- Email: [email protected]
Finite quadric segments with four conic boundary curves
Computer Aided Geometric North-Holland
Design
141
7 (1990) 141-150
Finite quadric segments with four conic boundary curves G. GEISE Sektion Mathematik,
Technische Universitiit Dresden, GDR
U. LANGBECKER Zentralinstitut
fti Molekularbiologie,
Berlin, GDR
Presented at Oberwolfach April 1989 Received July 1989; revised November
1989
Abstract. Finite quadric segments bounded by four plane curves and smooth in the sense of differential geometry are considered. Such a quadric segment which can swept by one conic possesses a representation x = x( U, v) on [0, I] x [O,l] as rational tensor product Btzier surface of degree (m. n) with m < 6 and n & 2. This is founded on known facts concerning rational Btzier representations of conks from the viewpoint of stereographic projection. Some special cases are investigated, especially patches on ruled quadrics bounded by four line segments. Keywords. Conic BCzier surfaces.
sections,
quadrics,
stereographic
projection,
rational
BCzier curves,
rational
tensor
product
0. Introduction 0.1. Quadric segments
bounded by four plane curve segments
Let Q be a non-reducible (unipartite) quadric of the 3-dimensional euclidean (or affine) space E3 (over R) and @ a (finite) surface segment on Q. Let @ have a boundary consisting of four conic segments cab, cbd, cdc, c,, intersecting only at their end points in the order ubdcu: The patch @ shall be smooth in the sense of differential geometry. a@=cllbUC~~UCdcuC,,. Wanted are admissible parametric representations x = x( u, v) defined on Z X I with Z = [0, l] such that the u-lines v = 0 and v = 1 and the v-lines u = 0 and u = 1 are the conic boundary curves, see Fig. 1. For the surface patch @ the following conditions are necessary: (1) If Q is a cone, the vertex does not belong to @. (2) The vertices of Q, are convex: Let e E (a, b, c, d} be one of the vertices; then in the tangent plane of @ at e the tangents of the curves of @ beginning at e fill an angle domain which has a measure < T and > 0. If four conic segments cab, cbd, cde, c,, on a quadric Q are given, satisfying the prescribed conditions, there is no need to determine a surface patch @ of the kind described. But that interesting problem: “Under what circumstances do four conks determine a surface as above?” will not be considered here. The aim of this article is to show that a quad& patch @ which can be swept by one conic possesses a parametric representation x = x(24, v)((u, v) E Z X I) as rational tensor product Bezier surface of degree (m, n) with 1 < m G 6, 1 < n G 2. The investigations are carried out in 0167-8396/90/$03.50
0 1990 - Elsevier Science Publishers
B.V. (North-Holland)
142
G. Geise, U. Lungbecker
/ Quadric segments
b Fig. 1. Surface segment
with four boundary
curves
the projectively closed euclidean (resp. affine) 3-space E3, which results by adding the points at infinity (or ideal or improper points) to E3 [Pedoe ‘70, Schaal ‘801. The essential instrument is the representation of conic sections by way of rational Betier curves of second degree from the viewpoint of stereographic projection. It enables for three points a, 6, z of a non-reducible conic c the description of both the segment C,b 3 z connecting a and b not passing through z and the segment cab 3 z connecting Q and b passing through z. If it is feasible to move the points a, b, z along conic segments on the quadric Q such that the curve segment cab or Zab sweeps the surface patch @ then the desired representation arises. The well-known stereographic projection of a quadric from a single quadric point is a special case. Additional special cases arise if Q is a ruled quadric and some or all bounding conic segments of @ are line segments. Subsets of hyperbolic paraboloids which are bounded by skew quadrangles (the sides are parts of line generators of the quadtic) are well known. Such surface segments on one-sheet hyperboloids are less familiar. Both of them completely comprise the rational tensor product BCzier patches of degree (1,l). Being part of both algebraic geometry and projective differential geometry the projective geometry used in this article is the classical basis for investigating rational curves and surfaces (i.e. described by means of polynomials). In the projective version an exterior phenomenon is the absence of the fraction bar but it is the essential natural way of including points at infinity (especially control points at infinity). Recent investigations on these topics, suggested by problems of CAGD, are generally in a pre-projective version; there are surveys in [Hoschek & Lasser ‘89, p. 2651 or [Farin ‘891, for instance, where the relevant literature is cited. 0.2. Definitions In E3 the quadric Q as a surface of second order has an equation Q:
O=x’Qx
withQ=(qij);+,
,___.s =Q’,rankQ~
(4, 3).
j=O,...,3
In this x := (x0, xi, xz, XJ E lR4\{O} with 0 = (0, 0, 0, 0)’ is a homogeneous (or projective) coordinate vector of a point x E E3 (same notation). If x E E3 (proper point of E3), x0 # 0 is
143
G. Geise, U. Langbecker / Quadric segments
characteristic; such points possess Cartesian (or affine) coordinates which are the elements of an inhomogeneous coordinate vector x := (x,/x,, x,/x,, x3/x0)’ =: (x, y, z)‘. Planes are denoted by lower case Greek letters, their coordinate vectors are marked by ? 2 E R4\ (0). The polar of a point x with respect to the quadric Q is the plane i = Qx. Note that = often stands for ‘projectively equal’, i.e. ‘equal apart from a factor # 0’, and therefore ‘unequal’ is equivalent to ‘linearly independent’. conic Let cw be one of the given conic segments. If cw belongs to a regular (unipartite) section c c Q, the other segment of c connecting p and q is denoted by Cm: c=c,Ui;,.The by sw; plane containing c is referred to as y. If cw is a line segment this will be distinguished three non-collinear points p, q, r, the line containing this segment is I,. The plane containing is denoted by cWr. For brevity the Bernstein polynomials are written by laying emphasis on the argument according to the model: T,” := B:(t).
1. Preliminaries I. I. The basic formular of stereographic projection Let 0 = X’QX be an equation different from z. The following
of the quadric Q in E3, z a point on the quadric lemma is well known:
and y a point
Lemma. Iff 2 y’Qzy - y’Qyz = 0 the line I, is a subset of Q. Otherwise the point of intersection of I, and Q different from z is x = 2 ytQr.y - y’Qvz.
(I)
1.2. Connecting conic segments Let a, b, z be three points of Q, not lying on one line. Let their plane eUbz intersect the quadric in a regular conic section c. The curve c is subdivided into two segments c,,~ 3 z and ~5,~3 z. Here the points a and b are assumed to be proper points with coordinate vectors of the form a = (1, a,, a2, a3)‘, b = (1, b,, b,, b,)’ (with no essential loss of generality). Then the following theorem holds (Fig. 2): Theorem 1. (1) A parametric s ab- *
y=y(t)
representation
= T;u+
T;b
of the euclidean
(affine)
line segment s,~ of I,, is
(tH).
The external segment connecting a and b, containing the point at infinity of lab, has the parametric representation S,,:
Y=y(t)=T,‘a-T,‘b
(2) A parametric c*b 3 z:
representation
+=I). of the finite conic segment
~=Z(t)=2T0*uu+T~s+2T2*~b withs:=&z+crb-{z,
in which u := u’Qz,
/3 := b’Qz,
5 := u’Qb,
C,, 3 z of c is
(tEI)
(2)
144
G. Geise, U. Langbecker
Fig. 2. Stereographic
and a parametric
projection
of a conic chord from a conic point onto a conic segment.
representation
C nb3Z:
/ Quadric segments
Autocollineation
of a conic.
of the other conic segment c,~ 3 z of c is
x=x(t)=2T02au-T;s+2T,2pb
(tE1).
(3)
Here s is the point of intersection of the tangents to c at a and b. The proof proceeds by verifying that x E c, si E c, and uses several geometric propositions. Points y E s,~ and jj E S,, belonging to the same parameter value t E (0, 1) form a harmonic point quadruple with the points u and b; the cross ratio is cr( y, JJ; 4, b) = - 1. Equation (2) gives the stereographic projection of s,~ from z onto the conic segment C,b 3 z.. The involutive central collineation of the plane eak with s as centre and I,, as axis exchanges the segments c ab and c,b- This is easily seen from X + x = T02aa + T,2Pb,
F-x=
T12s.
This autocollineation of c changes the stereographic projection of sab from z = pa + cub- s onto the segment ZUb3 z into the stereographic projection of sab from t = pa + ab + s onto cab 3 z. Now we can see c&, as the stereographic projection of iab from z and ZUb as the stereographic projection of Fab from Z. The inhomogeneous form of the Bernstein polynomials is a blemish. It is easily removable, but without influencing the representations (2) and (3) which only are of interest. We omit criteria needed to ensure that a conic segment cab or Cab is of finite extent. 1.3. Parametrization
of plane convex quadrangles
Let &by %d, sdc? %a be the sides of a plane convex quadrangle in E3, determined by the comers II, b, c, d with a = (1, a,, u2, a3)‘,.. . E R4. It then holds that (Fig. 3a) y=y(u,
v) = U;V;a+
U;V;b+
U;V;c+
U;V;d
((u,
v) ~1x1)
(4
G. Geise, U. Langbecker
Fig. 3. Plane quadrangle.
145
/ Quadric segments
(a) Convex. (b) Non-convex
is a rational, even an entire rational parametric representation of the convex surface with boundary a@ = s,,~ U sbd U sdc U s,,. The assumption of convexity is indispensible. The interpretation of (4) is obvious: The line segment sw with SW:
p=p(u)=
UJa +
U,‘b,
q=q(u)=U;c+
lJ;d
(uEI)
or the line segment s, with s TS:
r=r(u)=
VJa+
V:c,
s=s(u)=
I’2b-t
V;d
(uEI)
sweeps out the surface. In the case of non-convexity the line segment sw or s, sweeps out more then the surface bounded by the given line segments (Fig. 3b; the additional region is bounded by parts of the segments sbd and s,,~ and a parabolic segment). The corners of the convex quadrangle are suitable to determine in the plane of these points an internal affine coordinate system, for instance by the associations a - (0, 0), b * (1, 0), c - (0, l), d * (1, 1). Associating these points with coordinate vectors (1, 0, 0), (1, 1, 0), (1, 0, l), (1, 1, 1) respectively with an internal projective coordinate system one can see that convexity is equivalent to the condition that the line passing through (0, 1, 0) and (0, 0, 1) (the internal line at infinity) does not intersect one of the four line segments (Fig. 3).
146
G. Geise, U. Langbecker / Quadric segments
2. Finite quadric segments bounded by four plane curve segments 2.1. Segments
on ruled quadrics with a skew quadrangle
boundary
Let %b, %d? Sdey%abe the sides of a skew quadrangle in E3, determined by the comers a, Then the follow6, c, d with a = (1, a,, a2, a3)’ ,... E lR4, and let a@ = sab u sbd u sdcu se,,. ing holds: Theorem 2. (1) tit
w,,
wb,
w,, w, be positive real numbers and w := (w,, w,, w,, wd)‘. Then
v) = uo’v; wLIa + U;V;w,b
x=x(u,
+ U;V,‘w,c + U,‘V,‘w,d
((u,
v) E Ix
I)
(5)
is a parametric representation of a ruled quadric segment @, which is finite and has the sides of the skew quadrangle as boundary a@. The w are so called homogeneous weight(-coordinate) vectors. In other words: (l*) The non-planar rational tensor product Bizier surface of degree (1, 1) represented by (5) belongs to a ruled quadric and possesses the given skew quadrangle as boundary; the sides of the quadrangle belong to generators of the ruled quadric. (2) Different weights w and w’ provide different parametrizations of the same surface patch iff det
wd wa
I r
wd wa
zi$
1
=O.
(3) Iff w,w, - w,w, = 0, the patch belongs to a hyperbolic paraboloid, otherwise to a hyperboloid of one sheet. (4) As well as sharing the common boundary a@, all these surface patches have the same tangent planes at the corners a, b, c, d. (5) The point p is a point of the patch according to (5) iff the linear system [a b c d] w = p has a positive solution w.
Proof. Preliminary. By assumption the lines lob, I,,, lde, I,, do not lie in a plane. Therefore they determine a pencil of ruled quadrics. The pencil is spanned by the plane-pairs (zbCn, ekdE) and (edrrb, cdaC) which are reducible quadrics, each containing all four lines. Let 8, b, 2, d be
d
Fig. 4. A skew quadrangle defining a pencil of ruled quadrics.
G. Geise, U. Langbecker / Quadric segments
141
coordinate vectors of the planes ebca, ebcd, edllb, edpc respectively (Fig. 4). Then the pencil of quadrics has the representation Q=Q(x;
h, k):
O=ha^‘xd^‘x+k&c^‘x
(6)
with (h, k) as homogeneous pencil parameter; (h, k) E R X R \ { (0, 0)). A point p E E3 not lying on lab, I,,, I,, or I,, determines a unique surface of the pencil. The corresponding pencil parameter value is fixed by (h(p),
k(p))
= (i?pe’p,-atp,_‘p).
If p is not a point of one of the planes 8, f, P, 2, the quadric is regular. Two different points p and q determine the same quadric iff (h(p), k(p)) = (h(q), k(q)) ( f (0, 0)). Ad (1) and (l*): Because the weights are all positive the boundary of the surface patch represented by (5) is composed of the sides of the skew quadrangle a@. Substituting (5) into (6) one obtains the relation 0 = hw,w,&‘di?a
+ kw,w&c^‘b.
(7)
Hence the surface patch (5) is a subset of the ruled quadric determined by (h(w),
k(w))
= ( w,w,i@‘d,
- w~w,a’aS’d),
and since hk # 0, the quadric is regular. Ad (2): From (7) it follows that the different weights w and w’ determine the same surface patch iff the specified relation holds. Ad (3): In the pencil (6) there is exactly one hyperbolic paraboloid, and according to (4) the patch (5) is a subset of it if for instance w = (1, 1, 1, 1)’ is chosen. Ad (4): For a ruled quadric the tangent plane at a point is spanned by the generators passing through this point. 2.2. Quadric segments
swept by one conic
The conic segments cab, cbd, cdc, c,, constituting the boundary a@ of the patch @ (some of them or all may be line segments) are curves in the planes yr, y4, y2, y3 respectively. Distinguish two cases: Case 1: The planes are of rank 3, and the common point is a point of the quadric Q. Case 2: The planes are of rank 3 but the point of intersection is not a point of the quadric Q or the planes are of rank 4 (they are the lateral surfaces of a non-degenerate tetrahedron). 2.2. I. The classic stereographic projection For case 1 the planes intersect at the unique point Z of Q. Since Z is not a point on @, none of the conic segments on @ contains Z. The points a, b, c, d, Z are mutually different. If the line segments s,~, sbdl, sdc, s,, are contained in one plane, this plane does not include Z and the line segments must form the sides of a convex quadrangle. If the line segments are not contained in a plane, the point Z is not incident with the plane errbe, and the line 1, intersects the plane eabc at a point d’. (It is possible that d’ is a point at infinity, but we omit this case here.) The line segments sa6, sbdf, sdtc, s,, must form the sides of a convex quadrangle. In the first case a, 6, c, d, in the second case a, 6, c, d’ may be used to determine an internal affine coordinate system, associating a with (0, 0), b with (1, 0), c with (0, l), and d resp. d’ with (I, 1). In the non-planar case the line segments s,~, So,,, sdc, s,, define a hyperbolic paraboloid, and its patch, which is bounded by these line segments, has a parametric representation y(u,
u) = U;V;a+
U;V;b+
U&k+
U,‘V,‘d
((u, u) EZXZ).
(8)
G. Geise, U. Langbecker/ Quadricsegments
148
PO0
pro
Fig. 5. Control
points of a tensor product
Btzier patch of degree (2,2) representing
some quadric
segment
(schematic).
This may be considered as the result of a stereographic projection from Z of the parametrized convex quadrangle surface with a, b, c, d’ as comers. In the planar case (8) is the parametrization of the convex quadrangle surface with the sides sa6, sbd, sder s,,. If now the points y(u, u) of (8) are connected to Z and, by means of (2) the lines /,- are intersected with Q, that is with @, the parametric representation which results for the desired surface is 2
x=x(u,
2
c c
U)’
q2q2pij
((2.4, U)EIXI),
in which the points (control points) pij are described as follows by means of real numbers: LY:= a’Qf,
,l3:= b’Qt,
112:= a’Qb, 5;3 =
b’Qc,
poo := 2aa, pal := ya + ac PO2 := 2YC,
y := c’QF,
{,, := a’Qc,
12‘,:=
b’Qd,
plo := &35,
6 := d ‘QF,
cl4 := a’Qd, 5;, := c’Qd,
Pa + olb - S,,Z,
~20 := W,
~2~ := 6b + Bd - 12;4t,
p12 := SC + yd - &z,T,
~22 = 26d,
P 11=t[(~a+(Yd--,42)+(yb+Pc-5232)].
The points p ol, plo, p12, p21 and the points p14 := Sa + ad - l,,t and p23 := yb + PC - S23 have the simple geometric interpretations corresponding to the point s in Theorem 1 (Fig. 5). For P,~ it holds that either pl1 =p14 =p23 or pl, is a point on the connecting line through p14 and p23.
Obviously, (9) is a parametric representation of the quadric segment CDas a rational tensor product BCzier surface of degree (2,2). Some of the control points may be points at infinity (but not any of the comers pm, p20, p22, po2, by assumption). Note that for given points pi, in general the patch described by (9) is not a subset of a quad&, but a subset of an algebraic surface of order 4; the cyclides in particular have a
149
G. Geise, U. Langbecker / Quadric segmenis
representation like (9). In [Langbecker ‘881 there are necessary and sufficient conditions for the points p:, to define a quadric segment, but they are very cumbersome. Substituting I x I by R X IR one gets from (9) the classical parametric representation of the quadric Q as the stereographic projection from Z of the parametrized plane eobc, which is an injective representation with the exception of the generators passing through Z in the case when Q is a ruled quadric. This injective representation holds for the quadric segments of Q considered in every case. If the conic segments cab, c~,,, cde, c,, constituting the boundary a@ are line segments s,~, tensor product Bezier representation (5) does not always result from Sbd, sdcT scan the rational (9). One obtains 112 = lZ4 = 5r3 = 5;, = 0 immediately, but in particular the coefficient li4 + lZ3 of Z can be made zero by choosing suitable weights for a, b, c, d (i.e. by selecting a suitable projective coordinate system in E3 such that the quadric is a hyperbolic paraboloid with reference to it). If this coefficient is zero, then the point 5 occurs as the ‘ un-point’ 0 = (0, 0, 0, 0)’ and does not appear. Evidently it is decisive for this method that the convex quadrangle with a, 6, c, and d resp. d’ as comers may be stereographically projected onto the surface patch @. The interpretation that one of the conic boundary segments of @ as U- or u-line can be moved across the surface and sweeps it out is apparent.
2.2.2. Varying stereographic projection In the case just described sweeping of the surface @ may be regarded in a different way. Each u- or u-line of @ lies on a plane through t. The autocollineation of the conic containing this U- or u-line used in Theorem 1 maps Z onto a point z(u) resp. z(o) on the curve, and now each u- or u-line may be regarded as a connecting conic segment of its end points passing through z. This suggests a method for the remaining case mentioned at the beginning of subsection 2.2, suitable in principle also for the case just analysed. Each boundary segment c,,b, Cbd, Cd,, c,, permits a parametric representation corresponding to (3). Choosing points a, E c,, and b, E cbd (not end points) connect them by a conic segment caZbZwhich belongs to the inner region of @. With suitable weights for the points a = u1 =poo, cl=plo, b=b,=p,,, c=u2=po2, cZ=p12, d=b,=p,,, a, and bZ as well as for the intersection point cZ of the tangents in u, and in b, of the conic containing the segment c,_~_ parametric representations of these conic segments result in a form as follows (Fig. 6): _ 1 a(u)
=
c cd :
b(u)
= U&z, + U;c2 + U;b2
C r,b,
:
Uo2ul
+
Q2c,
+
U22b,
c ab. .
(UEZ).
z ( u) = U,a, + U,%, + U;Lb,
For each u E I the points u(u) and b(u) are connected by the conic through z(u). So, by using (3) of Theorem 1, the quadric patch representation which has the following structure: x=x(u,
u)
=2J$*a(u)u(u)=
5;
i=()
segment on @ passing @ gets a parametric
cJ6r$2q;j
V:[P(u)u(u)+cu(u)b(u)-C(u)z(u)] ((u,
+2Vt/l(u)b(u)
U)EIXI).
j=o
(The calculation of the control points q,, is realized indirectly by a program by means of which figures are produced.) Obviously, this is a representation of @ as a rational tensor product
G. Geise, U. Langbecker / Quadric segments
150
pooa
2
=b2
Fig. 6. Connecting conic segments sweeping a quadric segment and generating a tensor product BCz.ierpatch of degree (6,2) (schematic).
Btzier
domain
surface of degree (m, n) with 1 < m < 6, 1 < n < 2. In general, 1 x I to Iw x R yields a non-admissible representation of Q.
the extension
of the
References Farin, G. (1989), Rational curves and surfaces, in: T. Lyche and L.L. Schumaker, eds., Mathematical Merho& in Computer Aided Geometric Design, Academic Press, Boston. Hoschek, J. and Lasser, D. (1989), Grundlagen der geometrischen Daienuerarbeitung, Teubner. Stuttgart. Langbecker, U. (1988), Quadrik-Fllchenstiicke, die durch Kegelschnitte berandet sind, Diplomarbeit, Technische Universitat Dresden (GDR). Pedoe, D. (1970), A Course of Geometry, Cambridge Univ. Press, Cambridge. Schaal, H. (1980), Lineare Algebra und Analyiische Geomelrie, Bd. II, Vieweg, Braunschweig, Wiesbaden.
Design
141
7 (1990) 141-150
Finite quadric segments with four conic boundary curves G. GEISE Sektion Mathematik,
Technische Universitiit Dresden, GDR
U. LANGBECKER Zentralinstitut
fti Molekularbiologie,
Berlin, GDR
Presented at Oberwolfach April 1989 Received July 1989; revised November
1989
Abstract. Finite quadric segments bounded by four plane curves and smooth in the sense of differential geometry are considered. Such a quadric segment which can swept by one conic possesses a representation x = x( U, v) on [0, I] x [O,l] as rational tensor product Btzier surface of degree (m. n) with m < 6 and n & 2. This is founded on known facts concerning rational Btzier representations of conks from the viewpoint of stereographic projection. Some special cases are investigated, especially patches on ruled quadrics bounded by four line segments. Keywords. Conic BCzier surfaces.
sections,
quadrics,
stereographic
projection,
rational
BCzier curves,
rational
tensor
product
0. Introduction 0.1. Quadric segments
bounded by four plane curve segments
Let Q be a non-reducible (unipartite) quadric of the 3-dimensional euclidean (or affine) space E3 (over R) and @ a (finite) surface segment on Q. Let @ have a boundary consisting of four conic segments cab, cbd, cdc, c,, intersecting only at their end points in the order ubdcu: The patch @ shall be smooth in the sense of differential geometry. a@=cllbUC~~UCdcuC,,. Wanted are admissible parametric representations x = x( u, v) defined on Z X I with Z = [0, l] such that the u-lines v = 0 and v = 1 and the v-lines u = 0 and u = 1 are the conic boundary curves, see Fig. 1. For the surface patch @ the following conditions are necessary: (1) If Q is a cone, the vertex does not belong to @. (2) The vertices of Q, are convex: Let e E (a, b, c, d} be one of the vertices; then in the tangent plane of @ at e the tangents of the curves of @ beginning at e fill an angle domain which has a measure < T and > 0. If four conic segments cab, cbd, cde, c,, on a quadric Q are given, satisfying the prescribed conditions, there is no need to determine a surface patch @ of the kind described. But that interesting problem: “Under what circumstances do four conks determine a surface as above?” will not be considered here. The aim of this article is to show that a quad& patch @ which can be swept by one conic possesses a parametric representation x = x(24, v)((u, v) E Z X I) as rational tensor product Bezier surface of degree (m, n) with 1 < m G 6, 1 < n G 2. The investigations are carried out in 0167-8396/90/$03.50
0 1990 - Elsevier Science Publishers
B.V. (North-Holland)
142
G. Geise, U. Lungbecker
/ Quadric segments
b Fig. 1. Surface segment
with four boundary
curves
the projectively closed euclidean (resp. affine) 3-space E3, which results by adding the points at infinity (or ideal or improper points) to E3 [Pedoe ‘70, Schaal ‘801. The essential instrument is the representation of conic sections by way of rational Betier curves of second degree from the viewpoint of stereographic projection. It enables for three points a, 6, z of a non-reducible conic c the description of both the segment C,b 3 z connecting a and b not passing through z and the segment cab 3 z connecting Q and b passing through z. If it is feasible to move the points a, b, z along conic segments on the quadric Q such that the curve segment cab or Zab sweeps the surface patch @ then the desired representation arises. The well-known stereographic projection of a quadric from a single quadric point is a special case. Additional special cases arise if Q is a ruled quadric and some or all bounding conic segments of @ are line segments. Subsets of hyperbolic paraboloids which are bounded by skew quadrangles (the sides are parts of line generators of the quadtic) are well known. Such surface segments on one-sheet hyperboloids are less familiar. Both of them completely comprise the rational tensor product BCzier patches of degree (1,l). Being part of both algebraic geometry and projective differential geometry the projective geometry used in this article is the classical basis for investigating rational curves and surfaces (i.e. described by means of polynomials). In the projective version an exterior phenomenon is the absence of the fraction bar but it is the essential natural way of including points at infinity (especially control points at infinity). Recent investigations on these topics, suggested by problems of CAGD, are generally in a pre-projective version; there are surveys in [Hoschek & Lasser ‘89, p. 2651 or [Farin ‘891, for instance, where the relevant literature is cited. 0.2. Definitions In E3 the quadric Q as a surface of second order has an equation Q:
O=x’Qx
withQ=(qij);+,
,___.s =Q’,rankQ~
(4, 3).
j=O,...,3
In this x := (x0, xi, xz, XJ E lR4\{O} with 0 = (0, 0, 0, 0)’ is a homogeneous (or projective) coordinate vector of a point x E E3 (same notation). If x E E3 (proper point of E3), x0 # 0 is
143
G. Geise, U. Langbecker / Quadric segments
characteristic; such points possess Cartesian (or affine) coordinates which are the elements of an inhomogeneous coordinate vector x := (x,/x,, x,/x,, x3/x0)’ =: (x, y, z)‘. Planes are denoted by lower case Greek letters, their coordinate vectors are marked by ? 2 E R4\ (0). The polar of a point x with respect to the quadric Q is the plane i = Qx. Note that = often stands for ‘projectively equal’, i.e. ‘equal apart from a factor # 0’, and therefore ‘unequal’ is equivalent to ‘linearly independent’. conic Let cw be one of the given conic segments. If cw belongs to a regular (unipartite) section c c Q, the other segment of c connecting p and q is denoted by Cm: c=c,Ui;,.The by sw; plane containing c is referred to as y. If cw is a line segment this will be distinguished three non-collinear points p, q, r, the line containing this segment is I,. The plane containing is denoted by cWr. For brevity the Bernstein polynomials are written by laying emphasis on the argument according to the model: T,” := B:(t).
1. Preliminaries I. I. The basic formular of stereographic projection Let 0 = X’QX be an equation different from z. The following
of the quadric Q in E3, z a point on the quadric lemma is well known:
and y a point
Lemma. Iff 2 y’Qzy - y’Qyz = 0 the line I, is a subset of Q. Otherwise the point of intersection of I, and Q different from z is x = 2 ytQr.y - y’Qvz.
(I)
1.2. Connecting conic segments Let a, b, z be three points of Q, not lying on one line. Let their plane eUbz intersect the quadric in a regular conic section c. The curve c is subdivided into two segments c,,~ 3 z and ~5,~3 z. Here the points a and b are assumed to be proper points with coordinate vectors of the form a = (1, a,, a2, a3)‘, b = (1, b,, b,, b,)’ (with no essential loss of generality). Then the following theorem holds (Fig. 2): Theorem 1. (1) A parametric s ab- *
y=y(t)
representation
= T;u+
T;b
of the euclidean
(affine)
line segment s,~ of I,, is
(tH).
The external segment connecting a and b, containing the point at infinity of lab, has the parametric representation S,,:
Y=y(t)=T,‘a-T,‘b
(2) A parametric c*b 3 z:
representation
+=I). of the finite conic segment
~=Z(t)=2T0*uu+T~s+2T2*~b withs:=&z+crb-{z,
in which u := u’Qz,
/3 := b’Qz,
5 := u’Qb,
C,, 3 z of c is
(tEI)
(2)
144
G. Geise, U. Langbecker
Fig. 2. Stereographic
and a parametric
projection
of a conic chord from a conic point onto a conic segment.
representation
C nb3Z:
/ Quadric segments
Autocollineation
of a conic.
of the other conic segment c,~ 3 z of c is
x=x(t)=2T02au-T;s+2T,2pb
(tE1).
(3)
Here s is the point of intersection of the tangents to c at a and b. The proof proceeds by verifying that x E c, si E c, and uses several geometric propositions. Points y E s,~ and jj E S,, belonging to the same parameter value t E (0, 1) form a harmonic point quadruple with the points u and b; the cross ratio is cr( y, JJ; 4, b) = - 1. Equation (2) gives the stereographic projection of s,~ from z onto the conic segment C,b 3 z.. The involutive central collineation of the plane eak with s as centre and I,, as axis exchanges the segments c ab and c,b- This is easily seen from X + x = T02aa + T,2Pb,
F-x=
T12s.
This autocollineation of c changes the stereographic projection of sab from z = pa + cub- s onto the segment ZUb3 z into the stereographic projection of sab from t = pa + ab + s onto cab 3 z. Now we can see c&, as the stereographic projection of iab from z and ZUb as the stereographic projection of Fab from Z. The inhomogeneous form of the Bernstein polynomials is a blemish. It is easily removable, but without influencing the representations (2) and (3) which only are of interest. We omit criteria needed to ensure that a conic segment cab or Cab is of finite extent. 1.3. Parametrization
of plane convex quadrangles
Let &by %d, sdc? %a be the sides of a plane convex quadrangle in E3, determined by the comers II, b, c, d with a = (1, a,, u2, a3)‘,.. . E R4. It then holds that (Fig. 3a) y=y(u,
v) = U;V;a+
U;V;b+
U;V;c+
U;V;d
((u,
v) ~1x1)
(4
G. Geise, U. Langbecker
Fig. 3. Plane quadrangle.
145
/ Quadric segments
(a) Convex. (b) Non-convex
is a rational, even an entire rational parametric representation of the convex surface with boundary a@ = s,,~ U sbd U sdc U s,,. The assumption of convexity is indispensible. The interpretation of (4) is obvious: The line segment sw with SW:
p=p(u)=
UJa +
U,‘b,
q=q(u)=U;c+
lJ;d
(uEI)
or the line segment s, with s TS:
r=r(u)=
VJa+
V:c,
s=s(u)=
I’2b-t
V;d
(uEI)
sweeps out the surface. In the case of non-convexity the line segment sw or s, sweeps out more then the surface bounded by the given line segments (Fig. 3b; the additional region is bounded by parts of the segments sbd and s,,~ and a parabolic segment). The corners of the convex quadrangle are suitable to determine in the plane of these points an internal affine coordinate system, for instance by the associations a - (0, 0), b * (1, 0), c - (0, l), d * (1, 1). Associating these points with coordinate vectors (1, 0, 0), (1, 1, 0), (1, 0, l), (1, 1, 1) respectively with an internal projective coordinate system one can see that convexity is equivalent to the condition that the line passing through (0, 1, 0) and (0, 0, 1) (the internal line at infinity) does not intersect one of the four line segments (Fig. 3).
146
G. Geise, U. Langbecker / Quadric segments
2. Finite quadric segments bounded by four plane curve segments 2.1. Segments
on ruled quadrics with a skew quadrangle
boundary
Let %b, %d? Sdey%abe the sides of a skew quadrangle in E3, determined by the comers a, Then the follow6, c, d with a = (1, a,, a2, a3)’ ,... E lR4, and let a@ = sab u sbd u sdcu se,,. ing holds: Theorem 2. (1) tit
w,,
wb,
w,, w, be positive real numbers and w := (w,, w,, w,, wd)‘. Then
v) = uo’v; wLIa + U;V;w,b
x=x(u,
+ U;V,‘w,c + U,‘V,‘w,d
((u,
v) E Ix
I)
(5)
is a parametric representation of a ruled quadric segment @, which is finite and has the sides of the skew quadrangle as boundary a@. The w are so called homogeneous weight(-coordinate) vectors. In other words: (l*) The non-planar rational tensor product Bizier surface of degree (1, 1) represented by (5) belongs to a ruled quadric and possesses the given skew quadrangle as boundary; the sides of the quadrangle belong to generators of the ruled quadric. (2) Different weights w and w’ provide different parametrizations of the same surface patch iff det
wd wa
I r
wd wa
zi$
1
=O.
(3) Iff w,w, - w,w, = 0, the patch belongs to a hyperbolic paraboloid, otherwise to a hyperboloid of one sheet. (4) As well as sharing the common boundary a@, all these surface patches have the same tangent planes at the corners a, b, c, d. (5) The point p is a point of the patch according to (5) iff the linear system [a b c d] w = p has a positive solution w.
Proof. Preliminary. By assumption the lines lob, I,,, lde, I,, do not lie in a plane. Therefore they determine a pencil of ruled quadrics. The pencil is spanned by the plane-pairs (zbCn, ekdE) and (edrrb, cdaC) which are reducible quadrics, each containing all four lines. Let 8, b, 2, d be
d
Fig. 4. A skew quadrangle defining a pencil of ruled quadrics.
G. Geise, U. Langbecker / Quadric segments
141
coordinate vectors of the planes ebca, ebcd, edllb, edpc respectively (Fig. 4). Then the pencil of quadrics has the representation Q=Q(x;
h, k):
O=ha^‘xd^‘x+k&c^‘x
(6)
with (h, k) as homogeneous pencil parameter; (h, k) E R X R \ { (0, 0)). A point p E E3 not lying on lab, I,,, I,, or I,, determines a unique surface of the pencil. The corresponding pencil parameter value is fixed by (h(p),
k(p))
= (i?pe’p,-atp,_‘p).
If p is not a point of one of the planes 8, f, P, 2, the quadric is regular. Two different points p and q determine the same quadric iff (h(p), k(p)) = (h(q), k(q)) ( f (0, 0)). Ad (1) and (l*): Because the weights are all positive the boundary of the surface patch represented by (5) is composed of the sides of the skew quadrangle a@. Substituting (5) into (6) one obtains the relation 0 = hw,w,&‘di?a
+ kw,w&c^‘b.
(7)
Hence the surface patch (5) is a subset of the ruled quadric determined by (h(w),
k(w))
= ( w,w,i@‘d,
- w~w,a’aS’d),
and since hk # 0, the quadric is regular. Ad (2): From (7) it follows that the different weights w and w’ determine the same surface patch iff the specified relation holds. Ad (3): In the pencil (6) there is exactly one hyperbolic paraboloid, and according to (4) the patch (5) is a subset of it if for instance w = (1, 1, 1, 1)’ is chosen. Ad (4): For a ruled quadric the tangent plane at a point is spanned by the generators passing through this point. 2.2. Quadric segments
swept by one conic
The conic segments cab, cbd, cdc, c,, constituting the boundary a@ of the patch @ (some of them or all may be line segments) are curves in the planes yr, y4, y2, y3 respectively. Distinguish two cases: Case 1: The planes are of rank 3, and the common point is a point of the quadric Q. Case 2: The planes are of rank 3 but the point of intersection is not a point of the quadric Q or the planes are of rank 4 (they are the lateral surfaces of a non-degenerate tetrahedron). 2.2. I. The classic stereographic projection For case 1 the planes intersect at the unique point Z of Q. Since Z is not a point on @, none of the conic segments on @ contains Z. The points a, b, c, d, Z are mutually different. If the line segments s,~, sbdl, sdc, s,, are contained in one plane, this plane does not include Z and the line segments must form the sides of a convex quadrangle. If the line segments are not contained in a plane, the point Z is not incident with the plane errbe, and the line 1, intersects the plane eabc at a point d’. (It is possible that d’ is a point at infinity, but we omit this case here.) The line segments sa6, sbdf, sdtc, s,, must form the sides of a convex quadrangle. In the first case a, 6, c, d, in the second case a, 6, c, d’ may be used to determine an internal affine coordinate system, associating a with (0, 0), b with (1, 0), c with (0, l), and d resp. d’ with (I, 1). In the non-planar case the line segments s,~, So,,, sdc, s,, define a hyperbolic paraboloid, and its patch, which is bounded by these line segments, has a parametric representation y(u,
u) = U;V;a+
U;V;b+
U&k+
U,‘V,‘d
((u, u) EZXZ).
(8)
G. Geise, U. Langbecker/ Quadricsegments
148
PO0
pro
Fig. 5. Control
points of a tensor product
Btzier patch of degree (2,2) representing
some quadric
segment
(schematic).
This may be considered as the result of a stereographic projection from Z of the parametrized convex quadrangle surface with a, b, c, d’ as comers. In the planar case (8) is the parametrization of the convex quadrangle surface with the sides sa6, sbd, sder s,,. If now the points y(u, u) of (8) are connected to Z and, by means of (2) the lines /,- are intersected with Q, that is with @, the parametric representation which results for the desired surface is 2
x=x(u,
2
c c
U)’
q2q2pij
((2.4, U)EIXI),
in which the points (control points) pij are described as follows by means of real numbers: LY:= a’Qf,
,l3:= b’Qt,
112:= a’Qb, 5;3 =
b’Qc,
poo := 2aa, pal := ya + ac PO2 := 2YC,
y := c’QF,
{,, := a’Qc,
12‘,:=
b’Qd,
plo := &35,
6 := d ‘QF,
cl4 := a’Qd, 5;, := c’Qd,
Pa + olb - S,,Z,
~20 := W,
~2~ := 6b + Bd - 12;4t,
p12 := SC + yd - &z,T,
~22 = 26d,
P 11=t[(~a+(Yd--,42)+(yb+Pc-5232)].
The points p ol, plo, p12, p21 and the points p14 := Sa + ad - l,,t and p23 := yb + PC - S23 have the simple geometric interpretations corresponding to the point s in Theorem 1 (Fig. 5). For P,~ it holds that either pl1 =p14 =p23 or pl, is a point on the connecting line through p14 and p23.
Obviously, (9) is a parametric representation of the quadric segment CDas a rational tensor product BCzier surface of degree (2,2). Some of the control points may be points at infinity (but not any of the comers pm, p20, p22, po2, by assumption). Note that for given points pi, in general the patch described by (9) is not a subset of a quad&, but a subset of an algebraic surface of order 4; the cyclides in particular have a
149
G. Geise, U. Langbecker / Quadric segmenis
representation like (9). In [Langbecker ‘881 there are necessary and sufficient conditions for the points p:, to define a quadric segment, but they are very cumbersome. Substituting I x I by R X IR one gets from (9) the classical parametric representation of the quadric Q as the stereographic projection from Z of the parametrized plane eobc, which is an injective representation with the exception of the generators passing through Z in the case when Q is a ruled quadric. This injective representation holds for the quadric segments of Q considered in every case. If the conic segments cab, c~,,, cde, c,, constituting the boundary a@ are line segments s,~, tensor product Bezier representation (5) does not always result from Sbd, sdcT scan the rational (9). One obtains 112 = lZ4 = 5r3 = 5;, = 0 immediately, but in particular the coefficient li4 + lZ3 of Z can be made zero by choosing suitable weights for a, b, c, d (i.e. by selecting a suitable projective coordinate system in E3 such that the quadric is a hyperbolic paraboloid with reference to it). If this coefficient is zero, then the point 5 occurs as the ‘ un-point’ 0 = (0, 0, 0, 0)’ and does not appear. Evidently it is decisive for this method that the convex quadrangle with a, 6, c, and d resp. d’ as comers may be stereographically projected onto the surface patch @. The interpretation that one of the conic boundary segments of @ as U- or u-line can be moved across the surface and sweeps it out is apparent.
2.2.2. Varying stereographic projection In the case just described sweeping of the surface @ may be regarded in a different way. Each u- or u-line of @ lies on a plane through t. The autocollineation of the conic containing this U- or u-line used in Theorem 1 maps Z onto a point z(u) resp. z(o) on the curve, and now each u- or u-line may be regarded as a connecting conic segment of its end points passing through z. This suggests a method for the remaining case mentioned at the beginning of subsection 2.2, suitable in principle also for the case just analysed. Each boundary segment c,,b, Cbd, Cd,, c,, permits a parametric representation corresponding to (3). Choosing points a, E c,, and b, E cbd (not end points) connect them by a conic segment caZbZwhich belongs to the inner region of @. With suitable weights for the points a = u1 =poo, cl=plo, b=b,=p,,, c=u2=po2, cZ=p12, d=b,=p,,, a, and bZ as well as for the intersection point cZ of the tangents in u, and in b, of the conic containing the segment c,_~_ parametric representations of these conic segments result in a form as follows (Fig. 6): _ 1 a(u)
=
c cd :
b(u)
= U&z, + U;c2 + U;b2
C r,b,
:
Uo2ul
+
Q2c,
+
U22b,
c ab. .
(UEZ).
z ( u) = U,a, + U,%, + U;Lb,
For each u E I the points u(u) and b(u) are connected by the conic through z(u). So, by using (3) of Theorem 1, the quadric patch representation which has the following structure: x=x(u,
u)
=2J$*a(u)u(u)=
5;
i=()
segment on @ passing @ gets a parametric
cJ6r$2q;j
V:[P(u)u(u)+cu(u)b(u)-C(u)z(u)] ((u,
+2Vt/l(u)b(u)
U)EIXI).
j=o
(The calculation of the control points q,, is realized indirectly by a program by means of which figures are produced.) Obviously, this is a representation of @ as a rational tensor product
G. Geise, U. Langbecker / Quadric segments
150
pooa
2
=b2
Fig. 6. Connecting conic segments sweeping a quadric segment and generating a tensor product BCz.ierpatch of degree (6,2) (schematic).
Btzier
domain
surface of degree (m, n) with 1 < m < 6, 1 < n < 2. In general, 1 x I to Iw x R yields a non-admissible representation of Q.
the extension
of the
References Farin, G. (1989), Rational curves and surfaces, in: T. Lyche and L.L. Schumaker, eds., Mathematical Merho& in Computer Aided Geometric Design, Academic Press, Boston. Hoschek, J. and Lasser, D. (1989), Grundlagen der geometrischen Daienuerarbeitung, Teubner. Stuttgart. Langbecker, U. (1988), Quadrik-Fllchenstiicke, die durch Kegelschnitte berandet sind, Diplomarbeit, Technische Universitat Dresden (GDR). Pedoe, D. (1970), A Course of Geometry, Cambridge Univ. Press, Cambridge. Schaal, H. (1980), Lineare Algebra und Analyiische Geomelrie, Bd. II, Vieweg, Braunschweig, Wiesbaden.