VISUALIZATION OF DIFFERENTIAL GEOMETRY UDC 514.7(045) : : Eberhard Malkowsky 1, Vesna Veličković 2

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1 FACTA UNIVERSITATIS Srs: Mchancs, Automatc Control Robotcs Vol.3, N o, 00, pp VISUALIZATION OF DIFFERENTIAL GEOMETRY UDC 54.7(045) :59.688: Ebrhard Malkowsky, Vsna Vlčkovć Dpartmnt of Mathmatcs, Unvrsty of Gssn, Grmany Arndtstrass, D-3539 Gssn, Grmany Prrodno-matmatčk fakultt, Unvrztt u Nšu, Ćrla Mtodja, 8000 Nš Abstract. W apply our own opn softwar n PASCAL [6], [7] to vsuals som rsults n dffrntal gomtry. In partcular, w dal wth th graphc rprsntaton of paralll focal surfacs of surfacs of rvoluton, of pottal surfacs thr Gaussan man curvaturs. Ky words: Computr Graphcs, Gomtry, Dffrntal Gomtry. INTRODUCTION Throughout ths papr, w assum that surfacs ar gvn by a paramtrc rprsntaton! 3 x( = ( x ( u,, x ( u,, x ( u, ) (( u, D) () whr th componnt functons hav contnuous partal drvatvs of ordr r th surfac (unt) vctor N! xsts at vry pont. Th normal curvatur of a curv γ on a surfac s th projcton of th vctor of curvatur of γ nto th tangnt plan of th surfac n th drcton of th vctor product th surfac normal vctor th tangnt of γ. It s wll known that at vry pont of a surfac, thr corrsponds on only on valu of th normal curvatur to vry drcton. Th xtrm valus of th normal curvatur at a pont of a surfac ar calld prncpal curvaturs, dnotd by κ κ. Th Gaussan man curvaturs ar dfnd by K = κ κ H = (κ + κ )/ (cf. [], [4]). In gnral thy ar ral-valud functons of th paramtrs of th surfac. In ths papr, w apply our own softwar to rprsnt paralll focal surfacs of surfacs of rvoluton, potntal surfacs thr Gaussan man curvaturs. Rcvd August 5, Mathmatcs Subjct Classfcaton. 53A05, 68N05.

2 8 E. MALKOWSKY, V. VELIČKOVIĆ. PARALLEL SURFACES In ths scton, w consdr paralll surfacs thr Gaussan man curvaturs. Thy play an mportant rol n th thory of mnmal surfacs, that s surfacs wth H 0, du to th rlaton btwn thr man curvaturs. Lt S b a surfac wth paramtrc rprsntaton () normal vctor N! ( u ). Thn, for ε > 0, th surfac S *!!! wth paramtrc rprsntaton y( = x( + ε N( s calld paralll surfac of S. If K, H K *, H * dnot th Gaussan man curvaturs of S S *, rspctvly, thn t s wll known that * K * H εk K = H = εh + ε K εh + ε K ([,3.5, Problm, p. 6]). Frst w consdr surfacs of rvoluton. Thy ar gvn by! x( = ( r( cosu, r( sn u, h( ) (( u, I (0,π)) () whr r(u ) > 0 on I. Thr paralll surfacs agan ar surfacs of rvoluton wth paramtrc rprsntaton! y( = ( ρ( cosu, ρ( sn u, ψ( ), whr εh'( ρ( = r( φ( εr'( ψ( = h( + φ( whr φ( = ( r' ( ) + ( h'( ) for r( φ ( > εh'(. Furthrmor, th Gaussan man curvaturs of surfacs of rvoluton dpnd on th paramtr u only. Thrfor w may rprsnt K or H as surfacs of rvoluton wth r(u ) = u h(u ) = K(u ) or h(u ) = H(u ), rspctvly, n (). If th orgnal surfac of rvoluton s a catnod wth r(u ) = cosh u h(u ) = u for u R, thn th paralll surfacs ar gvn by ε ρ( = cosh u ψ( = u + ε tanhu cosh u for cosh u > ε that s, for all u R whnvr ε <. Furthrmor w hav K =, H = 0, cosh u K * = 4 cosh u ε * ε H = 4 cosh u ε for th Gaussan man curvaturs of th catnod ts paralll surfacs. As a scond xampl, w consdr th psudo-sphr, a surfac of rvoluton, gvn by r ( u u =, h( = du ( u > 0).

3 Vsualzaton of Dffrntal Gomtry 9 It s a surfac of constant Gaussan curvatur K =. Furthrmor w hav u H = ( ) u u * * H + ε K = H = εh ε εh ε for ts man curvatur th Gaussan man curvaturs of ts paralll surfacs. As a last xampl n ths scton, w consdr xplct surfacs wth paramtrc rprsntaton! x( = ( u, u, f ( u, ) (( u, D R ). Thr Gaussan man curvaturs ar gvn by f f f K = H = (( + f ) ( ) ) 4 f f 3 f f + + f f, φ φ whr φ = + ( f ) + ( f). Th fgurs blow show th Gaussan man curvaturs of a hyprbolc parabolod ts paralll surfac. Fg.. Catnod paralll surfac Fg.. Gaussan curvatur of a catnod Gaussan man curvatur of ts paralll surfac

4 30 E. MALKOWSKY, V. VELIČKOVIĆ Fg. 3. Psudo-sphr paralll surfac Fg. 4. Man curvatur of psudo-sphr Gaussan man curvatur of ts paralll surfac Fg. 5. Gaussan man curvatur of a hyprbolc parablod Fg. 6. Gaussan man curvatur of a paralll surfac of a hyprbolc parabolod 3. FOCAL OR CENTRAL SURFACES In ths scton, w consdr focal or cntral surfacs whch, n som cass, ar gnralzatons of paralll surfacs.

5 Vsualzaton of Dffrntal Gomtry 3 Lt S b a surfac wthout parabolc ponts, that s ponts of vanshng Gaussan curvatur, wthout umblcal ponts, that s ponts wth κ = κ for th prncpal curvaturs κ κ. If x! ( u ) s a paramtrc rprsntaton of S such that th lns of curvatur concd wth th paramtr lns, whch s th cas f only f th frst scond fundamntal coffcnts g k L k (,k =,) of S satsfy th condton g = L, thn th surfacs S S wth paramtrc rprsntatons!!!!!! y( = x( + N( z( = x( + N ( κ ( κ ( ar calld th focal or cntral surfacs of S. Th nam orgnats from th fact that! y z! ar th poston vctors of th cntrs of th osculatng crcls of th normal sctons of th surfac S that corrspond to th prncpal curvaturs κ κ. In th cas of surfacs of rvoluton, w hav g = L = 0, κ = g L κ = g L. Thus th focal surfacs of a surfac of rvoluton wthout parabolc umblcal ponts ar agan surfacs of rvoluton gvn by h' ρ j ( u ) = r ψ φκ j j r' ( u ) = h for j=,, φκ whr φ = ( r ') + ( h' ). If th surfac of rvoluton s th catnod of Scton, thn ρ( = coshu ψ ( = u snh u coshu. If th surfac of rvoluton s th psudo-sphr of Scton, thn ρ ( = snh u, u ψ( = u du, u u ρ ( u ) =, ( ) u u u = du + ψ. Fnally, for th hyprbolod of rotaton wth r ( = coshu, h ( = snh u j φ ( = cosh u + snh u, w obtan ρ( = cosh u 3 3 ψ ( = snh u. Fg. 7. Catnod ts focal surfac

6 3 E. MALKOWSKY, V. VELIČKOVIĆ Fg. 8. Hyprbolod st focal surfac Fg. 9. Psudosphr ts two focal surfacs 4. THE GAUSSIAN AND MEAN CURVATURES OF POTENTIAL SURFACES In ths scton w consdr th Gaussan man curvaturs of so-calld potntal surfacs gvn by a paramtrc rprsntaton!! y( u, = h( u, x( (( u, ( π, π ) (0,π)),! whr x( = (cosu cosu,cosu sn u,sn. Potntal surfacs play an mportant rol n physcs, chmstry crystallography. A lngthy but othrws straghtforward computaton ylds * * h = h g = h h g hh + h g = hh * g = h + h g = h + h cos u, * g = h (( h + h ) cos u + h ) g = +, =, * * L = h cosu ( h hh + h ), g * * L = h(cosu (hh hh) hh sn, g * L = h cosu (h hh + h cos u + hh sn u cosu ), g * for th frst scond fundamntal coffcnts of potntal surfacs thr Gaussan man curvaturs K * = L * /g * H * = / (g * * * * * * * ) ( g L g L + g ). L

7 Vsualzaton of Dffrntal Gomtry 33 REFERENCES. Carmo, M. P. do, Dffrntalgomtr von Kurvn und Flächn, Vwg Vrlag, Wsbadn, Braunschwg, 983. M. Falng, Entwcklung numrschr Algorthmn zur computrgrafschn Darstllung spzllr Problm dr Dffrntalgomtr und Krstallograph, Ph.D. Thss, Gssn, 996, Shakr Vrlag, Aachn, M. Falng, E. Malkowsky, En ffzntr Nullstllnalgorthmus zur computrgrafschn Darstllung spzllr Kurvn und Flächn, Mtt. Math. Sm. Gssn, 9, (996), J. Jost, Dffrntalgomtr dr Mnmalflächn, Sprngr-Vrlag, E. Kryszg, Dffrntalgomtr, Akadmsch Vrlagsgsllschaft Lpzg, E. Malkowsky, An opn softwar n OOP for computr graphcs som applcatons n dffrntal gomtry, Procdngs of th 0th South Afrcan Symposum on Numrcal Mathmatcs, (994), E. Malkowsky, W. Nckl, Computrgrafk n dr Dffrntalgomtr, Vwg-Vrlag Wsbadn, Braunschwg, 993 VIZUALIZACIJA DIFERENCIJALNE GEOMETRIJE Ebrhard Malkowsky, Vsna Vlčkovć Za prdstavljanj rzultata z dfrncjaln gomtrj prmnjujmo svoj sopstvn otvorn softvr psan u PASCAL-u [6,7]. Posbno stražujmo grafčk rprzntacj parallnh fokalnh površ rotaconh potncjalnh površ njhov Gausov srdnj krvn.