Now the inverse images of points are just the lines mentioned above. If we neglect metric properties of , an analogous map exists for all oval quadrics in P 3 , which are projectively equivalent to the sphere. It is now possible to dene an intrinsic control structure for NURBS curves on the sphere, as described in (Pottmann, 1995). Choose a sequence d0 , . . . of de Boor points and a sequence f0 , . . . of Farin points, choose p0 in 1 d0 and calculate 1 successively points q0 , p1 , q1 , p2 , . . . such that qi = 2 (pi + pi+1 ), (qi ) = fi and (pi ) = di . If we use the sequence p0 , p1 , . . . as a sequence of control points for a C k piecewise polynomial spline curve p(t) of degree n, (p(t)) will be a C k piecewise rational spline curve of degree 2n, which does not depend on the choice of p 0 , because an elliptic net admits a one-parameter group of automorphic collineations which map each line of the net onto itself. If we want to generate closed NURBS curves on the sphere described by the sequence d0 , f0 , . . . , dn1 , fn1 , dn = d0 , we extend the sequence dj by dening di+n = di and construct a corresponding control polygon pi . This does not have to 2

Figure 1: control structure for rational curves on the sphere be closed, but if we use a periodic knot vector, the -image of the resulting polynomial spline curve will be a periodic NURBS curve on the sphere, as is illustrated in gure 1.

The kinematic mapping of Blaschke and Gr nu wald

To perform similar constructions in the case of the unit cylinder Z 2 with equation x2 + y 2 = 1, we can do the following (see Blaschke and Mller, 1956): SO3 was u the group of invertible linear transformations automorphic for the unit sphere. We introduce an isotropic scalar product in R3 , that is a symmetric bilinear form of defect 1 and index 0, such as (x1 , y1 , z1 ) (x2 , y2 , z2 ) = x1 x2 + y1 y2 . Then we dene a norm by x = x.x and Z 2 is the unit sphere in this metric. The group G of those invertible linear transformations which are automorphic for Z 2 and leave the z-axis pointwise xed, is isomorphic to the planar euclidean motion group OA2 . An isomorphism is constructed as follows: Let x = (x1 , x2 , x3 ) Z 2 and consider the linear function fx (u, v) = x3 x2 u + x1 v. fx 0 denes a half-plane in R2 , whose boundary is the oriented line (x). The action of OA2 on oriented lines now equals the -image of G. The following construction is completely analogous to the spherical case: Let Q3 equal P 3 without the line x0 = x3 = 0. The kinematic mapping of Blaschke and Grunwald, which will be denoted by for simplicity, is a one-to-one mapping G = OA2 Q3 . In homogeneous coordinates, maps a euclidean rotation with center (xm , ym ) and angle to the point R( cot(/2), xm , ym , 1) and a euclidean translation with vector (u, v) to the point R(1, v, u, 0). This makes even smooth. We introduce a scalar product of defect 2 and index 0 in R4 by letting x y = x0 y0 + x3 y3 , which induces a distance d(., .) in Q3 by letting cos d(Ra, Rb) = a b/ (a a)(b b). Then has the following properties: 1. One-parameter subgroups of OA2 and their cosets are mapped to straight lines in Q3 . 2. The group-invariant distance in OA2 which equals the angle between two rotations or translations, is mapped to twice the distance in Q3 . 3. Fix a unit vector n Z 2 R3 . For all unit vectors v Z 2 the set of all L OA2 which map n to v has a straight line as image under . These lines form a bration of Q3 , which is known to geometers as a parabolic net.

A cylindrical control structure

We dene an intrinsic control structure for NURBS curves on a cylinder or a cone in projective three-space P 3 . First, it is sucient to consider the unit cylinder Z 2 : x2 + x2 = x2 , as all singular quadrics with 1 singular point are projectively 1 2 0 equivalent. From the Klein model of the Grassmann manifold of lines in threespace, where oval quadrics correspond to elliptic, ruled quadrics to hyperbolic and singular quadrics with one singular point to parabolic nets, we would suppose that there is a quadratic projection P 3 Z 2 with the lines of a parabolic net as inverse images of points.

Figure 2: convex hull property Figure 3: control points for trimlines This is accomplished in the following way: Let x Q3 and let L = 1 (x) be the preimage of x under the kinematic mapping. From (Strubecker, 1961) it follows that we can choose n Z 2 , such that the mapping : Q3 Z 2 : (x) = L(n), has the following representation in homogeneous coordinates: y0 y1 y2 y3 = x2 + x2 , 0 3 = x2 x2 , 3 0 = 2x3 x0 , = 2(x3 x1 x2 x0 ),

(2)

and thus has the desired properties. For more details on , see also (Dietz, 1995). The cylindrical control structure now is completely anologous to the spherical case. There are, however, some diculties that do not arise in the case of an elliptic net. The condition qi = 1 (pi + pi+1 ) can be fullled uniquely if e.g. the lines 1 di , 2 1 fi and 1 di+1 are skew and cannot be fullled if two of the lines intersect and the third is skew to them. In an elliptic net, all lines are pairwise skew. A parabolic net consists of line pencils, which correspond to lines on the cylinder. Lines of dierent pencils are skew. Therefore, if two of the points di , fi , di+1 lie on a common generator line l Z 2 , the third point must also be in l. This is also clear from the fact that the line segment pi pi+1 is mapped to a quadratic rational curve, which is part of a planar intersection of the cylinder, and if two points of a line lie in a plane, then the whole line is contained in the plane. It is possible to derive a variation diminishing property and a convex hull property for -images of segments of quadratic Bzier curves c: c is contained in some plane e which contains some line l belonging to the elliptic or parabolic net. Every line contained in does intersect l and therefore (c) contains the point (l). For all conic sections d on the sphere or cylinder through (l) there exists a line d = d. Therefore both the variation diminishing property and the convex with (d) hull property with respect to the conic sections containing (l) hold (see gure 2). A stereographic projection with center (l) maps all these conics to lines, so in a stereographic image this is just the classical case.

Construction of Blending surfaces

We can use the control structures dened above to nd rational blending surfaces between oval and singular quadrics. The linegeometric model of ruled quadrics, the hyperbolic net together with the hyperbolic motion group, would make it possible to perform analogous constructions, but ruled quadrics do not play such an important role in practice. The construction is as follows: 1. Dene four NURBS curves by their intrinsic control polygons di1 , ci1 , . . . , di4 , ci4 . (see gure 3) The two outer curves will be the trimlines of a blend. 2. Construct the corresponding polygons pi1 , . . . in R4 , such that the conditions given above are fullled, and use {pi1 , pi2 } and {pi3 , pi4 } as control net for (m, 1) piecewise TP-polynomial ruled spline surfaces.

Figure 4: Denition of the transition surface 3. Apply (the spherical in the case of a sphere, the cylindrical in the case of a cylinder). This gives us a (2m, 2)-NURBS patch on each of the two quadrics. Calculate the control points of the two surface patches. 4. Select the outermost two control point rows and dene them to be the control net of a (2n, 3)-NURBS transition surface (see gure 4). The two patches do not depend on the choice of the p0j , but the transition surface does. This could be used as a design parameter together with the choice of the trimlines themselves. As the boundary curve and the rst derivatives of a NURBS surface are completely determined by the rst two rows of control points, the so dened transition surface is an exact C 1 blend between the two quadrics. By letting m = 2 we achieve the lowest degree which is possible for a C 1 -blend constructed with this method: (4, 3). The trimlines could be given by the designer or be chosen automatically, e.g. the distance to the intersection curve could be chosen as a function of the angle between the surfaces. So far we have not dealt with the case of closed intersection curves and trimlines. Here some problems arise. As mentioned above, the curves x(t) do not have to be closed for their projections y(t) to be closed and to be of the same dierentiability class. For the transition surfaces, however, the situation is dierent. As C 1 - or G1 -boundary conditions for NURBS surfaces are rather complicated, it is desirable to have something closed in R4 to project. According to (Pottmann, 1995), a spherical polygon c denes a path in SO 3 : First we rotate along c1 , then along c2 , and so on. If we apply the spherical kinematic mapping to this path, we get just the polygon Rp0 , . . . , Rpn in P 3 . Analogously, a polygon consisting of elliptic segments on the cylinder denes a path in the planar euclidean motion group: The -image of an elliptic arc c i contained in Z 2 equals the set of oriented tangents of an oriented circular arc ki in the euclidean plane, or, as the degenerate case with radius 0, a sector of a line pencil. We rotate along k1 , then along k2 , and so on. Then we apply the kinematic mapping of Blaschke and Grnwald and get just the polygon Rp0 , . . . , Rpn in Q3 P 3 . u To formulate a closedness condition, it is necessary to dene the notion of angle between two oriented segments c1 and c2 of conic sections on the sphere or cylinder, where the endpoint p of c1 equals the starting point of c2 . In the case of the sphere the angle is dened to be the euclidean angle between tangent vectors. In the case of the cylinder, the angle is dened to be the isotropic angle, which is the dierence of the slopes of the tangent vectors to ci in p. The slope of the vector (x, y, z) equals k = z/ x2 + y 2 . It is important for us that this angle is an invariant under the groups SO3 and OA2 , respectively. Having dened the turning angle at each vertex of a spherical or cylindrical polygon, we dene the total turning number als the sum of the turning angles. Proposition 1: Let p0 , . . . , pn be the vertices of a polygon in R4 and let the spherical or cylindrical -image c, consisting of segments c1 , . . . , cn be closed. Then Rp0 = Rpn if and only if the total turning angle of c equals 2k for the spherical and 0 for the cylindrical case Proof: From the corresponding kinematic mapping it is clear that c is closed if and only if the motion corresponding to the endpoint of the last segment is the identity mapping. A motion with one point xed is uniquely determined by the image of one tangent vector. On the sphere this condition if fullled if and ony if 5

Figure 5: Example of blending surface between oval quadrics Figure 6: Example of blending surface between oval and singular quadric a tangent vector rotates about an angle of 2k during the motion. On the cylinder this is equivalent to the condition that after moving a tangent vector around its slope has not changed, which in our terms is expressed by a vanishing total turning angle. For design purposes it is best to leave the ci unchanged and change the fi such that the total turning angle will equal the desired value and the polygon Rp i will be closed. To achieve closedness of the polygon pi in R4 , we rst change the fi to be the group-midpoint of pi and pi+1 in the one-parameter subgroup determined by ci . Without loss of generality p0 = 1. Then pi = 1 for all i and qi = (pi + pi+1 ). The rotation along ci is mapped to a segment of a great circle starting at pi , having its midpoint at qi and ending at pi+1 . As both the unit spheres S 3 for the spherical and Z 3 for the cylindrical case are twofold coverings of P 3 and Q3 , resp., with bers consisting of pairs of antipodal points, for the case cn = c0 the endpoint pn will be either p0 or p0 . Let pi and qi be two such polygons. If the corresponding paths in the appropriate motion group are homotopic, p0 = pn if and only if q0 = qn . This gives us Proposition 2: Let c = c1 , . . . , cn and d = d1 , . . . , dn be two closed spherical polygons satisfying the angle condition. After changing the total turning angle of at most one of the two polygons there exist -preimages p0 , . . . and q0 , . . . such that the -image of a piecewise polynomial (m, 1) spline surface dened by the p i and the qi is a closed C 1 piecewise polynomial (2m, 2) spline surface in R4 . Proof: The homotopy relation will be denoted by . For details on homotopy and coverings, see e.g. (Bredon, 1993). If for the corresponding paths c and d in holds, either both pn = p0 and qn = q0 or both the spherical motion group c d pn = p0 and qn = q0 . As (x) = (x), the proposition then follows. If c d, w.l.o.g. we can assume pn = p0 and qn = q0 . Now we construct a spherical polygon d0 , . . . , dn with the same starting points and endpoints as d0 , . . . , dn by adding a total turning angle of 2. Let denote the rotation with angle 2 and axis R(p 0 ). Then 0 and d d imply d c. Proposition 3: Let c and d be two closed homotopic cylindrical polygons not containing a line segment and satisfying the angle condition. Then there exist preimages p0 , . . . and q0 , . . . with the same properties as in Prop. 2. Proof: If the cylindrical polygons are homotopic, the corresponding paths in the euclidean motion group are homotopic. Therefore either both pn = p0 and qn = q0 or both pn = p0 and qn = q0 . As (x) = (x), the proposition follows. Now let two oval or singular quadratic surfaces and on each of them two closed polygons satisfying the angle condition be given. It follows from propositions 2 and 3 that after changing the total turning number of at most one of them, the construction described in gure 4 can be applied. Figures 5 and 6 show some examples.

References
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