curvatures of conflict surfaces in euclidean 3-space

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CURVATURES OF CONFLICT SURFACES IN EUCLIDEAN 3-SPACE JORGE SOTOMAYOR Instituto de Matematica e Estatstica Universidade de S~ao Paulo Caixa Postal 66281 S~ao Paulo, SP, Brazil, Cep 05315-970 E-mail: [email protected] DIRK SIERSMA Mathematisch Instituut Universiteit Utrecht Postbus 80.010, 3508 TA Utrecht The Netherlands E-mail: [email protected] RONALDO GARCIA Instituto de Matematica e Estatstica Universidade Federal de Goias Caixa Postal 131 Goi^ania, GO, Brazil, Cep 74001-970 E -mail: [email protected]

Abstract. This article extends to three dimensions results on the curvature of the conflict curve for pairs of convex sets of the plane, established by Siersma 3]. In the present case a conflict surface arises, equidistant from the given convex sets. The Gaussian, Mean Curvatures and the location of Umbilic Points on the conflict surface are determined here. Initial results

on the Darbouxian type of Umbilic Points on conict surfaces are also established. The results are expressed in terms of the principal directions and on the curvatures of the borders of the given convex sets. 1991 Mathematics Subject Classication : Primary 53A04 Secondary 53C03 The paper is in nal form and no version of it will be published elsewhere. manuscript, December 01, 1998 This work was done under the project PRONEX/FINEP/MCT - Conv. 76.97.1080.00 Teoria Qualitativa das Eq. Dif. Ordinarias. 1]

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J. SOTOMAYOR, D. SIERSMA AND R. GARCIA

1. Introduction Let A1 and A2 be closed non empty sets in Euclidean space R3,

which is endowed with the standard orientation and the distance: d(p q) = jp ; qj = hp ; q p ; qi1=2 where h i is the standard Euclidean inner product in R3 . The conflict set C(A1 A2) between A1 and A2 is dened by C(A1 A2) = fp d(p A1) = d(p A2)g where, d(p A) = inf fd(p q) q 2 Ag. The set C(A1 A2) is also viewed as the common boundary between the Territory of A1 relative to A2 , dened by T err(A1 A2 ) = fp d(p A1) < d(p A2)g and Terr(A2 A1) which is the Territory of A2 relative to A1 . The set C(A1 A2) is also called the Bisector or the Equidistant Set between A1 and A2 7]: In this paper we consider only the case where Ai are convex closed sets, with disjoint interiors and smooth regular boundaries Bi = @Ai of class C k , k 2. Assume that the surfaces Bi are oriented by their unit normal vector elds Ni , pointing towards the Interior of Ai . To each point p on the closed set Ei = R3 nInt(Ai ), is associated its projection i (p) = pi onto Bi , characterized by d(p Ai) = h i (p) ; p Ni ( i (p))i: >From the tubular neighborhood properties and the convexity hypothesis, it follows that i is a retraction of class C r;1 of a neighborhood of Ei onto Bi see 4], Vol. 1. Therefore C = C(A1 A2) is dened implicitly by the zero level set of the function c(p) = d(p A2) ; d(p A1) which is regular of class C r;1 and satises: rc(p) = N1 ( 1 (p)) ; N2 ( 2 (p)): It is clear that this regularity property fails if the convexity and disjointness hypotheses are not impossed. In this work the conict surface C will be oriented by the unit normal N along rc i.e. pointing from A2 towards A1 : N = jN1( 1 (p)) ; N2 ( 2 (p))j;1 N1( 1 (p)) ; N2 ( 2 (p))]: To simplify the notation write (V ) = jV j;1V for the nomalization of a nonvanishing vector V . Therefore, N = (N1( 1 (p)) ; N2 ( 2 (p))): A positive moving frame attached to C is therefore given by fT1 T2 N g, where: T1 = (N1( 1 (p)) + N2 ( 2 (p)) T2 = (N1( 1 (p)) ^ N2 ( 2(p)):

CURVATURES OF CONFLICT SURFACES

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Notice that the vector elds Ti are singular at points p of the closed set M = M(A1 A2), where N1 ( 1 (p)) + N2 ( 2 (p)) = 0, which occurs when the distance from C to Ai : r(p) = d(p A1) = d(p A2) is minimal, assuming the value dm = 21 d(A1 A2). Under the condition of strict convexity of Bi at points of the sets i (M), M reduces to a unique point pm . Recall that strict convexity of Bi at pi means that DNi is an automorphism of the tangent space TBi , with the usual identication of the tangent space T Bi at pi with that of the unit sphere TS 2 , at Ni (pi ). In terms of the principal curvatures k1i k2i which are the eigenvalues of ;DNi , the condition of strict convexity (resp. convexity) means that both are positive (resp. non-negative). In terms of the Gaussian Curvature Ki = k1i :k2i = det(;DNi ) this means that Ki > 0(resp. Ki 0). Recall also that the Mean Curvature of Bi is given by Hi = 12 trace(;DNi ) = 12 (k1i + k2i ): The expression, p U i = 12 (k2i ; k1i ) = (Hi )2 ; Ki also called Skew Curvature, whose zeros locate the umbilic points of the surface, will be also of later of use here. Theorem 1 of his work establishes expressions for the Gaussian and Mean Curvatures of the surface C = C(A1 A2). These results can be regarded as the starting step for the investigation of the principal configuration of C 5]. Recall that this conguration is dened by the principal curvatures, the umbilic points, the elds of principal directions (which are the eigenspaces of ;DNi ), and their integral curves (including the periodic principal lines). The location of umbilic points and the slope of principal directions on C nM is established in Corollary 3. Initial results on the Darbouxian type of the umbilic points on M = M(A1 A2) are established in section 3. A complete study of the dependence on Bi of the principal conguration on conict surfaces will not be carried out in this work. However, the curvature formulas of this paper are expressed in terms of some elements of the principal configurations of the surfaces Bi . Global topological properties of conict sets, such as their connectivity, have been studied in 7]. The basic smoothness and curvature expressions of the conict curve for convex sets in the plane have been established in 3], where interesting connections with Euclidean Geometry of conics can also be found. Acknowledgement This work was prepared with the partial support from CNPq, FINEP and FUNAPE.

2. Curvatures of conict surfaces Let p 2 C nM be such that pi = i(p) are non-umbilic points in Bi . Let fE1i E2i Ni g be positive principal frames on Bi around pi.

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This means that

DNi :Eji = ;kji Eji where k1i < k2i i = 1 2 are the principal curvatures of Bi . Denote by i the angle between the vectors E1i and F1i = (D i (p)T1 ). Write gi = gi (i ) = (k2i ; k1i ) sin i cos i for the geodesic torsion along the unit vector F1i on Bi . Write kni = kni (i ) = k1i cos2 (i) + k2i sin2 (i) which, by Euler's formula, is the normal curvature of Bi in the direction of F1i = (D i (p)T1 ). Similarly, the normal curvature of Bi in the direction of F2i = (D i (p)T2 ) is given by kni ? = kni ? (i) = kni (i + 2 ) = k2i sin2 (i ) + k1i cos2 (i) : Denote by Ari the convex set of points at distance r 0 from Ai its border is the surface Bir obtained by moving each point pi on Bi to r (pi ) = pi ; rNi (pi ). By restriction , i denes the dieomorphism (r );1 of Bir onto Bi and, by composition, it denes the retraction ri = i (r );1 onto Bir . The principal frames fE1i E2i Ni g on Bi are parallel translated along the normals to principal frames on Bir . this follows from the fact that r preserves the principal direction elds as well as the umbilic points. The principal curvatures however change into kji (r) = 1+krk , 4], Vol. 3. The above expressions for gi kni and kni ? on Bi , being dened in terms of principal curvatures, can be obviously modied to be valid on Bir and denoted respectively by gi (r) kni (r) and kni ?(r). For instance: i j

i j

gi (r) = gi (r i) = (k2i (r) ; k1i (r)) sin i cos i kni (r) = k1i (r) cos2 (i) + k2i (r) sin2 (i) kni ?(r) = k2i (r) sin2 (i ) + k1i (r) cos2 (i) : Denote by the angle between Ni and T1 . For future reference, notice that sin = 12 jN1( 1 (p)) ; N2 ( 2 (p))j = hF11 T1 i = hF12 T1i Theorem 1. Let Ki = Ki(r) = k1i (r)k2i (r), Hi = Hi (r) = 21 (k1i (r) + k2i (r)) and U i = U i (r) = 12 (k2i (r) ; k1i (r)) where kji (r) = 1+krk . a) The Gaussian Curvature of the conict surface C is given by: K = 12 21 (K1 + K2) ; H1 H2 + cos(2(2 ; 1))U 1 U 2 ] b) The Mean Curvature of the conict surface C is given by: 2 2 H = ( 1 + sin2 ) (H1 ; H2 ) + ( cos 2 )(cos(21)U 1 ; cos(22)U 2 )] 4 sin

1 + sin

i j

i j

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The proof of this theorem will follow from the next proposition. Proposition 2. With the notation above at points of C nM , it holds that DN:T1 = sin2 kn2 (r) ; kn1 (r)]T1 + 21 g2(r) ; g1(r)]T2 1 k2 (r) ; k1 (r)]T DN:T2 = 21 g2(r) ; g1 (r)]T1 + 2 sin

2 n? n? Proof. The conclusion will follow from the calculation of the inner products in DN (T1 ) = hDN (T1 ) T1 iT1 + hDN (T1 ) T2iT2 DN (T2 ) = hDN (T2 ) T1 iT1 + hDN (T2 ) T2iT2 Dierentiation of N gives: r ; DN2 :D r 2 + 2D (sin );1 ]: N1( r (p)) ; N2 ( r (p))] DN = DN1 :D 21sin 1 2

This shows that for the present analysis the contribution of the second term is null. The basis fT1 T2g on C projects along ri onto the othonormal bases fF1i F2ig on Bir . Calculation shows that: fF11 = v(N2 ; hN1 N2 iN1 ) F21 = T2g fF12 = v(;N1 + hN1 N2iN2 ) F22 = T2 g Therefore, D ri (T1 ) = sin F1i and D ri (T2 ) = F2i i i The bases E and F are related by: F1i = cos(i )E1i + sin(i)E2i F2i = ; sin(i)E1i + cos(i )E2i DNi :D ri (T1 ) = = = =

Analogously,

;

sin DNi cos(i )E1i + sin(i)E2i = sin cos(i )DNi (E1i ) + sin(i)DNi (E2i )] = sin ;k1i (r) cos(i)E1i ; k2i (r) sin(i)E2i ] = sin f;k1i (r) cos(i) cos(i)F1i ; sin(i )F2i] ;k2i (r) sin(i) cos(i)F1i + sin(i )F2i]g = = sin f ;k1i (r) cos2 (i ) ; k2i (r) sin2(i )]F1i + (k1i (r) ; k2i (r)) sin(i) cos(i)]F2ig = = sin f ;kni (r)]F1i + ;gi (r)]F2ig:

DNi :D ri (T2) = DNi cos(i + 2 )E1i + sin(i + 2 )E2i = = k1i (r) sin(i)E1i ; k2i (r) cos(i )E2i ] = = fk1i (r) sin(i) cos(i)F1i ; sin(i )F2i] ;k2i (r) cos(i ) sin(i )F1i + cos(i )F2i]g =

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Therefore,

= fk1i (r) sin(i) cos(i )F1i ; k1i (r) sin2 (i)F2i ;k2i (r) cos(i ) sin(i)F1i ; k2i (r) cos2 (i)F2i g = = f (k1i (r) ; k2i (r)) sin(i) cos(i )]F1i ; k1i (r) sin2 (i) + k2i (r) cos2(i )]F2ig = = f ;gi (r)]F1i ; kni ?(r)]F2ig:

1 1 1 1 2 2 2 2 DN (T1 ) = 2sin

sin f ;kn(r)F1 ; g (r)F2 ] ; ;kn(r)F1 ; g (r)F2 ]g = = 21 f kn2 (r)F12 ; kn1 (r)F11] + g2(r)]F22 ; g1(r)F21]g

1 f ; 1(r)]F 1 ; k1 (r)]F 1 + 2(r)]F 2 + k2 (r)]F 2g = DN (T2 ) = 2 sin

g 1 n? 2 g 1 n? 2 1 f k2 (r)F 2 ; k1 (r)F 1] + 2(r)]F 2 ; 1(r)]F 1g: = 2 sin

n? 2 n? 2 g 1 g 1

Performing the inner products, taking into account that sin = cos( 2 ; ) = hF11 T1 i = hF12 T1i nishes the proof.

Remark 1. Notice that the proof of the last proposition uses only the principal frames at points (and not on open sets) of B i . Therefore the calculations for DN hold also at points of C nM whose projections on either one (or both) of the surfaces B i are umbilic points, in that case the corresponding expressions gi (r) vanish. Proposition 4 will deal with a preliminary analysis of the case of points on M , where T1 and the angles i are not dened.

Proof of Theorem 1.

a) The Gaussian Curvature of C is given by det(;DN): Therefore, taking into account that : kn2 (r) = H2 ; U 2 cos 22 kn1 (r) = H1 ; U 1 cos 21 kn2 ?(r) = H2 + U 2 cos 22 1 kn?(r) = H1 + U 1 cos 21 g2 (r) ; g1 (r) = U 2 sin 22 ; U 1 sin 21 and (U i )2 = (Hi )2 ; Ki, obtain that

K = 41 f kn2 (r) ; kn1 (r)] kn2 ? (r) ; kn1 ? (r)] ; g2(r) ; g1(r)]2g = 1 f(H2 ; U 2 cos 2 ) ; (H1 ; U 1 cos 2 )] (H2 + U 2 cos 2 ) 2 1 2 4 ;(H1 + U 1 cos 21)] ; U 2 sin 22 ; U 1 sin 21]2g = = 21 12 (K1 + K2) ; H1 H2 + cos(2(2 ; 1))U 1 U 2 ] b) The Mean Curvature is given by trace(; 12 DN). From the relations kn2 (r) ; kn1 (r) = H2 ; H1 + U 1 cos 21 ; U 2 cos 22 and

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kn2 ?(r); kn1 ?(r) = H2 ; H1 + U 2 cos 22 ; U 1 cos 21, get using also the expressions in part a) that:

H = ; 14 fsin kn2 (r) ; kn1 (r)] + sin1 r kn2 ? ; r kn1 ? ]g = 2 2 = ( 1 + sin2 ) (H1 ; H2 ) + ( cos 2 )(cos(21)U 1 ; cos(22)U 2 )] 4 sin

1 + sin

;

;

Corollary 3. 1) The point p 2 C nM is umbilic if and only if = = 0, where: = 2(r) ; 1 (r)] = U 2 sin 22 ; U 1 sin21 g g 2 2 1 = sin kn (r) ; kn (r)] ; kn2 ? (r) ; kn1 ? (r)] = (H2 ; H1 ) cos2 + (1 + sin2 )(U 1 cos 21 ; U 2 cos 22)

2 ) (H1 ; H2 ) (1 + cos 2 ) + (U 1 cos 2 ; U 2 cos 2 )] = (3 ; cos 1 2 2 (3 ; cos 2 ) 2) The principal directions at a non-umbilic point p in C nM are characterized by the condition of making an angle with T1 , given by: 2 g2(r) ; g1 (r)] sin

tan 2 = 2 2 = 2 sin

sin kn (r) ; kn1 (r)] ; kn2 ? (r) ; kn1 ? (r)] Proof. Imediate from Proposition 1. Proposition 4. Assume that p = pm 2 M and that pi = i (p) are non-umbilic points in Bi . The normal curvature kn = kn( ) of C in the direction of an angle i with the rst vector of the parallel basis fE1i E2i Ni gis given by

2 1 kn( ) = ; 21 1 +knrk( 22( ) ) ; 1 +knrk( 11( ) ) ] n 2 n 1 Proof. Similar to that of Proposition 1, replacing the frame fT1 T2 N g by any tangent frame to C at p: Remark 2. Taking into account that the two tangent frames di er by an angle and therefore 2 = 1 + the above equation can be simplied to: 2 kn1 ( 1 ) ] ; kn( ) = ; 21 1 +knrk( 21( + ) n 1 + ) 1 + rkn1 ( 1 ) Remark 3. If both pi are umbilic then the point p = pm is umbilic other case leading to umbilic points on M is when the principal frames at pi are paralell and the principal curvatures verify k22 ; k21 = k12 ; k11 = ;k: The case k = 0 is studied in partial detail in next section. Pertinent changes for k 6= 0 are made in Remark 4.

3. An Introduction to Umbilics on Conict Surfaces In this section it will be shown how to determine the Darbouxian type of an umbilic point pm 2 M . Recall from 1], 5], that this type, (D1 , D2 , D3 ) depends on the 3-jet of the surface at the point

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and it determines the behavior of lines of curvature near the umbilic point. To this end in what follows, the 3-jet of the conict surface at p, in Monge form will be calculated in terms of the corresponding 3-jets of the surfaces B1 and B2 at p1 and p2: Let the convex surfaces B1 and B2 be locally given in Monge charts (x y) and (u v) by the graphs of: f1 (x y) = r + a2 x2 + 2b y2 + 61 (a30x3 + 3a21x2y + 3a12xy2 + a30y3 + :::) f2 (u v) = ;r ; a2 u2 ; 2b v2 ; 16 (b30u3 + 3b21u2v + 3b12uv2 + b30v3 + :::) Let P = (X Y Z) be a point of R3 . By the considerations above it follows that p = (0 0 0) 2 C. The coordinate expressions of the second order jets at p of the projections 1(P ) and 2 (P ), with targets expressed in the Monge coordinates (x y) and (u v) will be computed now. The projection 1 (X Y Z) = (x y), in coordinate expression, is dened implicitly by the following equations: @F = @F = 0 @x @y 2 where F (x y X Y Z) = d(P B1) = (X ; x)2 + (Y ; y)2 + (Z ; f1 (x y))2 : Using the Implicit Funcion Theorem, after extensive calculation, it is obtained that the solution of the system of equations above is given ra by: 1 ra 30 2 x = 1 + ra X ; 2(1 + ra)3 X ; (1 + rb)(121+ ra)2 XY ra12 a 2 ; 2(1 + ra)(1 + rb)2 Y + (1 + ra)2 XZ + ::: ra21 ra12 2 y = 1 +1 rb Y ; 2(1 + rb)(1 + ra)2 X ; (1 + ra)(1 + rb)2 XY ; 2(1ra+03rb)3 Y 2 + (1 +brb)2 Y Z + ::: Similarly, considering the function G(x y X Y Z) = d(p B2)2 = (X ; u)2 + (Y ; v)2 + (Z ; f2)2 it is obtained that the projection 2 (X Y Z) = (u v) in coordinate expression, is given by rb21 u = 1 +1 ra X + 2(1rb+30ra)3 X 2 + (1 + rb)(1 + ra)2 XY rb12 a 2 + 2(1 + ra)(1 + rb)2 Y ; (1 + ra)2 XZ + ::: rb21 rb12 2+ v = 1 +1 rb Y + 2(1 + rb)(1 X 2 + ra) (1 + ra)(1 + rb)2 XY + 2(1rb+03rb)3 Y 2 ; (1 +brb)2 Y Z + ::: Recall from the Introduction that the Conict Surface C is dened by the equation c(X Y Z) = d((X Y Z) (u v f2(u v)) ; d((X Y Z) (x y) f1(x y)) = 0


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with (x y) and (u v) representing the projections 1 (X Y Z) and 2(X Y Z), given by the expressions above. Extensive calculation leads the following expression for the 3-jet of the Monge representation of C in a neighborhood of p = (0 0 0) : (a21 ; b21) X 2 Y 30 ; b30) X 3 + Z = (a 3 2(1 + ar) 6 2(1 + br)(1 + ar)2 2 (a12 ; b12) XY 2 + (a03 ; b03) Y 3 + ::: + 2(1 + ar)(1 + br)2 2 2(1 + br)3 6

This follows by using the Implicit Function Theorem applied to the function c(X Y Z), @c (0 0 0) = 4r 6= 0: observing that c(0 0 0) = 0 and @Z Since the Darbouxian type of the umbilic is dened by semialgebraic conditions on the coecients of j 3 Z(0 0) the behavior of lines of curvature near p can be expressed in terms of the coecients of the 3-jets of the surfaces Bi at pi:

Remark 4. Under generic conditions on the coe cients, the cubic form above determines the Index ( 21 ) as well as the Singularity (Hyperbolic, Elliptic) and Darbouxian (D1 , D2 , D3 ) Types of the umbilic of the Conict Surface C . In the example discussed above the Caustic Umbilic (on the Focal Surface)is located at innity. By taking a + k and b + k, k 6= 0, instead of a and b in f1 and keeping f2 unchanged, the Caustic Umbilic of the Conict Surface moves to (0 0 k1 ). The Index of the umbilic, as well as the Darbouxian and Singularity types of the Umbilic will be also determined by the coe cients of the cubic form, changed accordingly to the presence of constant k. The generic conditions referred invoked above are expressed by the nonvanishing of a quadratic form, for the Index, to which the non-vanishing of pertinent cubic forms should be added for the Singularity and Darbouxian classications 2].

References 1] G. Darboux, Lecons sur la Teorie des Surfaces, Vol 4, Note 7, Sur la forme des lignes de courboure dans la la voisinage d 'un ombilic, Gauthier Villars, Paris, 1896. 2] I. Porteous, Geometric Di erentiation , Cambridge Univ. Press, (1995) 3] D. Siersma, Properties of conict sets in the plane, This Proceedings, Banach Center Publications, Polish Acad. Sciences, Warzaw, (1998). 4] M. Spivak, A Comprehensive Introduction to Di erential Geometry, Vols 1, 3, Publish or Perish , Berkeley, (1979). 5] J. Sotomayor, C. Gutierrez, Structurally stable congurations of lines of principal curvature, Asterisque, 98-99 (1982). 6] J. Sotomayor, C.Gutierrez, Lines of Curvature and Umbilic Points on Surfaces, XVIII Coloquio Brasileiro de Matematica, IMPA, Rio de Janeiro, (1991). 7] J.B.Wilker, Equidistant sets and their connectivity properties, Proc. of Amer. Math Soc., 47, (1975).