Envelopes – Computational Theory and Applications

Category: survey

Abstract for points, whose tangent plane maps to a line under the projection. These points form the so-called Based on classical geometric concepts we discuss contour of the surface with respect to the given the computational geometry of envelopes. The projection; the image of the contour is the silhou- main focus is on envelopes of planes and natural ette. Since the tangent t at a point p of the silhou- quadrics. There, it turns out that projective duality ette is the projection of the tangent plane at a sur- and sphere geometry are powerful tools for devel- face point, both the u-curve and the v-curve passing oping efﬁcient algorithms. The general concepts through p have t as tangent there. This shows the are illustrated at hand of applications in geometric envelope property. Points on the envelope are also modeling. Those include the design and NURBS called characteristic points. representation of developable surfaces, canal surfaces and offsets. Moreover, we present applica- X c tions in NC machining, geometrical optics, geo- v metric tolerancing and error analysis in CAD constructions. Keywords: envelope, duality, sphere geometry, NURBS surface, developable surface, canal surface, geometric tolerancing.

s 1 Introduction p

2 Deﬁnition and computa- Figure 1: The silhouette of a surface as envelope of tion of envelopes the projections of the parameter lines

2.1 First order analysis For an analytic representation we note that the tangent plane of the surface is spanned by the tan- We will deﬁne and discuss envelopes at hand of a gent vectors to the parameter lines, which are the spatial interpretation. Let us ﬁrst illustrate this ap- partial derivatives with respect to u and v, proach with a simple special case. Consider a sur- ∂ ∂ face in Euclidean 3-space R3, which has the regular X X ∂ =: Xu, ∂ =: Xv. parameterization u v

Under the normal projection onto the plane x3 = 0, X(u,v)=(x1(u,v),x2(u,v),x3(u,v)). the tangent plane has a line as image iff it is parallel to the x -axis. In other words, the projections Here and in the sequel, we always assume suffi- 3 xu,xv of the two vectors Xu,Xv must be linearly de- cient differentiability. The curves u = const and pendent, i.e., v = const form two one-parameter families of curves (‘parameter lines’) on the surface. Under x1,u x2,u det(xu,xv)= 0 ⇐⇒ = 0. (1) a projection, e.g. the normal projection onto the x1 v x2 v , , plane x3 = 0, each family of parameter lines is mapped onto a one-parameter set of planar curves. The computation of the silhouette (envelope) s The envelope of each of these two curve families is therefore equivalent to finding the zero set of is the silhouette s of the surface X (see Fig.1). It a bivariate function. This is done with a stan- is defined as follows: On the surface X, we search dard contouring algorithm, a special case of a surface/surface intersection algorithm (see [21]). The ui are the parameters of the n-dimensional sur- Let us recall that (1) characterizes the envelope faces and the λk are the set parameters. We inter- of two curve families, described by x(u,v) = pret this set of surfaces as normal projection of a k+n+1 d (x1(u,v),x2(u,v)): the set of u-lines (where u is the hypersurface Φ in R onto R (xd+1 = ... = curve parameter and v is the set parameter) and the xk+n+1 = 0). At any point of Φ, the tangent hyper- set of v-lines (where v is the curve parameter and u plane is spanned by the partial derivatives with re- is the set parameter). spect to u1,...,un,λ1,...,λk. Their projections are Given any one-parameter set of curves in the the partial derivatives of f . At a point of the sil- plane with a parametric representation x(u,v), we houette (= envelope), these derivative vectors must can interpret it as projection of a surface onto that span at most a hyperplane in Rd. This yields the plane; it is sufficient to set x3(u,v)= v. If the curve envelope condition family is given in implicit form, ∂ f ∂ f ∂ f ∂ f rank( ,..., , ,..., ) < d. (4) ∂u1 ∂un ∂λ1 ∂λk f (x1,x2,v)= 0, It characterizes points on the envelope by a rank we interpret it again as projection of a surface with deficiency of the Jacobian matrix of f . the level curves f (x1,x2,v)= 0 in the planes x3 = v. Analogously, we find the following known re- The tangent planes of this surface f (x1,x2,x3)= 0 sult. The envelope of a k-parameter set of hyper- are normal to the gradient vectors surfaces in Rd, analytically represented by λ λ ∂ f ∂ f ∂ f f (x1,...,xd, 1,..., k)= 0, (5) ∇ f =( , , ). ∂x1 ∂x2 ∂x3 is the solution of the system

Hence, a tangent plane is projected onto a line iff f (x1,...,λk)= 0, (6) the gradient vector is normal to the x3-axis, i.e. ∂ f ∂ f (x1,...,λk)= 0,..., (x1,...,λk)= 0. ∂λ1 ∂λk ∂ f = 0. The envelope conditions elucidate the computa- ∂x 3 tional difﬁculties: we have to solve a nonlinear sys- We see that the silhouette (= envelope) is the solu- tem of equations. tion of the system (we set again x3 = v), In special cases, a variety of methods are avail- able. Let us give an important example. For a one- ∂ f parameter set of surfaces in R3 (n = 2,k = 1), equa- f (x ,x ,v)= 0, (x ,x ,v)= 0. (2) 1 2 ∂v 1 2 tion (4) can be formulated as ∂ ∂ ∂ This is a well-known result. Its computational f f f det(∂ , ∂ , ∂λ )= 0. treatment can follow our derivation: one has to u1 u2 1 compute a projection of the intersection of two Hence, we have to compute the zero set of a trivari- implicit surfaces (see [21]). An implicit rep- ate function, which can be done, for example, with resentation of the envelope follows by elimina- marching cubes [31]. The solution is the preim- tion of v from (1). A parametric representation age of the envelope in the parameter domain, and (x1(v),x2(v)) is obtained by solving for x1 and x2. application of f to it results in the envelope itself. By the spatial interpretation, various algorithms for For systems of polynomial or rational equations, computing silhouettes may also be applied [1, 21]. one can use algorithms which employ the subdi- With the understanding of this special case, it vision property of Bezier´ and NURBS representa- is straightforward to extend the results to the gen- tions [10, 25, 56]. In more general situations one eral case. It concerns envelopes of k-parameter sets realizes the lack of algorithms for geometric proof n-dimensional surfaces in Rd (n < d). We set cessing in higher dimensions. k + n ≥ d, since otherwise there is in general no envelope. 2.2 Singularities, curvature com- Let us start with the parametric representation in putation and approximation of the form envelopes λ λ (u1,...,un, 1,..., k) 7→ Let us continue with the special case of a one- d f (u1,...,un,λ1,...,λk) ∈ R . (3) parameter family of curves in the plane and its en-

2 velope, viewed as silhouette of a surface in R3. It also shows cusps at the curve of regression. They is well known in classical constructive geometry are additionally satisfying [6] that the silhouette of a surface possesses a cusp ∂3 if the projection ray is an osculating tangent at the f (x1,x2,x3,v)= 0. corresponding point of the contour. Therefore, the ∂v3 condition for cusps follows by expressing that the The transition to higher dimensions is straightfor- x3-parallel projection ray has contact of order two ward: A one-parameter family of hypersurfaces in with the surface. Rd, written as Let us work out this condition in case that the 3 curve family (surface in R ) is given in implicit f (x1,...,xd,v)= 0, 0 0 0 form f (x1,x2,v)= 0. We pick a point (x1,x2,v ) on the surface and express that the projection ray has an envelope hypersurface which also satisﬁes 0 0 0 R (x1,x2,v + t), t ∈ , has contact of order two ∂ f at this point. We use the Taylor expansion at (x1,...,xd,v)= 0. 0 0 0 ∂v (x1,x2,v ), On this envelope, we have a (d − 2)-dimensional f (x1,x2,v)= (7) surface of singular points characterized by ∂ ∂ ∂ f 0 f 0 f 0 2 (x1 − x1)+ (x2 − x2)+ (v − v )+ ∂ f ∂x1 ∂x2 ∂v (x ,...,x ,v)= 0. ∂ 2 1 d ∂2 ∂2 v 1 f 0 2 f 0 2 (x1 − x ) + ... + (v − v ) +(∗). 2 ∂x2 1 ∂v2 The singularities of this surface possess vanishing 1 third derivative of f , and so on. Clearly, the enve- Here, all partial derivatives are evaluated at lope and the singular sets need not be real. 0 0 0 (x1,x2,v ) and (*) denotes terms of order ≥ 3. In- For a reliable approximation of envelope serting v0 +t for v, we must get a threefold zero at curves/surfaces, the computation of singularities t = 0 in order to have second order contact. Using is an important subtask. More generally, we can 0 0 0 the fact that (x1,x2,v ) is a point on the contour, use results of constructive differential geometry to this requires in addition compute curvatures. These results concern curvature constructions for silhouettes [6]. An example ∂2 f of an application of such a formula to envelope (x0,x0,v0)= 0. (8) ∂v2 1 2 curves has recently been given by Pottmann et al [50]. We will brieﬂy address it in section 7. Analogously, we can investigate singular points There is still a lot of room for future research on the envelope surface of a one-parameter family in this area. Reliable approximation of envelopes R3 of surfaces in . These points, if they exist at all, is one of our current research topics. It involves form in general a curve, which is called an edge a segmentation according to special points of the of regression. Planar intersections of the envelope envelope, curvature computations and approxima- surface possess (in general) a cusp at points of the tion schemes based on derivative information up to edge of regression. Points of the edge of regression second order at discrete points. solve the system

f (x1,x2,x3,v)= 0, (9) 3 Kinematical applications ∂ f ∂2 f (x ,x ,x ,v)= 0, (x ,x ,x ,v)= 0. ∂v 1 2 3 ∂v2 1 2 3 Classical kinematical geometry studies rigid body motions in the plane and in 3-space [5, 23, 24]. The characteristic curves, along which the surfaces Because of their appearance in applications such f (x1,x2,x3,v)= 0 touch the envelope, are tangent as the construction of gears, simulation and veri- to the regression curve. Thus, the regression curve fication for NC machining, collision avoidance in may be seen as envelope of the family of character- robot motion planning, etc., envelopes have re- istic curves on the envelope surface. As an exam- ceived a lot of attention in this area. We will briefly ple, Fig. 6 shows the envelope surface of a family address some main ideas and point to the literature. of planes. The characteristic curves are lines which Let us start with a one-parameter motion in the touch the curve of regression. Note also the cusp plane R2 and consider a curve c(u) in the mov- in a planar intersection of the surface. The figure ing plane Σ; its position in the fixed plane Σ0 at

3 time t shall be c(u,t). Any point c(u0) of the curve Thus, we search on the curves a(t) and c(u) for generates a path c(u0,t) in the fixed plane Σ0. By points a(t0),c(u0) with parallel tangents and per- equation (1) we get an envelope point, if the tan- form the addition a(t0)+c(u0). Since the curvature gent vector ∂c/∂t to the path (velocity vector) is computation depends just on derivatives up to sec- tangent to the curve position (linearly dependent to ond order, we can replace the curve a by its osculat- ∂c/∂u (see Fig. 2). The construction is simplified ing circle at a(t0), and analogously we compute the by looking at the velocity distribution at a time in- osculating circle of c at c(u0). If we translate two stant t. We either have the same velocity vector circles along each other, the envelope is formed of at all points (instantaneous translation) or we have two circles. Taking orientations into account and ρ0 ρ0 an instantaneous rotation, where the velocity vec- respecting signs of the curvature radii a and c tor of a point x is normal to the connection with the (defined as the sign of the curvature), we see that instantaneous rotation center p (pole) and its length the envelope has curvature radius is proportional to kp−xk (Fig. 2). With two veloc- ρ = ρ0 + ρ0. (10) ity vectors the pole can be constructed and then the a c envelope points are the footpoints of the normals This is illustrated in Fig.3. Recall that the envelope from the pole to the curve position. For derivations is the silhouette of a surface, which is in the present and further studies of this approach we refer to the special case a translational surface. In case that the literature [5, 24]. There, one also finds a variety of two given curves are the boundaries of convex do- applications, in particular the construction of gears. mains A and C, the outer part of the envelope is convex. It is the boundary of the Minkowski sum A + B of the two domains A,B. The Minkowski sum is defined as locus of all points x + y with Σ dx x ∈ A and y ∈ B. For the computational treatment dt x of Minkowski sums, we refer to Kaul and Farouki c(u) [27]. In the non convex case, the extraction of those parts of the envelope, which lie on the boundary ∂c/∂u = ∂c/∂t of the sum A + B, is a subtle task (see Lee et al. [30]). Farouki et al. [12] transformed another envelope problem occurring at so-called Minkowski products to the present one.

ρ0 a Figure 2: Envelope construction for a motion in the a(t) a˙ t plane ( ) ρ0 c c(u) Example: Let us consider the special case of a translatory motion in the plane. It is completely deﬁned by prescribing the trajectory of one point, say the path a(t) of the origin of the moving sys- Figure 3: Envelope construction for a translatory tem. A curve c(u) in the moving system then has motion in the plane at time t the position For an instant of a one-parameter motion in 3- c(u,t)= a(t)+ c(u). space, the velocity distribution is also quite simple [5, 24]. The velocity vector v(x) of a point x can be At any instant t, we have an instantaneous transla- expressed with two vectors c,c¯ in the form tion parallel to the derivative vector da/dt. Enve- lope points are characterized by parallelity of the v(x)= c¯+ c × x. (11) vectors d d This is in general the velocity ﬁeld of a helical a(t), c(u). dt du motion, whose axis is parallel to the vector c. In

4 special cases we have an instantaneous rotation or Π translation. In order to construct the envelope surface of a surface in the moving system, one has to ﬁnd on the positions Φ(t) of the moving surface those points, where the velocity vector touches Σ Φ(t). The computation of the envelope of a moving c surface occurs for example in NC machining simulation and veriﬁcation [35]. The cutting tool generates under its fast spinning motion around its axis a rotational surface (surface of revolution), which is the ’cutter’ from the geometric viewpoint. Un- der the milling operation, the cutter removes ma- terial from the workpiece. The thereby generated Φ(t) surfaces are (parts of the) envelopes of the moving surface of revolution (see Fig. 4).

Figure 4: Envelopes in NC simulation Figure 5: Constructing characteristic points on a moving surface of revolution

A simple geometric method for the computation of points on the envelope is as follows (see Fig. 4 Bezier´ representations 5): Consider a position Φ(t) of the moving surface of revolution and pick a circle c on it. We search A wide class of practically interesting cases of en- for the envelope points (characteristic points) on c. velopes can be discussed if we assume representa- Along the circle, the surface is touched by a sphere tions in Bezier´ form. These work in arbitrary di- Σ. Hence, along c, the surfaces Σ and Φ(t) have mensions, but we are confining here to envelopes the same tangent planes and thus the same charac- in R3. Moreover, it is clear that the transition to teristic points (points where the velocity vector is B-spline representations is straightforward. tangent to the surface). All characteristic points of We first consider a one-parameter family of sur- the sphere form a great circle in the plane Π nor- faces, written in tensor product Bezier´ form mal to the velocity vector of the sphere center (see Π l m n also section 6). Hence, this normal plane inter- l m n sects c in its characteristic points. Of course, they f (u,v,t)= ∑ ∑ ∑ Bi(u)B j (v)Bk(t)bijk. (12) need not be real, and one has to take those circles i=0 j=0 k=0 Φ of (t), where one gets real characteristic points. We do not necessarily restrict the evaluation to the For more details on this approach and on a graph- standard interval [0,1], as is usually done when dis- ics related method (extension of the z-buffer), we cussing Bezier´ solids [21]. Depending on which refer to Glaeser and Groller¨ [16]. A survey on the variables we consider as set parameters and which computation of envelope surfaces has been given as curve or surface parameters, equation (12) is a by Blackmore et al. [2]. representation of Kinematical geometry has also investigated motions where the positions of the moving system • three one-parameter families of surfaces, ob- are not congruent to an initial position, but simi- tained by viewing u or v or t as set parameter lar or affine to it (see, e.g., the survey [44]). One ((v,t) or (u,t) or (u,v) as surface parameters, has again a linear velocity field, and this simpli- respectively) fies the envelope computation. We focussed here on one-parameter motions. However, k-parameter • three two-parameter families of curves, ob- motions, where the positions of the moving system tained by viewing u or v or t as curve param- depend on k > 1 real parameters, have been studied eter ((v,t) or (u,t) or (u,v) as set parameters, as well [44]. respectively)

5 All these families have the same envelope surface. Figure 6: Tangent surface of a curve on an ellipsoid It is obtained via equation (4), ∂ f ∂ f ∂ f det( , , )= 0. (13) characteristic curve c(t) along which the surface ∂u ∂v ∂t Φ(t) touches the envelope. By Bezout’s theorem, Inserting representation (12) and using the linear- the characteristic curve is in general an algebraic ity of a determinant in each of its variables, we curve of order 2m. see: The preimage of the envelope in the (u,v,t)- In the following sections we will discuss a vari- parameter space is an algebraic surface Ω of or- ety of interesting examples for envelopes computed der 3(l +m+n−1). Of course, the algebraic order from Bezier´ type representations. may reduce in special cases. It might be convenient to write the surface Ω as zero set of a TP Bezier´ polynomial of degrees (3l,3m,3n) and then apply 5 Envelopes of planes a subdivision based approach of finding the solution [10, 25]. The simplest case of a one-parameter family of al- It has to be pointed out that part of the envelope gebraic surfaces is that of a family of planes U(t), can appear at the boundary of the corresponding written in the form TP Bezier´ solid (parameterized over [0,1]3), see [25]. U(t) : u0(t)+ u1(t)x + u2(t)y + u3(t)z = 0. (16) There are other trivariate Bezier´ representations Its envelope is found by intersecting planes U(t) as well [21]. For our purposes, a one-parameter with the first derivative planes family of surfaces in triangular Bezier´ form is interesting, U˙ (t) :u ˙0(t)+ u˙1(t)x + u˙2(t)y + u˙3(t)z = 0. (17) f (u,v,w,t)= ∑ Bm (u,v,w)Bn(t)b . (14) ijk l ijkl For each t, this yields a straight line r(t). Hence, i, j,k,l the envelope surface Φ is a ruled surface which Here, u,v,w, (u + v + w = 1) are barycentric coor- is tangent along each ruling r(t) to a single plane dinates for the triangular surface representations, (namely U(t)). It is well-known in differential ge- and t is the set parameter. Again, we do not ometry [9] that this characterizes the surface as de- just have a one-parameter family of surfaces (set velopable surface. It can be mapped isometrically parameter t), but also a two-parameter family of into the Euclidean plane. Because of this property, curves (curve parameter t), which have the same developable surfaces possess a variety of applica- envelope. The envelope computation is performed tions, for example in sheet metal based industries. as above. Restricting the evaluation to (u,v,w) in Continuing the general concepts from section 2, the domain triangle (u,v,w ≥ 0) and to t ∈ [0,1], we compute the curve of regression. With the sec- equation (14) parameterizes a ‘deformed pentahe- ond derivative plane dron’, which is called a pentahedral Bezier´ solid of ¨ degree (m,n) [21]. Part of the envelope can appear U(t) :u ¨0(t)+ u¨1(t)x + u¨2(t)y + u¨3(t)z = 0, (18) at the boundary of this solid. its points are the intersection points c t U t In case we are working with implicit represen- ( )= ( ) ∩ U˙ t U¨ t In special cases c t can be fixed (Φ tations, a Bezier´ form may be present in the co- ( ) ∩ ( ). ( ) is a general cone) or the system does not have a so- efficients of the surface equations. For example, lution (c t is a fixed ideal point and Φ is a general we can discuss a one-parameter family of algebraic ( ) cylinder surface). In the general case, the tangents surfaces Φ(t) of order m, of c(t) are the rulings r(t) and thus Φ is the tan- i j k f (x,y,z,t)= ∑ aijkx y z = 0, i + j + k ≤ m, gent surface of the regression curve c(t) (cf. the i, j,k example in Fig. 6). n n aijk = ∑ Bl (t)bijkl. (15) l=0 5.1 Dual representation of curves and surfaces We then have to form the partial derivative with respect to t, which is in general again an algebraic The coefficients (u0,...,u3) in the plane equation surface Φ˙ (t) of order m, and intersect with the orig- (16) are the so-called homogeneous plane coordi- inal surface Φ(t). This results (for each t) in the nates of the plane U(t). Here and in the sequel,

6 we will also denote the vector (u ,...,u ) ∈ R4 by 0 3 Figure 7: Dual Bezier´ curve U(t). Within projective geometry, we can view planes as points of dual projective space. Thus, in dual curve segment we are interested in, is the envelope space a one-parameter family of planes is seen as a of the lines U(t), t ∈ [0,1]. curve and we can apply curve algorithms to com- As an example for dualization, let us first discuss pute with them. This point of view has been very the dual control structure (see Fig.7). It consists of fruitful for computing with developable surfaces in the Bezier´ lines Ui, i = 0,...,m, and the frame lines NURBS form [3, 4, 22, 48, 52]. Fi, whose line coordinate vectors are A planar curve, represented as envelope of its tangents, is said to be given in dual representa- Fi = Ui +Ui+1, i = 0,...,m − 1. (21) tion (curve in the dual projective plane). Analo- gously, a surface in 3-space, viewed as envelope of From (21) we see that the frame line Fi is concur- its tangent planes, is said to be in dual representa- rent with the Bezier´ lines Ui and Ui+1. This is dual tion. If the set of tangent planes is one-dimensional to the collinearity of a frame point with the adja- (curve in dual projective space), the surface is a cent two Bezier´ points. We use frame lines – dual developable surface. Otherwise, we have a two- to the frame points – instead of weights, since the dimensional set of tangent planes and thus also a latter are not projectively invariant. A projective surface in dual space. formulation is important for application of projec- Dual representations in the context of NURBS tive duality. curves and surfaces, have been first used by The complete geometric input of a dual Bezier´ J. Hoschek [20]. To illustrate some essential ideas, curve consists of the m + 1 Bezier´ lines and m we briefly discuss the dual representation of planar frame lines. Given these lines, each of it has a one- rational Bezier´ curves. dimensional subspace of homogeneous coordinate A rational Bezier´ curve possesses a polyno- vectors with respect to a given coordinate system. mial parameterization in homogeneous coordinates It is possible to choose the Bezier´ line coordinate C(t)=(c0(t),c1(t),c2(t)), which is expressed in vectors Ui such that (21) holds. This choice is terms of the Bernstein polynomials, unique up to an unimportant common factor of the vectors U0,...,Um. Now (20) uniquely defines the n n corresponding curve segment. C(t)= ∑ Bi (t)Pi. (19) i=0 For a Bezier´ curve, the control point P0 is an end point of the curve segment and the line P0 ∨ P1 is The coefficients Pi are the homogeneous coordi- the tangent there. Dual to this property, the end nate vectors of the control points. The tangents tangents of a dual Bezier´ curve are U0 and Um, U(t) of the curve connect the curve points C(t) and the end points are the intersections U0 ∩U1 and ˙ with the derivate points C(t). This shows that Um−1 ∩Um, respectively. the set of tangents U(t) also possesses a polyno- To evaluate the polynomial (20), one can use the mial parameterization in line coordinates, i.e., a well-known recursive de Casteljau algorithm. It dual polynomial parameterization. It can be ex- 0 starts with the control lines Ui = Ui and constructs pressed in the Bernstein basis, which leads to the k recursively lines Ui (t) via dual Bezier´ representation, k k−1 k−1 m Ui (t)=(1 −t)Ui (t)+tUi+1 (t). (22) m U(t)= ∑ Bi (t)Ui. (20) i=0 At the end of the resulting triangular array, we get the line U(t). However, usually one would like to We may interpret this set of lines as a curve in get the curve point C(t). It is also delivered by the the dual projective plane. Thus, by duality we ob- de Casteljau algorithm, if we note one of its prop- tain properties of dual Bezier´ curves, i.e. rational erties in case of the standard representation: in step curves in the dual Bezier´ representation. m−1 m−1 m − 1, we get two points C0 (t),C1 (t), which When speaking of a Bezier´ curve we usually lie on the tangent at C(t). Dual to that, the lines mean a curve segment. In the form we have writ- m−1 m−1 U0 (t) and U1 (t) intersect in the curve point ten the Bernstein polynomials, the curve segment C(t). Hence, we have is parameterized over the interval [0,1]. For any m−1 m−1 t ∈ [0,1], equation (20) yields a line U(t). The C(t)= U0 (t) ×U1 (t).

7 Note that the curve point computation based on the P dual form has the same efﬁciency as the computa- 1 tion based on the standard form. The above procedure leads via

m−1 m−1 m−1 U2 U3 U0 (t) = ∑ Bi (t)Ui, i=0 m−1 U0 U1 m−1 m−1 P0 U1 (t) = ∑ B j (t)Uj+1, j=0 to the following formula for conversion from the dual Bezier´ form U(t) of a rational curve segment to its standard point representation C(t), Figure 8: Control planes of a developable Bezier´ 2m−2 surface 2m−2 C(t)= ∑ Bk (t)Pk (23) k=0 are expressed as with m U(t)= ∑ Bm(t)U , t ∈ [0,1]. (26) 1 m 1 m 1 i i − − i 0 P = ∑ Ui ×Uj 1. = k 2m−2 i j + k i+ j=k Like in the study of planar rational curves in dual Bezier´ form, we can easily obtain insight into the We see that C(t) possesses in general the degree behaviour of the dual control structure by dualiza- 2m − 2. For more details, degree reductions, fur- tion. Let us start with the behaviour at the interval ther properties and applications, we refer to the lit- end points t = 0 and t = 1. A rational Bezier´ curve erature [20, 43, 46, 45, 53]. possesses at its end points the tangents P0 ∨ P1 and Pm−1 ∨ Pm. Dual to that, the end rulings of the 5.2 Developable NURBS surfaces developable surface (26) are r(0)= U0 ∩ U1 and as envelopes of planes r(1)= Um−1 ∩Um (Fig. 8). The osculating plane of a Bezier´ curve at an end point is spanned by The fact that developable surfaces appear as curves this point and the adjacent two control points. Dual in dual space indicates the computational advan- to an osculating plane (connections of curve point tages of the dual approach. We will briefly describe with the first two derivative points) is a point of re- a few aspects of computing with the dual form. gression (intersection of tangent plane with the first A developable NURBS surface can be written as two derivative planes). Hence, the end points of the envelope of a family of planes, whose plane coor- curve of regression c(t) of the developable Bezier´ dinate vectors U t R4 possess the form ( ) ∈ surface are n m c(0)= U0 ∩U1 ∩U2, c(1)= Um−2 ∩Um−1 ∩Um. U(t)= ∑ Ni (t)Ui. (24) i=0 The computation of rulings and points of regres- 4 sion is based on the algorithm of de Casteljau ap- The vectors Ui ∈ R are homogeneous plane coordinate vectors of the control planes, which we plied to the dual form (26). Stopping this algorithm one step before the last one, there are still also call Ui (Fig. 8). To get a geometric input, we additionally use the frame planes with coordinate two planes in the triangular array. Their intersection is a ruling. One step earlier, we still have three vectors Fi, planes, which intersect in the point of regression. Fi = Ui +Ui+1, i = 0,...,n − 1. (25) Finally, we note the dual property to the well- known result that any central projection of a ratio- For several algorithms it is convenient to con- nal Bezier´ curve c yields a rational Bezier´ curve c′ vert (24) into piecewise Bezier´ form and then per- whose control points and frame points are the im- form geometric processing on these Bezier´ seg- ages of the control points and frame points of c. ments. Therefore, we will in the following restrict our discussion to such segments, i.e., to de- Proposition: The intersection curve of a devel- velopable Bezier´ surfaces. In the dual form, they opable Bezier´ surface U(t) with a plane P is a

8 rational Bezier´ curve C(t). The control lines and B2 frame lines of C(t) are the intersections of the con- A2 trol planes and frame planes of U(t) with P.

C2 B(t) Figure 9: Control net of the surface patch in Fig. 8 A(t) C1 A1 This property is very useful for converting the B1 dual representation into a standard tensor product C(t) form (see Fig. 9 and [48]). For other algorithms in connection with developable NURBS surfaces B0 (interpolation, approximation (see Fig. 10), con- trolling the curve of regression, etc.) we refer the A0 C reader to the literature [3, 4, 22, 48, 52]. 0

Figure 11: Family of planes generated by a penta- Figure 10: Approximation of a set of planes by a hedral Bezier´ solid of degree (1,2) developable surface family (28), 3n 5.3 Developable surfaces via pen- 3n U(t)= A(t) × B(t) ×C(t)= ∑ Bl (t)Ul, (29) tahedral Bezier´ solids l=0 A pentahedral Bezier´ (PB) solid of degree (1,n) with is formed by a one-parameter family of triangles, 1 n n n U := ∑ (Ai × B j ×C ). whose planes envelope a developable surface. We l 3n i jk k l i+ j+k=l now show how to derive the dual representation of the envelope. Thus, the generated envelope is in general of In the special case m = 1 of (14), we set dual degree 3n. Degree reductions are possible. They occur if for some t0 the vector U(t0) is zero. b100l =: Al, b010l =: Bl, b001l =: Cl, Then, all polynomial coordinate functions of U(t) are divisible by (t −t0), which explains the degree and obtain the representation of the PB solid, reduction. We omit a detailed study of these reductions and their geometric meaning. n F(u,v,w,t)= ∑ Bn(t)(uA + vB + wC ). (27) l l l l Example: The simplest example belongs to n = 1. l=0 As envelope Φ we get in general a developable sur- We assume a rational representation and therefore face of dual degree 3. It is well known that these the involved vectors F,Al,Bl,Cl are homogeneous surfaces are tangent surfaces of rational cubics. In coordinate vectors, i.e., vectors in R4. The solid is case of degree reductions, Φ may be a quadratic generated by a one-parameter family of triangles in cone, or it degenerates to a line (all planes of the planes U(t), family pass through this line). If we use an integral representation (equation (14) with inhomogeneous U(t) := A(t) ∨ B(t) ∨C(t), (28) coordinates and n = 1), Φ is either the tangent sur- n n n n face of a polynomial cubic, a parabolic cylinder or with A(t)= ∑ Bl (t)Al,...,C(t)= ∑ Bl (t)Cl. a line (which may be at inﬁnity, i.e., all planes of l=0 l=0 the family are parallel). Note that the rational Bezier´ curves A(t),B(t),C(t) are edge curves of the pentahedral solid (see Fig. 11). 6 Envelopes of spheres and Given the homogeneous coordinate vectors natural quadrics A,B,C of three points, the plane coordinates of the spanning plane are given by the vector prod- We will study envelopes of spheres and cones and uct A × B ×C (in R4!). This yields for the plane cylinders of revolution, called natural quadrics. In

9 particular we focus on rational families. The enve- their radius function r(t) will not be a square root lope of a one-parameter family of spheres is called of a rational function. canal surface. If the family consists of congru- Corollary 6.1 The envelope Φ of a rational fam- ent spheres (constant radius), the envelope is called ily of spheres is rationally parametrizable. If the pipe surface. family has rational radius function then all offset surfaces Φd of Φ are rational, too. 6.1 One-parameter families of Φ spheres S S˙ First we start discussing one parameter families of c D spheres. Let S(t) and S˙(t) denote the spheres and D c M M˙ v its derivatives,

3 2 2 S(t) : ∑(xi − mi(t)) − r(t) = 0, (30) i=1 3 Figure 12: Geometric properties of a canal surface S˙(t) : ∑(xi − mi(t))m˙ i(t) − r(t)r˙(t)= 0, i=1 Figure 12 shows also that the envelope Φ is tangent where M(t)=(m ,m ,m )(t) and r(t) denote cen- to a cone of revolution D(t) at the characteristic 1 2 3 Φ ters and radii of S(t). The envelope Φ is tan- circle c(t). Thus, is also part of the envelope of gent to S(t) in points of the characteristic curves the one-parameter family of cones D(t). c(t)= S(t) ∩ S˙(t). Since S˙ are planes, c(t) consists of circles. Φ consists of real points if and only if ˙ 2 2 kM(t)k − r˙(t) ≥ 0 M Φ holds. The envelope possesses singular points if S¨(t) ∩ c(t) consists of real points. S¨(t) is again a plane, such that each circle c(t) can possess at most two singular points. If M(t) is a rational center curve and r(t) is rational, S(t) shall be called rational family of spheres with rational radius function. It is proved in [38] Figure 13: Rational canal surface that the envelope Φ possesses rational parametrizations. Thus it is representable as a NURBS surface. Additionally, all offset surfaces Φd at distance d 6.2 One-parameter families of are rational, since they are also canal surfaces with cylinders and cones of revolu- rational center curve and rational radius function. tion Let Q(t) be a one-parameter family of spheres represented in the form (15). Equivalently, we have Next we will study envelopes of a one-parameter family of cylinders or cones of revolution. In the 2 2 2 Q(t) : a(x1 + x2 + x3)+ bx1 + cx2 + dx3 + e = 0, following it is not necessary to distinguish between (31) cones and cylinders and we speak simply of cones where the coefﬁcients a,...,e are rational func- of revolution. Let D(t) be such a family. Its derivations or polynomials in t. Q(t) shall be called ra- tive D˙ (t) deﬁnes in general a family of regular tional family of spheres. One can verify that the quadrics. It can be proved that each surface D˙ (t) centers M(t) of spheres Q(t) form a rational curve contains the vertex v(t) of D(t). The characteris- but its radius r(t) is just the square root of a ra- tic curves c(t)= D(t)∩D˙ (t) are in general rational tional function. But also in this case it is proved curves of order four with v(t) as singular point. Let Φ ([38]) that the envelope of the family Q(t) pos- D(t) be given by an implicit quadratic equation in sesses rational parametrizations. This implies that the coordinates xi, the general cyclides (see [37]), which are algebraic 3 3 surfaces of order four, are rational. By the way, D(t) : ∑ aij(t)xix j + ∑ bi(t)xi + c(t)= 0. (32) the offset surfaces are in general not rational since i, j=1 i=1

10 The coefficients aij,bi,c are functions of the pa- Theorem 6.1 The envelope Φ of a rational one- rameter t. Additionally we always can assume that parameter family of natural quadrics D(t) pos- A =(aij) is a symmetric 3×3-matrix. D(t) is a sesses rational parametrizations. If additionally, cone of revolution if the matrix aij possesses a D(t) has rational radius function then all off- twofold eigenvalue and the extended matrix sets Φd of Φ at distance d are also rationally parametrizable. c b1 b2 b3  b1 a11 a12 a13  A one-parameter family of cones of revolution b a a a D(t) enveloping a canal surface is a special case,  2 12 22 23   b3 a13 a23 a33  since the characteristic curve c(t) of order four de-   composes into a circle and a not necessarily real has rank 3. D(t) defines cylinders of revolution if pair of lines. additionally A has rank 2. D(t) is called a ratio- Since the property of possessing rational nal family, if the coefficients in (32) are rational parametrizations is invariant under projective functions of t. In particular, its vertices v(t) form a transformations, first part of theorem 6.1 holds for rational curve. one-parameter families of general quadratic cones In view of rational parametrizations of offset (we drop the double eigenvalue condition). The surfaces, special rational families of cones of rev- property of having rational offsets is not preserved olution are of interest. Let v(t) be a rational curve by projective transformations but by similarities and S(t) be a rational family of spheres with ra- and more generally by sphere-preserving geomet- tional radius function, as discussed in section 6.1. ric transformations, see [38]. The unique family of cones of revolution D(t) join- In section 5 we introduced the dual representa- ing v(t) and S(t) shall be called rational family tion of curves in the plane and surfaces in space. with rational radius function. By the way, there Since a natural quadric D is a developable surface, is a one-parameter family of spheres (with ratio- it is enveloped by a one parameter family of planes nal radius function) being inscribed in each cone. Note that the general rational family (32) has ratio- U(s) : u0(s)+ u1(s)x1 + u2(s)x2 + u3(s)x3 = 0, nal vertices v(t) but no further spheres S(t) with (33) rational radii. If a surface Φ is enveloped by a where we can assume that ui(s) are quadratic polynomials. Thus, the problem of constructing rational parametrization of a family D(t) is equivalent m(t) to the problem of finding a representation of the form S(t) U(s,t) : u0 + u1x1 + u2x2 + u3x3 = 0 (34)

with bivariate polynomials ui(s,t) such that U(s,t ) represents the tangent planes of the cone D(t) 0 D(t0). The envelope Φ of the family D(t) is enveloped by the tangent planes U(s,t). The points of contact v(t) are obtained by

∂U ∂U p(s,t)= U(s,t) ∩ (s,t) ∩ (s,t). (35) ∂s ∂t

6.3 Offset surfaces of regular Figure 14: Rational family of cones with rational quadrics radius function It is known that the offset curves of a conic in the one-parameter family of cones (cylinders) of rev- plane are in general not rational algebraic curves olution D(t), its offset surface Φd at distance d of order 8. Besides the trivial case of a circle, only is also enveloped by a one-parameter family of the offset curves of parabolas are rational curves of cones(cylinders) of revolution Dd(t), the offset sur- order 6. Surprisingly it can be proved the following faces of D(t). Then, the following theorem holds. result, see also [32].

11 Theorem 6.2 The offset surfaces of all regular regression c. The reason is that the characteristic quadrics in space can be rationally parametrized. curve c(t) of order four decomposes into a circle and two lines. The circle is located in the normal The reason is that any regular quadric can be repre- plane to the curve of regression c. Thus, it gener- sented as envelope of a rational family of cones of ates the pipe surface. The lines generate the offset revolution with rational radius function. Details on surface Rd of R which is again a developable sur- the parametrization and low degree representations face. of the offsets can be found in [38]. Let D(t) be a rational one-parameter family of congruent cylinders. Theorem 6.1 says that their envelope Φ is rational and possesses rationally parametrizable offsets. More precisely, if the axes surface R is a rational non-developable ruled surface then Φ and its offsets are rational. If R is a rational developable surface, its offsets Rd need not to be rational, but the pipe surface, centered at the line of regression c of R, and its offset pipes are rational. Details on the computation can be found in the survey [55] and in [26]).

Figure 15: Quadric of revolution and outer offset Corollary 6.2 The envelope Φ of a one-parameter surface family of congruent cylinders of revolution is an offset surface of the axes surface R. If R is rational and non-developable then Φ admits rational 6.4 Envelope of congruent cylin- parametrizations. ders of revolution A special example of a surface enveloped by a one- parameter family of natural quadrics is the envelope of a moving cylinder of revolution. These sur- Φ faces appear in applications as results of a (five- D(t) axis) milling procedure with a cylindrical cutter. The axes of the one-parameter family of cylinders (cutting tool) generate a ruled surface R, the axes surface. In general, R is a non-developable ruled surface which says that the tangent planes at points of a fixed generating line of R vary. In particu- R lar, the relation between contact points and tangent planes of a fixed generator is linearly. A developable ruled surface possesses a fixed tangent plane along a generating line. Figure 16: Envelope of a one-parameter family of A cylinder of revolution D can be considered as congruent cylinders of revolution envelope of a one-parameter family of congruent spheres, centered at the axis of D. Then it is ob- vious that the envelope Φ of a moving cylinder of 6.5 Two-parameter families of revolution D(t) is an offset surface of R. The dis- spheres tance between R and Φ = Rd equals the radius of the cylinders D(t). In general, the characteristic One-parameter families of spheres can envelop a curve where D(t) and Φ are in contact, are rational canal surface, but also two-parameter families of curves of order four, as mentioned in section 6.2. spheres may have an envelope. Let S(u,v) be such If the axis surface R is developable such that the a family and let Su(u,v) and Sv(u,v) be its partial axes are tangent lines of the curve of regression c of derivatives with respect to u,v. Thus, we have R, the envelope Φ of the family D(t) decomposes 3 into the offset surface Rd of R and into the pipe sur- 2 2 S(u,v) : ∑(xi − mi(u,v)) − r(u,v) = 0, face, interpreted as offset surface of the curve of i=1

12 3 Given a point p ∈ R3, the circle ζ−1(p) is the S (u,v) : ∑(x − m )m − rr = 0, u i i iu u intersection of the cone of revolution γ(p) with R2, i=1 γ 3 where (p) has p as vertex and its inclination angle with R2 is π/4 = 45◦. Thus, all tangent planes of Sv(u,v) : ∑(xi − mi)miv − rrv = 0. (36) 2 i=1 γ(p) enclose an angle of π/4 with R . −1 3 Points p of the envelope Φ have to satisfy all equa- Applying ζ to a curve p(t) ∈ R , one ob- ζ−1 tions. Since Su and Sv are planes, p has to lie on tains a one-parameter family of circles (p(t)). the line of intersection g = Su ∩Sv. There might be The envelope of the one-parameter family of cones two, one or no intersection point. If the number of γ(p(t)) is a developable surface Γ(p(t)). Thus, the −1 intersection points is two or zero, then Φ consists envelope of the family of circles ζ (p(t)) is the 2 locally of two or no components. The case of g intersection of Γ(p(t)) with R . being tangent to S can hold for isolated points or a Given a two-parametric family of circles c(u,v), two dimensional set of points; usually it is satisfied it may have an envelope. Its cyclographic image 3 for a one-parametric family of points. ζ(c(u,v)) is a surface in R . If this surface pos- We want to address a special case. If the ra- sesses tangent planes τ enclosing an angle of π/4 2 dius function r(u,v) is constant, then the envelope with R , then c(u,v) will possess an envelope. of S(u,v) is the offset surface of the surface M Let us assume that c(u,v) has an envelope. Then ζ traced out by the centers (m1,m2,m3) at distance there exists a curve p(t) ⊂ (c(u,v)) whose tan- τ π r. If mi(u,v) are rational functions, then M is a ra- gent planes (p(t)) have slope /4. These planes tional surface. In general, its offset surfaces will τ(p(t)) envelope a developable surface Γ(c). Thus, not be rational. Of course, there are several sur- we finally obtain the envelope of c(u,v) by inter- 2 faces possessing rational offsets; see section 6.2. secting Γ(c) with R . Since a cone or cylinder of revolution can be en- We will discuss some examples. At first, let veloped by a one-parameter family of spheres, one- p(u,v) ∈ R3 be the parametrization of a plane g, parametric families of cones are special cases of enclosing an angle of π/4 with R2. The envelope two-parametric families of spheres. A sphere ge- of the family of circles ζ−1(p(u,v)) is simply the ometric approach to rational offsets and envelopes intersection g ∩ R2. of special two-parameter families of spheres can be At second, let p(u,v) ∈ R3 be a sphere. The set found in [38] and [41]. of points q(t) possessing tangent planes with slope π/4 are two circles. These tangent planes envelope 6.6 Two-parameter families of cir- two cones of revolution, being tangent to the sphere in points of these circles. Finally, the envelope of cles in the plane ζ−1(p(u,v)) consists of two circles if p(u,v) is not A one-parameter family of circles in the plane can centered in R2. Otherwise it is one circle. possess an envelope. In particular, congruent circles will envelope the offset curve of its centers. 6.7 Families of spheres in space But also a two-parametric family of circles c(u,v) can have an envelope. Let its centers and radii be Analogously to two-parametric families of cir- m(u,v)=(m1,m2)(u,v) and r(u,v), respectively. cles in the plane are three-parametric families of To enlighten this geometrically, we will use a spheres in space. Since a sphere is determined by 3 mapping ζ which associates a point ζ(c) ∈ R to its center M =(m1,m2,m3) and its radius r, we will each circle in the following way, define a cyclographic mapping ζ, 3 c 7→ ζ(c)=(m1,m2,r) ∈ R . (37) 4 S 7→ ζ(S)=(m1,m2,m3,r) ∈ R , (38) We will assume r(u,v) being a signed radius. The transformation ζ is called cyclographic map- which maps the spheres in R3 onto points of R4. ping. The inverse mapping ζ−1 maps points p = We will again assume that r is a signed radius. A 3 2 R3 (p1, p2, p3) ∈ R to circles in R , three-parametric family of spheres in is mapped to a hypersurface in R4. If we apply an analogous ζ−1 p : x p 2 x p 2 p2 ( ) ( 1 − 1) +( 2 − 2) = 3. construction as done in section 6.6 we arrive at a We identify points in R2 with circles with zero geometric interpretation of the computation of the radii. The cyclographic image ζ(c) of one or two- envelope of a three-parametric family of spheres. parametric families of circles are curves or surfaces Let their image hypersurface in R4 be denoted by in R3. p(u,v,w). If this hypersurface possesses tangent

13 ∂ ∂ hyperplanes (3-spaces) enclosing an angle of π/4 M E Nv = ( × E + w × E). with R3, then there exists an envelope of the family ∂v ∂v of spheres ζ−1(p). We will not go into details here. The tangent planes of the surfaces Ru and Rv co- The cyclographic mapping is also helpful for incide if and only if the normals Nu,Nv (which are constructing envelopes of two-parameter families assumed to be not zero) are linearly dependent, of spheres. Details about these constructions with an emphasis on rational families of spheres and Nu × Nv =(0,0,0). surfaces possessing rational offset surfaces can be found in [41, 51]. In connection with tolerance Elaborating this one obtains the following analysis, the cyclographic mapping was fruitfully quadratic equation applied in [50]. ∂E ∂E ∂E ∂M w2 det( ,E, )+ w[det( ,E, )+ ∂u ∂v ∂u ∂v 6.8 Two-parameter families of ∂M ∂E ∂M ∂M det( ,E, )]+( ,E, )= 0. (40) lines and cylinders ∂u ∂v ∂u ∂v We have discussed in section 6.4 that the axes of Depending on the number 2,1 or 0 of real solu- a one-parameter family of congruent cylinders of tions, the line L is called hyperbolic, parabolic or revolution form a ruled surface and the envelope elliptic. If L is hyperbolic or elliptic this property of the cylinders itself is an offset surface of this holds for lines in the congruence C being close to ruled axis-surface. In case that we have a two- L. So we might say that the congruence C itself parameter family of congruent cylinders of revolu- is hyperbolic or elliptic. There might be isolated tion, the situation is much more involved. First of parabolic lines as well as one or two-dimensional all we will study two-parameter families of lines domains of parabolic lines of the congruence. In (axes), since this will partially answer the question the following we will focus at the hyperbolic case. how to compute the envelope of a two-parameter We want to note that if the congruence C consists family of cylinders D(u,v). of normals of a smooth surface (different from a A two-parameter family of straight lines C plane or a sphere), then C is always hyperbolic. is called line congruence. It shall be defined If (40) has two real solutions w1,w2, they cor- by the following parametrization. Let M = respond to points with coinciding tangent planes. (m1,m2,m3)(u,v) a smooth surface in 3-space and These points generate the so called focal surfaces E =(e1,e2,e3)(u,v) a vector field. We may assume that kEk = 1 for all (u,v) ⊂ U. Then, Fi = M(u,v)+ wiE(u,v), i = 1,2. R M(u,v)+ wE(u,v), w ∈ (39) We return to the previous problem of computing parametrizes a two-parameter family of lines. The the envelope of a two-parameter family of congru- points of a fixed line are parametrized by w ∈ ent cylinders of revolution D(u,v). These cylinders R. Since the direction vectors E(u,v) are nor- have axes A(u,v) which form a line congruence. malized, E(u,v) parametrizes a domain in the unit If the congruence is hyperbolic, the axes envelop sphere. We will assume that this domain is two- the focal surfaces F1,F2. Thus, the envelope of the dimensional and does not degenerate to a single cylinders itself contains the offset surfaces G1,G2 curve or a single point. Mathematically this is of F1,F2 at distance d, which equals the radii of guaranteed by linearly independence of the partial the cylinders D(u,v). The offset surfaces admit the derivatives ∂E/∂u and ∂E/∂v. following representation Let L = M(u0,v0)+wE(u0,v0) be a fixed line in 0 Gi = Fi(u,v) ± dN (u,v) the congruence and let Ru : M(u,v0)+wE(u,v0) be i a ruled u-surface passing through the fixed chosen 0 0 0 with Ni (u,v)= Nu (u,v,wi)= Nv (u,v,wi) as unit line L. Analogously, let Rv : M(u0,v)+ wE(u0,v) normal of the corresponding focal surface Fi. If the be the ruled v-surface passing through L. We want congruence is parabolic, the envelope contains just to study the distribution of the tangent planes of Ru one surface G being the offset of the single focal and Rv along L. It is an elementary computation surface F. In case of an elliptic congruence there that the surface normals Nu and Nv of the ruled sur- exists no real envelope. faces Ru, Rv are ∂M ∂E Theorem 6.3 The envelope of a two-parameter N = ( × E + w × E), u ∂u ∂u family of congruent cylinders of revolution, whose

14 Malus-Dupin is of great interest which states the following.

A φ2 Φ1 φ M 1 E p N y 2

n2 T2 n1 L1 p1 φ1 s φ Φ2 2 Φ r2 2

Figure 17: Focal surfaces of the line congruence Φ S Ts 1 T L2 r1 formed by the normals of a surface 1 axes lie in a hyperbolic (parabolic?) congruence Figure 18: Theorem of Malus-Dupin of lines, consists of the offset surfaces of the focal surfaces of the congruence. Theorem 6.4 (Malus-Dupin) Given a two- A Bezier´ representation of a rational line congru- parameter family of lines L1 being perpendicular ence is obtained by taking a trivariate tensor prod- to a surface Φ1. If this family of lines is refracted uct Bezier´ representation at an arbitrary smooth surface S in such a way that this refraction satisfies Snellius law, then 1 m n 1 m n the refracted two-parameter family of lines L2 is P(u,v,w)= ∑ ∑ ∑ Bk(w)Bi (u)B j (v)bijk, again perpendicular to a surface Φ2. k=0 i=0 j=0 which is linear in the parameter w. Evaluating There are several methods to prove this theorem. P at w = 0 gives the surface M, evaluation at We will give a sphere-geometric proof, since it is w = 1 gives M + E, which are both (m,n)-tensor closely related to envelopes. Actually, the proof is product surfaces. Bezier´ representations of a two- done by a careful study of figure 18. Let L1 be a Φ parametric family of congruent cylinders can be line which intersects 1 perpendicularly in a point found in [61]. p1. This line shall be refracted at s ∈ S according Remark: Since parameter values w are com- to Snellius law. It says that the sin-values of the i φ φ puted as solutions of a quadratic equation, the fo- angles 1, 2 of incoming and refracted rays L1,L2 cal surfaces are in general not rational. It seems formed with the normal N at s ∈ S are proportional, to be quite difficult to study rational line congru- sinφ ksinφ k const ences with rationally parametrizeable focal sur- 1 = 2, = . faces, whose offset surfaces are also rationally Let S be a sphere, centered at s and tangent to parametrizeable. 1 Φ1 at p1. Its radius is r1. The tangent planes T1,Ts of Φ1 and S intersect in a line A. The angle 6.9 Geometrical optics φ1 = ∠(L1,N) equals the angle ∠(T1,Ts). We cen- tre a second sphere at s whose radius is r = r /k. Geometrical optics has a close relation to line con- 2 1 Further, let T be a plane which is tangent to S and gruences and sphere geometry. The geometric 2 2 passes through A. The angles φ between L and properties of light rays are found by studying ge- i i N also occur between the tangent planes T and T . ometric properties of two-parametric families of i s Thus it follows that lines (line congruences). In particular, those families are important which intersect a given surface sinφ1 r1 Φ = = k. 1 orthogonally. In this context, the theorem of sinφ2 r2

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18 Envelopes – Computational Theory and Applications

Helmut Pottmann and Martin Peternell

Institut fur¨ Geometrie, Technische Universitat¨ Wien Wiedner Hauptstrasse 8-10, A-1040 Wien

Category: survey Format: print

Contact: Helmut Pottmann Wiedner Hauptstr. 8-10 A-1040 Wien Austria phone: +43-01-58801-11310 fax: +43-01-58801-11399 email: [email protected]

Estimated # of pages: 20

Keywords: envelope, duality, sphere geometry, NURBS surface, developable surface, canal surface, geometric tolerancing

Based on classical geometric concepts we discuss the computational geometry of envelopes. The main focus is on envelopes of planes and natural quadrics. There, it turns out that projective duality and sphere geometry are powerful tools for developing efﬁcient algorithms. The general concepts are illustrated at hand of applications in geometric modeling. Those include the design and NURBS representation of developable surfaces, canal surfaces and offsets. Moreover, we present applications in NC machining, geometrical optics, geometric tolerancing and error analysis in CAD constructions. Keywords: envelope, duality, sphere geometry, NURBS surface, developable surface, canal surface, geometric tolerancing.