Many algorithms in Computer Aided Geometric Design are comer cutting algorithms like de Casteljau's, degree elevation, or knot insertion. Comer cutting is attractive since it provides simple geometric constructions of curves. It means to produce a sequence (Pn) of polygons, i.e., piecewise linear curves, such that Pn+] is obtained from T'n by one or several cuts as illustrated in Fig. 1.
In this paper we assume that any three consecutive vertices of a polygon are distinct and consider only local cuts. Following de Boor (1990, Section 3) we will call a cut local if it removes exactly one vertex and adds two new ones. Thus if 79n+1 is obtained
* Corresponding author. E-mail: [email protected]. The work of this author has been partially supported by the DAAD and CDCH-UCV. I The work of this author has been partially supported by the DFG and CDCH-UCV.
from 79n by m local cuts, then "~r~q-I has m more vertices than 7vn and each edge of 7v~ contains an edge of Pn+l.
Local comer cutting has been studied by de Rham (1947, 1956), Gregory and Qu (1996), and de Boor (1990): Let (79n) be a sequence of polygons generated by local cuts such that all comers of Pn are cut simultaneously, i.e., Pn+l divides each edge of P,~ into three segments as illustrated in Fig. 2. Let the lengths of these 3 segments have the ratios ~: 1 - ~ - ~7 : ~/where ~ and r/may be different for each n and each edge of 79n.
Further, let ~, 7, ~, ~ be the supremum and infimum of all ~'s and ~7's. Then Gregory and Qu and later de Boor, with a different proof, showed that the sequence (Pn) con- verges to a C 1-curve, if the points (~, 7) and (~_, 77) lie in the interior of the shaded region shown in Fig. 3.
Not every comer cutting algorithm which produces C 1-curves satisfies the Gregory-Qu condition. In this paper we will present a necessary and sufficient condition for a local comer cutting scheme to produce C '~-curves. Our result also leads to a new proof of the result by Gregory and Qu. While the proofs in (Gregory and Qu, 1996) and (de Boor, 1990) are analytic, our proofs are geometric.
Furthermore we will consider the dual statement of our result. A comer cutting process is dual to an interpolatory refinement scheme, i.e., a scheme which produces a sequence
M. Paluszny et al. / Computer Aided Geometric Design 14 (1997) 421-447
1
1/2
0 1/2 1
Fig. 3. The Gregory-Qu region.
423
of polygons Qn such that for all n the vertices of Qn are also vertices of Q,~+l as illustrated in Fig. 6.
Finally we present a convexity preserving local interpolation scheme with geometric shape handles well suited for interactive geometric design. The method is very much related to the matrix subdivision scheme studied in, e.g., (Prautzsch, 1991).
2. Preliminaries
Here we recall some simple well-known geometric facts which are crucial for the entire paper.
Throughout we will work in an affine plane A over I~ equipped with the standard topology and extend A if needed by the ideal line. Often we will need a scalar product in A and thus consider a Euclidean plane instead.
In this paper we define a closed convex curve in an affine plane as the boundary of a bounded convex subset with nonempty interior. Note that a convex set without interior points is either a point, line, or a line segment, see, e.g., (Limaye, 1981, Lemma 4, p. 330).
For the rest of this section let .A4 be an open or closed convex subset of A with
nonempty interior .M and let x be some boundary point of .A4. Then the lines through
x which do not intersect .A~ are the supporting lines of .A4 and its boundary a M at x. Note that the supporting lines at x form a closed cone which is complementary to the
cone x + R ( x - .A4). If there is more than one supporting line at x, we say that .M and a M have a comer at x.
Further, let I? be the projective extension of A. Then the polar set M v is defined as the subset of the dual plane ]?* formed by all lines in P which do not intersect M , see (Schneider, 1993, Section 1.6). Since every point p E A \ .M carries a line which does not meet M (cf. (Schneider, 1993, Theorem 1.3.4)), M consists of all points (~ lines
424 M. Paluszny et aL / Computer Aided Geometric Design 14 (1997) 421~147
in IP*) which do not meet .AA p. Hence A,4 is the polar set of ,At p (Schneider, 1993, Theorem 1.6.1) and .Ad p lies in an(y) affine plane (of It ~* whose ideal line is a point of .AA).
Note that the above polarization reverses set operations. Namely if .A4 is a subset of another convex set A/" in A, then .M p contains A/"p and if the union of some convex sets .Mi in A with nonempty interior is convex, then this union is polar to the intersection of the polar sets .A4i v, cf. also (Schneider, 1993, Theorem 1.6.2).
Next we observe that Adv is also convex. Namely every line in ~* (= pencil of lines in ~), which intersects .A,4 p, does so in a connected line segment. This line segment forms a cone in ~ complementary to some cone p + ~ ( . M - p ) . Further, if AA is bounded, A//p has a nonempty interior and is also bounded. This follows from the fact that the polar set of an open (closed) triangle of A is a closed (open) triangle in some affine plane of ~* and since a bounded set with nonempty interior is contained in a triangle and contains a triangle.
Let .A4 be bounded. Then the closure of A4 can be obtained by intersecting all closed triangles (= intersection of three half spaces) containing .A4, while the interior of .A4 is formed by the union of all open triangles contained in AA. Hence the closure (interior) of AA is the polar set of the interior (closure) of A l p. This enables us to prove the following lemma.
Lemma 2.1. Let AA be a bounded convex set of the affine plane with nonempty interior. Then the supporting lines of .M form the boundary of .A4 v.
Proof. A line E in ~ supports AA iff it meets the closure but not the interior of M or iff E as a point of ~* does not lie in the interior but in the closure of .AAP which proves the lemma. []
The lemma shows that the supporting lines of a convex curve C form a convex curve C a in the dual plane. It is the envelope of C. Note that with the usual identification of and I~** we further obtain that C is the envelope of C a.
There is another crucial consequence of the lemma above: Namely, the supporting lines/4 at some point x of C are the points L / o f a line segment of C a which lies on the line x in 79.. Similarly, a line segment of C corresponds to a comer of C a.
Remark 2.2. Consider a correlation (i.e, a projective map) between ? and IP*. On view- ing the image of the ideal line as the ideal line in ]P*, the correlation is an extended affine map. Hence the image of C under a correlation is again a convex curve. Moreover, the image of a comer or line segment is also a comer or line segment respectively.
3. Open curves
So far we considered only closed curves. But it is easy to include also open curves: An open curve C in an affine plane A is called convex if there is a second curve ~D such that the union of C and 79 forms a closed convex curve.
Let C be an open convex curve and C t2 79 be a closed convex curve. Then the support- ing lines of C t_J/9 contacting C form an open convex curve C a in the dual plane. Let
M. Paluszny et al. / Computer Aided Geometric Design 14 (1997) 421-447 4 2 5
£
a b
t I I 1
\ \
\
I I I
I /
/ D /
a b
p°
I I i I
\ / \ i
Fig. 4. An open convex curve and its envelope.
a and b be the end points of C. If C U 79 has a comer at a (or b), then the envelope C ~ starts (or ends) with a line segment whose length depends on 79. We will say that C starts or ends with a corner if C U 79 has a comer at a or b.
Let 9 r and £ be the first and last point of C ~ in ~*, then C 1~ is completely determined by C and the lines .T" and / : in ~' (without need of 79). We will call the lines ~" and £ the end tangents of C. Moreover, C is the envelope of C 1~ with end tangents a and b in ~*. Fig. 4 gives an illustration, where C with end tangents .T" and £ has no end corners while C 1~ with end tangents a and b does. The right side of Fig. 4 is a dual image of the left side obtained by the polarity with respect to the circle 79, cf., e.g., (Brieskorn and Kn~rrer, 1986, pp. 580-582) for a description of this construction.
Further we recall, e.g., from (Schneider, 1993, Corollary 1.4.5), that a compact convex set is the convex hull of its boundary. We employ this relationship to define the limit o f a
sequence o f open convex curves C,~ with the same end points and end tangents: Assume that the end tangents are specified by some common arc 79 as described above where 79 is disjoint with all Cn. Further let the convex hulls 7-/n of the convex curves Cn U 79 be
, o o monotonely decreasing, 7-/0 __D 7-/1 _3 . . . such that n,~=0 n has a nonempty interior. Then we define lira tTn as the boundary of N~=0 ~n without 79.
If the sets 7-/n are monotonely increasing such that Un~__0 7-/,~ is bounded, we define lim Cn analogously as the boundary of Un~__0 7"/n without 79. Under polarization we ob- tain then monotonely decreasing sets, 7-/~ _3 7"/p _3 7-/~ _3 . . . . From Section 2 it follows that UT-/n is the polar set of r) 7-/p. Thus, Lemma 2.1 implies (limCn) v = limC~ p. In both cases for in- and decreasing sequences (7-/n) we define the end tangents of limCn also by 79.
Example 3.1. Even if all Cn do not start with a comer, the limiting curve may do so: Consider three affinely independent points a, c, b and the open polygons tTn with vertices
a , a + , c + b b n + n + l '
without end corners. Then limCn is the line segment ab with end tangents a c and cb
as illustrated in Fig. 5.
4 2 6 M. Paluszny et al. Computer Aided Geometric Design 14 (1997) 421~147
c
a ¢ ~ .
Fig. 5. An example.
Fig. 6. Contact points on 790 and 791 and the polygons Qo and Qi.
4. Fill-in schemes
Let 10,~ be a sequence of polygons such that 10n is obtained from 1on-1 by one or several successive local cuts. Then there is at least one point on each edge of every 10n which is never cut away. Here we will call such points contact points. They are marked by solid dots • in all the figures.
A local comer cutting algorithm is completely described by a sequence of cuts. Here, however, we will also choose one contact point on each initial and every new edge and restrict further cuts such that the already chosen contact points are not cut away. Then it follows that one can describe any local comer cutting scheme in this way.
The chosen contact points also form a polygon. Thus with the above extension a local comer cutting algorithm generates a sequence of polygon pairs ion, Qn where Qn connects the contact points on the edges of ion. We will call this extended procedure where we obtain a sequence of polygon pairs 10n, Qn as described above, afill-in scheme, see Fig. 6 for an illustration. Further we will say that 10n circumscribes Qn.
Let a and b be two contact points on successive edges of 100 as illustrated in Fig. 6. In order to investigate the differentiability of the limiting curve between a and b one may truncate all 10,~ at a and b and assume that io0 consists of just the three vertices a , c, b, see Fig. 6. Thus without loss of generality we will always assume in the sequel that
• the starting polygons 100 and Q0 for a fill-in scheme form a nondegenerate triangle a c b and
• a c and cb determine the end tangents for all 10n and Qn.
M. Paluszny et al. / Computer Aided Geometric Design 14 (1997) 421-447 427
Then the enveloping curves Q~ and 79~ are again polygons which determine a fill-in scheme. We will call it the polar fill-in-scheme. Note that under polarization 79n and Qn change their roles: Q~ circumscribes 79~. However, note that under a correlation 79n and Qn do not change their roles.
R e m a r k 4.1. In general 79~ and Qn converge to different curves. However, the condi- tions under which we study fill-in schemes in the sequel will always guarantee that the 79n and Qn have the same limiting curve.
5. The basic iemma
From Lemma 2.1, it follows that a fill-in scheme produces a CI-curve if the polar scheme produces a curve without line segments and vice versa. The following simple lemma provides a sufficient and necessary condition.
Lemma 5.1. Let 79n, Qn be a sequence of polygon pairs in a Euclidean plane obtained by a fill-in scheme from acb. I f the maximum edge length of the polygons 79n goes to zero as n --4 oo, then IP~ = lim 79n has no line segments. The converse holds for any monotone sequence of convex polygons 79n converging to a convex curve without line segments.
Proof. Assume that ~ has a line segment S. Under local comer cutting any three vertices of any 79,~ are not collinear and therefore for any n the line segment S lies in some triangle formed by two successive edges of 79,~. Hence not all edges become arbitrarily small.
For the converse assume that there are arbitrarily large indices n such that 79n has an edge which is longer than some positive e. Then there is a sequence of long edges in the convex hull of 790 if the sequence is decreasing or in the convex hull of lim 79,, in case the sequence is increasing. Since under our definition the convex hull of a convex curve is compact, a subsequence of these long edges converges to a line segment of 79~. []
Remark 5.2. If the maximum edge length of the 79n goes to zero, then the maximum edge length of the Qn also goes to zero. Hence, in this case the polygons 79,~ and Qn converge to the same curve.
The pairs of antipodal points on the unit sphere in a Euclidean 3-space form a model of the projective plane. If one introduces angles between and distances on (geodesic) lines, one has the elliptic plane, see (Hilbert and Cohn-Vossen, 1932, pp. 238-242). Under the correlation (actually it is even a polarity) which relates points and lines with the same coordinates the angle of a comer equals the length of the dual line segment. Moreover, projecting a plane C which does not contain the origin onto the sphere with the origin as center of projection as illustrated in Fig. 7 one has that every angle and length in E corresponds to an angle or length on the sphere, respectively. Comparing all angles or lengths in a compact subset of E with their images on the sphere, we obtain ratios which
sphere J
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Fig. 7. Central projection from C onto the unit sphere.
Fig. 8. C~-interpolation on the sphere.
are bounded from above and below by positive constants. Hence we can take Lemma 5.1 to the elliptic plane, by polarization further to the dual plane, then by the polarity above to the elliptic point plane again, and finally back to a Euclidean plane. Then Lemma 5.1 assumes the following form:
Corollary 5.3 (de Boor, 1990). Let 79n, Q~ be a sequence o f polygon pairs in a Eu- clidean plane obtained by a fill-in scheme from aeb. I f the sharpest comer o f the poly- gons Qn flattens out completely as n --+ c~, then Qc~ = lim Qn has no comers (and no endcorners). The converse holds for any monotone sequence o f convex polygons Qn converging to a convex curve without corners and end comers (Example 3.1 shows that end comers must not be disregarded).
In particular, one should note that besides the above all results of this paper hold for spherical cutting schemes which produce a sequence of geodesic polygons on a sphere.
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r
Fig. 9. Cutting a comer of "Pn and Q~.
For example the Cl-interpolant shown in Fig. 8 was produced by the scheme presented in Section 10.
Remark 5.4. The comers of the Qn flatten out if and only if the comers of the 79n flatten out.
6. Main results
In this section we will apply Lemma 5.1 to derive simple fill-in strategies which produce curves without comers or line segments. A major result is the following.
Theorem 6.1. Consider a sequence o f polygon pairs 79n, Qn obtained by a fill-in scheme from acb. Assume that 790 is a polygon in an affine plane. Further suppose that
• all comers o f every 79n are cut away eventually, • there is some e > 0 such that all edges o f 79n+l with endpoints s and t which cut a
comer between two contact points p , q and a vertex r o f 79n satisfy or, 7- 6 [e, 1 - e ] where s = r + cr(p - r ) and t = r + r (q - r), cf. Fig. 9 for an illustration.
Then the limiting curve 7900 has no line segments.
Remark 6.2. The contact points can lie anywhere in the interior of the corresponding edge.
Remark 6.3. The limiting curve 7900 interpolates 790 at the endpoints and has no line segments there.
Proof. Without loss of generality, assume that a, e, b are the points ( - 1,0), (0, 1), and (1,0) of the Euclidean plane R 2, respectively, as illustrated in Fig. 10. Because of
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c 1
430
Fig. 10. T'0 and a comer of "Pn with two further cuts.
Lemma 5.1 it suffices to show that the longest edge of the polygons 79n shrinks to zero. Note that for all n the slope of any edge of 79n lies between the slopes of the two edges of 790. Thus the length of any edge is bounded by the length of its orthogonal projection onto the line ab multiplied by v~.
Furthermore, if the projections of all contact points become dense in ab for n ~ c~, then the projections of the edges and thus the edges themselves become arbitrarily small. Hence we just need to consider two successive contact points of some 79n. Fig. 10 shows 790 and an arbitrary comer with two further cuts where a, b , . . . , f denote lengths of certain edges after their projections onto ab. Because of symmetry reasons we can restrict ourselves to the case a ~> b. Under this provision it is easy to see that
C 2 c > d/> ~2a ~> -~-(a + b)
and that
c ~ < e + ( 1 - - ~ ) f
~ < e + ( l - e ) ( a - e + b )
= ~ e + (l - ~ ) ( a + b)
~<~(1 - ~ ) a + (1 - ~ ) ( a + b )
~< (1 - s 2 ) ( a + b),
which implies that the maximum distance between successive contact points becomes arbitrarily small for n --+ c~. []
Note that a represents s with respect to the affine system r , p - r. It is called the affine scale with respect to r , p . Moreover, s has the projective coordinates 1 ,a with respect to the frame r , x ; p where x is the ideal point of the line and p is chosen as the unit point. Thus a is the projective scale with respect to r , x; p or equivalently the cross ratio of s , p with respect to r , x, see, e.g., (Boehm and Prautzsch, 1994, pp. 210, 227).
M. Paluszny et aL / Computer Aided Geometric Design 14 (1997) 421-447 431
Next let the envelopes Q~ and P~ satisfy the premises of Theorem 6.1 where Q~ circumscribes 79~. Then the limit of these envelopes has no line segments in ~* which
implies that 79oo - Qoo = lim Qn has no corners. Recall from Section 2 that @0 lies in an affine space of ~* where the ideal line is a point m of ~ which does not lie on any line of Q~0. Using Lemma 2.1, the conditions of Theorem 6.1 for the polar scheme can be expressed in terms of 79n and Qn. Then Theorem 6.1 takes on the following form:
Theorem 6.4. Consider a sequence o f polygon pairs 79n, Qn obtained by a fill-in scheme from acb. Assume that Qo is a polygon in the projective plane and let ra be a point which does not meet any supporting line o f Qo. Further suppose that
• all comers o f every 79n are cut away eventually, • there is some e > 0 such that all comers o f Qn+ i with end tangents ,.q and 7- which
replace an edge R o f Qn between two contact lines 79, Q satisfy ~r, 7" E Is, 1 - e] where ~ and 7" are the projective scales o f $ and 7- with respect to ~ , X; 7 ~ or ~ , y ; Q respectively, and X and y are the lines through m , c f Fig. 9 for an illustration.
Then the limiting curve 79oo has no comers.
Remark 6.5. The new contact line s t in Fig. 9 can be any line which does not intersect the secant pq.
Remark 6.6. The limiting curve 79oo is tangent to 790 at the endpoints and has no corners there.
As described before Corollary 5.3 we can take Theorem 6.1 to the elliptic plane: Theorem 6. l constrains the ratios <7 and 7", i.e., we have conditions of the form a : b = <r/> e for certain lengths a and b. Now let a and/3 be the corresponding lengths on the sphere. Then there are, as we already observed, two positive constants k and K independent of a and b such that a >t ka and/3 ~< Kb. Hence we obtain a :/3/> ka : K b >~ e k / K , i.e., if we replace Euclidean by spherical distances, Theorem 6.1 is also valid on the elliptic plane.
Again we can use the polarity which maps points onto lines with the same coordinates. Then the conditions of Theorem 6.1 constrain ratios of angles for a fill-in scheme in the dual plane. As before Theorem 6.4 we express these conditions in terms of the polar- fill-in scheme in the primal (point) plane and bring them back to a Euclidean plane in the same way as we took the conditions for ratios of lengths to the elliptic plane. Then, finally, Theorem 6.1 assumes the following form:
Theorem 6.7. Consider a sequence o f polygon pairs 7an, Qn obtained by a fill-in scheme from aeb. Assume that Q0 is a polygon in a Euclidean plane and let m be a point which does not meet any supporting line o f Qo. Further suppose that
• all comers o f every 79n are cut away eventually, • there is some ~ > 0 such that all comers o f Qn+l with end tangents ,_q and T which
replace an edge T~ o f Qn between two contact lines 7 9, Q satisfy or, 7- E [~, 1 - ~] where cr = angleT4.~:angleT~79 and 7" = angleR"]-: angleT"C.Q as illustrated in Fig. 1 1.
Then the limiting curve is C I.
432 M. Paluszny et al. / Computer Aided Geometric Design 14 (1997) 421-447
F
P Q
p q
Fig. 11. Cutting a comer of 79n.
p ~ ~ ~ q
7~
Fig. 12. The fill-in region of Dyn et al.
A weaker form of Theorem 6.7 was first proved in (Dyn et al., 1991). Fig. 12 shows successive contact points • of Qn connected by secants and a newly introduced contact point ® which lies on Qn+l. The interpolatory refinement scheme of Dyn et al allows to build only sequences of polygons such that the interior angle ~/is flatter than a certain average of qo and ~. Their conditions imply that the maximum interior angle formed by the secants of each Qn+l is flatter than some constant in (0, 1) times the maximum interior angle of Qn.
This is not the case under the assumptions of Theorem 6.7. Consequently the fill-in region of the scheme by Dyn et al is smaller than the one presented here. Fig. 12 shows their fill-in region, the shaded area, for this example.
M. Paluszny et al. / Computer Aided Geometric Design 14 (1997) 421--447 433
A special case of Theorem 6.7 was also studied in (Hejna, 1988). There the lines S, T are given by the bisectors of ;o and R, and Q and R, respectively. The new contact line is given by the bisector of the lines $ and T as just described.
7. The Gregory-Qu result
The Gregory-Qu comer cutting scheme also satisfies the conditions of Theorems 6.4 and 6.7. In order to show this, we need a property first derived by Gregory and Qu. Let ;on be a sequence of polygons in the Euclidean plane. If, for all n, ;on+l arises from ;on by cutting all comers simultaneously such that (~, 7) and (~, 77) lie in the Gregory-Qu region shown in Fig. 3, then there is a positive constant R not depending on n such that the lengths r; and A of any two successive edges of any ;on satisfy I / R <<. n /A <<. R.
Now consider a triangle on ;on formed by two consecutive contact points p, q and the vertex r between them. This triangle with a next cut s t is illustrated in Fig. 13.
Comer cutting under the Gregory-Qu condition means that for each edge one has
contact point = (1 - c) • left end point .4. c . right end point
where c > E :-- min{~, rT}. Hence the ratio k : l of the distance k of p to r and the distance l of q to r lies between e / R and R / e . Furthermore, let s and t be such that s -- (1 - s )p + s r and t -- (1 - t )q + tr . Then the Gregory-Qu conditions imply
s : (1 - s) > e . A /3 : (1 - e)A > e / 3 : 1 - e/3,
s : ( 1 - s ) < ( 1 - e ) ( 1 - 2e)A:eA < 1 - e : e
and analogous inequalitites for t. Thus s and t lie between e /3 and 1 - e. Now we can write the new contact point e on a t as
c = (1 - c)s .4. ct = lop -4- (1 - p)((1 - q ) r .4. qq),
where (1 - q) :q = (s - cs + c t ) : ( c - ct) =: x, which shows that q = 1 / ( I + x) lies in a closed interval of (0, 1) depending only on e. Further one has
sin act sincct sinfl ( l - q ) l k
sin-----~ - sin fl sin ct f l "
r
Fig. 13. Cutting a comer of T',~.
434 M. Paluszny et al. / Computer Aided Geometric Design 14 (1997) 421-447
~7
1'
G 0 1
Fig. 14. The region of Theorem 7.1.
cf. Fig. 13. Since comer cutting under the Gregory-Qu condition produces Cl-curves, we can, without loss of generality, assume 3' >/ 90°, see Corollary 5.3. Then we have f < k + ql <<. k(1 + qR/e) . Thus s i n a a / s i n a is bounded away from 0 and 1 for all possible q. Therefore, cr must lie in some compact subinterval of (0, 1) and the result of Gregory and Qu follows from Theorem 6.7.
The converse, however, is not true since the Gregory-Qu condition restricts the cut, i.e., the choice of the new contact line st, while there is no such restriction in Theorems 6.4 and 6.1. Furthermore, a cutting scheme where all comers are cut simultaneously and where all ( 's and all ~/'s are equal and greater than 1/3 produces a curve with comers, see (Gregory and Qu, 1996), but without line segments since Theorem 6.1 applies. Thus, the polar scheme satisfies the premises of Theorems 6.4 and 6.7 but not the Gregory-Qu condition in ]?*. Moreover one has:
Theorem 7,1. Any local comer cutting scheme in an affine plane where all comers are cut simultaneously by local cuts produces a limiting curve without line segments if the accumulation points of the (~, r/) lie in the closure of the triangle shaded in Fig. 14 without its three vertices.
Remark 7.2. In particular, if ~ and ~ > 0 and ( + ~ < 1, then the conditions of Theorem 7.1 are satisfied.
Proof. We introduce any scalar product in the affine plane, consider all pairs of two consecutive edges of Pn and denote the maximum length of these pairs by ran. Then proceeding similarly as in the proof of Theorem 6.1 one shows that there is some q < 1 such that ran+2 < q •mn for all n. Hence the theorem follows from Lemma 5.1. []
8. The interpolation schemes by Carnicer and Le M6haut6--Utreras
Very recently Le M6haut6 and Utreras (1994) presented an interpolatory refinement scheme which produces a sequence of polygons Qn in a Euclidean plane converging
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Fig. 15. Construction by Le M6haut6 and Utreras.
lmO3
sub, porting line
435
Fig. 16. New formulation of the construction by Le M6haut6 and Utreras.
to a Cl-interpolant of Q0. Le M6haut6 and Utreras assume that the Qn are graphs of piecewise linear functions. In order to construct Q,~+I from Qn they choose at each vertex of Qn a supporting line. Then they determine a new contact point ® between each pair of consecutive vertices • indirectly by the ratio ~v: 1 - w as illustrated in Fig. 15. For the entire scheme w is a fixed number in (1/2, 1), while the supporting lines at any fixed vertex can be different for all Qn with this vertex.
This scheme is polar to a Gregory-Qu scheme in 1t~*: Consider two consecutive edges .T" and R of Q,~ and two consecutive edges G and S of Qn+l with a common point q as illustrated in Fig. 16. 'Let ~ : 1 - ~ - ~7 : ~/denote the ratios in which the slopes of G and S divide the slopes of R and 3 r . Then, obviously ~ + r / = 1 - ~ E (0, 1/2) which means that (~, ~) lies in the region shaded in Fig. 17.
Figs. 15 and 16 show a special case of the situation shown in Fig. 9. Here m is the ideal point of the vertical axis. Thus if one views the ideal point of the vertical axis as the ideal line of 11 ~*, the projective scales ~ ---- ~(~) and ~ /= ~7(S) become affine scales on the line q of ~* with respect to the affine systems ~', T~ - ~" and ~ , ~" - R . Hence
436 M. Paluszny et al. / Computer Aided Geometric Design 14 (1997) 421-447
r \ , , \ ' \
1/2 \ "
< " . " ,,
[///x - - , O '~" '% 73
0 1/2
Fig. 17. The Le Mrhautr-Utreras region.
the envelopes Q~ determine a Gregory-Qu scheme and are covered by Theorem 7.1 (compare Figs. 14 and 17).
In particular, one may run the scheme by Le Mrhautd and Utreras with w = 1/2 and = ~/ = 1/4 fixed for the entire algorithm. Then one still obtains a cl- interpolant .
This particular scheme which is excluded by Le Mdhautd and Utreras and was proposed in (Camicer, 1992). The polar scheme is Chaikin's construction of piecewise quadratic splines (Chaikin, 1974). Hence it follows from Remarks 10.2 and 13.2 that Camicer ' s interpolant consists of segments of hyperbolas and parabolas.
9. Choosing a good contact line
Using the terminology of Theorem 6.7 a fill-in scheme in a Euclidean plane is com- pletely defined by a sequence of numbers a and T and the choices of the tangents. I f the a ' s and 7-'s satisfy the conditions of Theorem 6.7, the limiting curve P ~ has no comers but it can have line segments. We show that for any sequence of a ' s and T'S satisfying the conditions in Theorem 6.7 one can find tangents such that the cuts also satisfy the conditions of Theorem 6.1.
Fig. 18 shows the triangle of Figs. 9 and 11 again with further notation. It is the comer of some Pn. We will see that there is some 5 > 0 such that for every P,~ and any of its comers 5 <<. g/h ~< 1 - ~ where g and h are given by Fig. 18. Thus if the tangent at the new contact point ® is chosen to be parallel to the base of the triangle, the condition of Theorem 6.1 is satisfied. Other, although less simple choices, are also possible.
The ratio g/h depends continuously on a , /3, a and T. Since comer cutting does not create comers which are sharper than the ones being cut, we have 180 - a - / 3 / > qa > 0 where ~p is the interior angle defined by the two edges of 790 shown in Fig. 10. Hence we consider g/h only over the compact domain e ~< a , T ~< 1 - e, 0 ~< a, /3 ~< 180 - qo. For a , /3 > 0 the ratio g/h is by construction in (0, 1). Thus we need to show that g/h >1 constant > 0 for (a,/3) --+ (0, 0). With the law of sine it follows
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r
P Q
P q
Fig. 18. Choosing a tangent.
437
g sin a a . sin r/3 sin(1 80 - a - / 3 )
h sin(180 - a a - ~-/3) sinc~ • sin/3
sin crc~- sin Tfl sin(c~ +/3)
sin ~ - sin/3 sin(ac~ + ~-/3)"
Hence if (c~,/3) --+ (0, 0) the inferior and superior limits of the ratio 9 / h are (positive) values between cr and ~-. This concludes the proof.
I0. A convexity preserving interpolation scheme
As an application of Theorem 6.4 we describe a convexity preserving interpolation scheme where the shape of the curve can be modified by a very direct shape handle. The input for the scheme is a sequence of contact elements in the affine plane, i.e., line segments with one distinguished point a~ on each segment, i = 0 , . . . , m, as illustrated in Fig. 19. Let ci be the point where the lines through a~ and a i+l intersect. Then a0cocl " " C m - l a m and a o a l . " a m are two polygons 79o and Q0 as needed for a fill-in scheme where 79o circumscribes Q0.
Note that the tangent at a~ has to be chosen such that it intersects the tangents at a~_ 1 and ai+j on opposite sides of a~. Thus if the tangents are not given properly, one must explicitly introduce inflection points with tangents - - ® - - as further contact elements, see Fig. 19.
The fill-in scheme is specified by a standard cut in a standard comer a c b as illustrated in Fig. 20. A user can choose the cutting line d e and the contact point g arbitrarily inside the standard triangle. The standard cut is then mapped by projectivities onto each comer of 79o and further iterates 79n. The images of the standard cut determine all cuts on every 79n. The underlying projective maps are determined by the four points m , the comer vertex c, and the two contact points a , b of the standard comer and their images. Hence for each comer of 79n formed by two successive contact points and the vertex between one needs to specify the image of m . We choose for all comers of 79n between a~ and a~+l the same image for m and denote it by m~. Since cross ratios are invariant under projective maps, the fill-in scheme above satisfies the conditions of Theorem 6.4 provided that m does not meet a supporting line of the polygon a b with end tangents
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f" m--l
Fig. 19. Polygon to be interpolated.
C . • C i
O m
Fig. 20. Applying a standard cut.
a c and be and similarly m i does not meet a supporting line aiaiq_ l with end tangents a~c~ and a i + l c i . Hence the limiting curve is a Cl-interpolant.
If one applies the fill-in scheme above to the standard comer itself, one obtains the s t a n d a r d curve. Mapping the standard curve to each comer of 79o gives the same result as if 79o is cut iteratively by the standard cut since every comer a i c i a i + l m i is a projective image of the standard comer. Another option is to map the standard curve onto each comer of 790 by an affine map which would yield a different curve but eliminate the need for the extra points m i . This is discussed in Section 13.
In the following we describe a very simple algorithm to produce the standard curve: Applying the standard cut to the standard comer itself generates a left and a right comer. Further applications of the standard cut produce a left left, a left right, etc., comer. We will label these comers by sequences X l ' ' ' X n of binary digits xi E {0, 1} where 0 stands for "left" and 1 for "right".
Let A0 and A1 be the pro jec t i ve m a p s which map the standard comer onto the left and right comer, respectively, i.e., A0 and A1 map a, c, b, m onto a, d, g, m and g, e, b, m respectively. Since the cutting scheme under consideration is projectively invariant, the standard curve restricted to the left or right comer is a projective image of the entire curve under A0 and AI, respectively. Hence Ax, • • • Axn maps the standard comer onto
M. Paluszny et al. / Computer Aided Geometric Design 14 (1997) 421--447
w v
Fig. 21. Interpolant produced by the algorithm.
439
the xl .. • x,~-comer. Note, e.g., that AoA1 maps the standard comer via the right comer onto the left right comer.
l e t a(x) be the straightforward 1-1 correspondence between the dyadics x = ~-'~i~=1 xi2 - i in [0, 1] and the contact points on the standard curve, i.e., let a(1) = b and a(y'~i~ l xi2 - i ) be the left end point of the xl . . .xn-comer. Then, from what is said above, we can derive the following algorithm to compute all comer end points after some n iterations.
Algorithm. Given: Standard comer as in Fig. 20, number of iterations n (1) a(0) := a, a(1) := b. (2) For l -- 1 , 2 , . . . , n .
For i = 1 , 3 , 5 , . . . , 2 t - 1
Aoa 2i i ( ~ ) ' i f2 i ~< 2t'
a ( ~ ) := A ' a ( ~ t - 1) , i f 2 i > 2 z.
Figs. 8 and 21 were obtained by this algorithm and subsequent projective maps. The figures show the starting sequence of contact elements and the standard cut which defines the fill-in scheme used to generate the interpolating curve which is also shown. Further we used rn~ = ai - - ai+l + ai+2, for i = 0, 2, 4 , . . . , m -- 2. These points are suppressed in the figures. In Section 12 we present an explicit matrix representation for the maps A0 and A1.
Remark 10.1. The algorithm above is formally identically to the matrix subdivision algorithm introduced and studied in (Micchelli and Prautzsch, 1987, Prantzsch, 1991). However, here A0, Al are not affine, but projective maps. Hence, the analysis there does not apply to our scheme here.
Remark 10.2. If the standard cut with its contact point is a contact element of the conic through the points a, b, rn with the tangents a e and be, then the projective maps .A0 and .,41 map this conic onto itself since any conic is determined by three points and two
440 M. Paluszny et al. / Computer Aided Geometric Design 14 (1997) 421-447
tangents. Thus the cutting scheme above generalizes Haase's alias de Casteljau's rational algorithm for conics, see, e.g., (Boehm and Prautzsch, 1994, p. 243).
11. Standard curves without line segments
Theorem 6.4 guarantees that the standard curve is C 1 but not necessarily the absence of line segments. In the following, we will investigate when a standard cut leads to a curve with or without line segments. A first result is the following theorem.
Theorem 11.1. The standard curve S has no line segments if and only if the projective maps Ao and Al as given by the standard cut, see Section 10, have no fixed points in the standard triangle besides a and b, respectively.
Proof. Assume that the map A0 has a fixed point f -¢ a in the standard triangle. Then each 0 . . . 0-corner contains a and f . As already mentioned in Section 10 the standard curve has no comers because of Theorem 6.4. Hence it follows from Corollary 5.3 that the 0 . . . 0-corners become arbitrarily fiat after sufficiently many iterations. Thus S contains the line segment a f . Similarly the standard curve has a line segment ending at b if Al has a second fixed point besides b in the standard triangle.
In order to prove the converse we observe first that the construction of the stan- dard curve implies, since the cuts are local, that no three contact points---denoted by a(~-~ xi2 - i ) in Section 10--are collinear. Therefore if we can show that the set of all contact points is dense in S, then S has no line segment.
Now assume that A0 and A~ possess no fixed points besides a or b, in the standard triangle, respectively, and let q be an arbitrary point on S. We need to show that q is a limit point of the contact points on the left and also of the contact points on the right side of q:
(1) Let q = a, then the 0 . . . 0-corners form a strictly monotone sequence of triangles a, A'~c, A'~b. Hence the sequence A~b cannot have two accumulation points and con- verges to a point f in the standard triangle. Since f must be a fixed point of A0, it follows
m that a = f . (2) One can proceed similarly i f q = b. (3) Let q = a(y'~i= j xi2 - i ) ¢ a be a contact point of S where xm = 1. Then the fight endpoints of the xl • .. xm0- • - 0-comers and the left endpoints of the xl - . . xm_101 . . . 1-corners converge to q:
A '~ lim A x , . . . xmAo b = A x , . . . A x m a = q , n - - + ( x ~
lim A~:, . . . A ,m_ , A o A ~ a = A , , . . . Axm_, Aob = A~, . . . Axm_ , AI a = q. n --a, o 0
(4) In all other cases there is a sequence xn containing infinitely many pairs xnxn+~ = 01 such that the xj • • • xn-comers contain q.
Fig. 22 shows on the right side in bold the left right comer of some xl .. • Xn_l-comer a(x) , c', a ( x + 2 l -n ) where x = }"~in-i I xi2 - i . It is the projective image of the 01-corner also shown in bold on the left side.
Consider the central projection with r a as center onto the line ab. Let x, y, z be the affine scales of the projections of a (1 / 4 ) , c and a ( l / 2 ) with respect to a and b and let
M. Paluszny et al. / Computer Aided Geometric Design 14 (1997) 421-447 441
c I -( (~ + 2-") Z: ~NN~a(1/2 ) ~ ~ ~ x + 21_n)
m u 1~ Z: Fig. 22. The standard left right comer and some projective image of it.
w ' C
Fig. 23. Fixed points in the standard triangle.
x ' , y ' , z ' be the affine scales of the projections of a(x + 2 - l - n ) , c ' and a(z + 2 - n ) with respect to the projections of a(z) and a(z + 21-'~). Since the zl -. - x n - l - c o r n e r is a projective image of the 01-corner under the map Ax, . . . Axn_,, one has
~ : = c r [ x , y I 1,0] = c r [x ' , y ' [ 1 0] = x ' - 1 y ' , x I yt _ 1 '
¢ : : cr[z, V ] 1,0] = cr[z', y' [ l, 0],
where cr[a, b [ c, d] denotes the cross ratio of a, b with respect to c, d. These two identities imply for y' >>. y (or y' <~ y)
yl
z ' - /> ( < ) y ( 1 y u'(1 - ~ ) + ~ _~) + ~ - x
and analogously z ' / > (or ~< )z. Hence z ' - x ' ~< max{ 1 - x, z - 0}.
442 M. Paluszny et al. / Computer Aided Geometric Design 14 (1997) 421-447
Therefore the central projection of any x I " ' " x,~Ol-comer onto the line ab is shorter than the projection of the xl . . .Xn-corner by some uniform factor q < 1. This means that the end points of the comer sequence above converge to the same point. []
Further assume that A0 has a fixed point f besides a in the standard triangle illustrated in Fig. 23. Then the line m f is fixed under A0 but not pointwise since the intersection d with the line ab is mapped onto a different point e. Hence A0 restricted onto m f can have at most two fixed points, namely m and the intersection with the fixed line ac. Thus if Ao has a fixed point besides a in the standard triangle, it lies between on the edge ac . The analogous result holds for A1.
12. Computing the fixed points
In order to compute the fixed points of A0 and A1 as given by the standard triangle shown in Fig. 20 one may choose any suitable coordinate system: Let c, a , b and m be represented by the homogeneous coordinates
i] !1 and
1
1 ,
1
then the endpoints d and e of the cutting edge and the contact point g are represented by
and
where a, 3, x, y E (0, 1) and x / a + y//3 -- 1, and the projective maps A0 and Al have the matrix representations
Ao [i 1 1
0
lJ 1 - x
x
Y
][: :]_1 1 1 1
y - x - a y + a 1 ,
1 - a 0
A~ [i 1 0
1
11 1 -y x
Y
x - y - / 3 x + ~ : ]L 1 1 1
0 1
1 0
M. Paluszny et al. / Computer Aided Geometric Design 14 (1997) 421-447 443
b
oe ",4 I \ :
i ", , ,d 0 x a 1
i14
0 0
b
/ I I
c I
I i a
w 1/4 1 x
Fig. 24. Each point in the shaded area carries a continuum of lines such that the associated standard cut produces C I-curves without line segments.
With the aid of a computer algebra system one can now easily show that A0 and A~ have no fixed points in the standard triangle except for a and b if and only if
1 - y z ...</3...< y l + z . . . . , w h e r e z = l - x - y .
x l + z a 1 - x z
Thus for any given contact point (x, y) there exists a continuum of slopes - / 3 / a such that the standard cut produces curves without line segments if and only if
x 3 -}- x 2 y -k- x y 2 -k- y3 _ 3x 2 _ 3 x y - 3y 2 + 3x + 3y - 1 ~> O.
The region of all contact points satisfying this cubic inequality is shaded in Fig. 24 (right).
R e m a r k 12.1. For later use we observe that the cubic boundary has the parametrization
[:] ] 2a 2 -- 2 a + 1 (1 -- C~) 3 "
In order to get the region of all contact lines having a contact point such that the standard cut does not produce line segments, we transform the first two inequalities above into
( 1 - / 3 ) 2 <~ x ~< 1 ( l - a ) 2
a + / 3 2 _ a /3 ~ a2 + / 3 _ a /3"
Note that these inequalities imply 0 ~< x / a ~< 1. Hence we drop the term x / a and transform the remaining inequality into a + / 3 /> 1. Thus whenever a + / 3 >/ 1, one can find a continuum of contact points on the corresponding contact line such that the standard cut produces no line segments. The lines whose intercepts a and/3 sum to 1 envelope the parabola
Z2 @ y2 _ 2xg - 2x - 2y + 1 = 0.
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Y
1 i
1/2
b O
0 1/2 T x
m
Fig. 25. Each line meeting the shaded area but not the line segment ab carries a continuum of contact points such that the associated standard cut produces C J-curves without line segments.
This parabola forms the boundary of the shaded region shown in Fig. 25. It means that exactly all contact lines which touch or go through the shaded area of Fig. 25 but not through the line segment ab have a contact point c leading to a standard curve without line segments.
Remark 12.2. The parabola above lies in the shaded area of Fig. 24.
13. Affine standard cuts
The interpolation scheme in Section 10 could also be based on a standard cut which is mapped affinely onto the comers of 790. Then the extra points rni and m are not needed and one can apply the Gregory-Qu result. However, such a scheme applied to the standard triangle falls under the Gregory-Qu condition, see Fig. 3, only if the standard cut with the notation defined by Fig. 26 is such that 0 < c~ = fl and
/3a c~b +277< 1 and 2 ~ + ~ < l, w h e r e ~ - a + b ' ~1- a + b '
i.e., the standard cut must be parallel to the line ab while the contact point can only lie in the interior of the dotted area shown in Fig. 26, right.
However, as we will show in the sequel there are many more affine standard cuts producing Cl-curves. Hence the Gregory-Qu condition is too restrictive.
Consider the scheme given by the algorithm in Section 10, where A0 and Ai are affine maps mapping a , c, b onto a, d, g and g, e, b respectively. Since Ao and Ai map lines onto lines, they also describe maps on ~*. These are the inverse dual maps/30 = (A~) - l and B1 = (A~') -1 which map the ideal line onto itself and the lines (~ points of ]P*)ca, ab, bc onto da, ag, de and de, gb, be, respectively. Thus the envelope of the standard curve is obtained by successive applications of B0 and Bl to ca and bc. In other words,
M. Paluszny et al. / Computer Aided Geometric Design 14 (1997) 421-447 445
C
a ~ a b
Fig. 26. Standard cut with notation (left) and the set of all feasible contact elements based on the Gregory-Qu condition (right).
Y
b
0 1 X
Fig. 27. Each line intersecting the standard triangle only in the shaded area carries a continuum of contact points such that the associated affine standard cut produces CJ-curves without line segments.
the polar scheme is given by the standard triangle c a , ab, be in l?*, the contact line g and the contact point de while the ideal line of ~ assumes the role of m .
Next we use the coordinates of Section 12 and the polarity with respect to the circle around rn through a and b. Then the ideal line and the lines ca, ab, be are mapped onto the points m , a , c, b. Further, the points on the cubic given in Remark 12.1 correspond to the tangents of the curve
y(t) J (t 2 - t + 1) 2 t3(2 - t) '
while the points in the interior of the shaded area shown in Fig. 24 correspond to cuts which do not intersect this curve. This is illustrated in Fig. 27: If the affine standard cut intersects the standard triangle only in the shaded area, then there is a continuum of
446 M. Paluszny et al. / Computer Aided Geometric Design 14 (1997) 421--447
b I n
v o 1 x
Fig. 28. Each point in the shaded area carries a continuum of lines such that the associated affine standard cut produces C l-curves without line segments.
contact points on the cut such that the associated standard curve has no corners and line segments.
Next consider the parabola of Fig. 25. Its envelope corresponds to the ellipse through m, a, b with tangents a c and cb. It is shown in Fig. 28 and in analogy to above we obtain the following: If the contact point lies in the shaded area of Fig. 28 there is a continuum of contact lines through it such that affine applications of this standard cut produces no corners and line segments. Fig. 28 also shows the cubic of Fig. 25.
Remark 13.1. The union of the shaded areas in Figs. 24 and 27 as well as the union of the shaded areas in Figs. 25 and 28 fills the standard triangle. Hence for each contact point (line) one can find a continuum of contact lines (points) such that either the asso- ciated projective or affine standard cutting procedure produces a Cl-curve without line segments.
Remark 13.2. Dual to Remark 10.2 one has that the affine standard cut produces parabo- las if it lies on the parabola through a and b with tangents a c and be.
Addendum
After finishing this paper, Bernd Mulansky showed us an unpublished manuscript with a result and proof by himself similar to Theorem 6.1.
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