Differential Geometry of Surfaces with Mathcad: A Virtual Learning Approach

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1 The 4 h Inernaional Conference on Virual Learning Gheorghe Asachi Technical Universiy of Iaşi, Oc 30-Nov, 009 Differenial Geomery of Surfaces wih Mahca: A Virual Learning Approach Nicolae Dăneţ Technical Universiy of Civil Engineering of Buchares Faculy of Railways, Roas an Briges Since 88

The esign of he roofs of some moern builings use Differenial Geomery.

2 Differenial Geomery of Surfaces wih Mahca Why Differenial Geomery? Differenial Geomery is an imporan opic for he suens in Civil Engineering. The esign of he roofs of some moern builings use Differenial Geomery. Ahens Olympic Saium Tenerife Opera House The 4 h Inernaional Conference on Virual Learning, Iaşi, Oc 30 Nov, 009 Pag. /

3 Differenial Geomery of Surfaces wih Mahca Why Differenial Geomery? In he acual syllabus we have a shor ime o each his opic. Before 005/006 Afer 005/006 (Bologna Process) Year Semeser (4 weeks) Semeser (4 weeks) Year Semeser (4 weeks) Semeser (4 weeks) Mahemaical Analysis (I) 3 C S Mahemaical Analysis (II) 3 C S Mahemaical Analysis (I) C S Mahemaical Analysis (II) C S Linear Algebra, Analyical an Differenial Geomery (I), C S Linear Algebra, Analyical an Differenial Geomery (II), C S Linear Algebra, Analyical an Differenial Geomery 3 C S Orinary an Parially Differenial Equaions C S Using Compuers C L Using Compuers C L Avance Mahemaics C S Numerical Analysis C L Legen: C = Course S = Seminar L = Laboraory The 4 h Inernaional Conference on Virual Learning, Iaşi, Oc 30 Nov, 009 Pag. 3 /

4 Teaching Differenial Geomery of surfaces for suens in engineering in such a shor ime is a ifficul ask for every eacher. For goo unersaning of Differenial Geomery he suens mus have soli knowlege of: Geomery Calculus Linear Algebra Bu he firs ifficuly in eaching Differenial Geomery is ha he suens mus have a goo 3D imaginaion. The 4 h Inernaional Conference on Virual Learning, Iaşi, Oc 30 Nov, 009 Pag. 4 /

5 Traiionally he eacher raws on he blackboar: surfaces angen planes normal lines curves on surfaces angles beween curves ec. In his paper I propose an alernaive o raiional eaching echniques of Differenial Geomery. The concep is o creae a virual learning environmen by using moern sofware wih goo capabiliies for ploing curves an surfaces. For his aim I propose Mahca. The 4 h Inernaional Conference on Virual Learning, Iaşi, Oc 30 Nov, 009 Pag. 5 /

6 Differenial Geomery of Surfacea wih Mahca Why Mahca? Is easy o be learne an can be learne simulaneously wih he Differenial Geomery. The equaions are wrien in Mahca like on he blackboar on in MS Wor. The normal vecor a he surface The equaion of he normal line In MS Wor N( u, v) r u ( u, v) r ( u, v) ( v0 In Mahca Nuv ( ) r v u ( uv ) r v ( uv ) L ) r( u, v ) N( u, ) L N u 0 v 0 () r u 0 v 0 Mahca has a Wha-You-See-Is-Wha-You-Ge user inerface. In his inerface is easy o combine: mahemaics regions (for equaions) graphical represenaions The 4 h Inernaional Conference on Virual Learning, Iaşi, Oc 30 Nov, 009 Pag. 6 /

7 ex regions (for explanaions) Here is an example of here ifferen regions in he inerface of Mahca. The 4 h Inernaional Conference on Virual Learning, Iaşi, Oc 30 Nov, 009 Pag. 7 /

8 How can be use his virual environmen? The moels iniially creae by eacher for his lecures can be laer use by suens for: he visualizaion of new surfaces (by changing he equaions) or for compuaion of some numerical characerisics associae o he surfaces. All hese facs are possible because he environmen is an ineracive Mahca e-book. The suens can make heir own changes an can see immeiaely he answer o hese moificaions. The 4 h Inernaional Conference on Virual Learning, Iaşi, Oc 30 Nov, 009 Pag. 8 /

9 Some Lecures in Mahca See he pages afer References The 4 h Inernaional Conference on Virual Learning, Iaşi, Oc 30 Nov, 009 Pag. 9 /

10 Conclusions Using his Mahca environmen he suens: learn o plo poins, space curves an surfaces in he same raw relae graphical objecs o heir analyical efiniions see quickly he graphical effecs of varying parameers focus heir aenion on he properies of he surfaces an no on har han compuaions can pu an easy verify wha if quesions The 4 h Inernaional Conference on Virual Learning, Iaşi, Oc 30 Nov, 009 Pag. 0 /

11 References. Aams, J.A., Graphics an Geomery Ineracive Examples, vol. an vol.; available a hp:// Birkelan, B., Creaing Amazing Images wih Mahca, free Mahca e-book; available a hp:// images.asp 3. Gray, A., Abbena, E., Salamon, S., Moern Differenial Geomery of Curves an Surfaces wih Mahemaica, Thir Eiion, Chapmann & Hall/CRC, Lipschuz, M. M.: Differenial Geomery. Schaum s Ouline Series, McGraw-Hill, New York, San Francisco, Lorczak, P.R.: 3D Ploing from he Mahca Treasury. Upae o Mahca 00. MahSof Engineering an Eucaion, Inc., Palais, R.S., A Moern Course on Curves an Surfaces; available a hp:// 7. Rovenski, V., Geomery of Curves an Surfaces wih MAPLE, Birkhäuser, Boson, Basel, Berlin, 000. The 4 h Inernaional Conference on Virual Learning, Iaşi, Oc 30 Nov, 009 Pag. /

12 Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces Ploing surfaces in vecor or parameric form In orer o have he saring inex of all arays equal o we efine ORIGIN To plo a surface we usually use he vecor parameric equaion ruv (, ) u + v u v raher hen he hree scalar parameric equaions of he componens xuv (, ) := r( u, v) yuv (, ) := r( u, v) zuv (, ) := r( u, v) 3 To plo he surface inser he 3D plo operaor in he workshee using he pah Inser/Graph/Surface Plo an ener in he placeholer he name of he vecor funcion r or he riple of he scalar funcios (x,y,z). := uv r ( xy,, z) Then ouble click on he graph an change in he convenien moe he appererance of he plo. For example, push he buon Appearance an a Fill Opions choose Fill Surface an a Color Opions choose Colormap. Also push he buon Special an a Line Syle choose ashe. The surface will have he appearanbe like in he figure below. Surfaces_angen_plane.mc Pag. /

Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces r Ploing a poin on a surface Firs efine he poin T u 0 := 3 v 0 := P 0 := ru 0, v 0 P 0 = ( 3 5 6 ) Then efine a consan vecor funcion

13 Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces r Ploing a poin on a surface Firs efine he poin T u 0 := 3 v 0 := P 0 := ru 0, v 0 P 0 = ( ) Then efine a consan vecor funcion P ():= P 0 an use he CreaeSpace funcion o plo he poin P0 := CreaeSpace( P) Plo he poin firs using Inser/Graph/3D Scaer Plo. Then ouble click he graph, push he buon Appearance an a he Poin Opions increase he sixe of he symbol o o 3 an a Color Opions choose Soli Color Re. Then a he surface r an o he same operaion iscribe above. We wiil have he following p P0, r Surfaces_angen_plane.mc Pag. /

Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces Ploing he angen plane o a surface a a poin Firsly we efine he parial erivaive vecors r u ( u, v) r v ( u, v) u ruv (, ) := u ruv (,

14 Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces Ploing he angen plane o a surface a a poin Firsly we efine he parial erivaive vecors r u ( u, v) r v ( u, v) u ruv (, ) := u ruv (, ) r u ( uv, ) u ruv (, ) 3 v ruv (, ) := v ruv (, ) r v ( uv, ) v ruv (, ) 3 u u v v v The equaion of he angen plane o he surface a he poin P0 is TP( a, b) := r u 0, v 0 ar u ( u 0, v 0 ) + + br v u 0, v 0 u P0, r, TP The angen plane epens of wo variable a an b. The efaul values for hese variables are in he inerval [-5, 5]. To change hese values ouble click on graph, push he buons QuickPlo Daa an Plo3 an change he sar value o - an he en value o for every range variable a an b. Also push he buon Appearance an choose FillSurface an Colormap a Fill Opions an a Line Opions choose Soli Color Turqoise. Surfaces_angen_plane.mc Pag. 3 /

Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces A secon way o plo he angen plane Define he normal vecor o he surface Nuv (, ) := r u ( uv, ) r v ( uv, ) Nuv (, ) u + v v u 8 u v or

15 Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces A secon way o plo he angen plane Define he normal vecor o he surface Nuv (, ) := r u ( uv, ) r v ( uv, ) Nuv (, ) u + v v u 8 u v or he unui nornal vecor nuv (, ) := r u ( uv, ) r v ( uv, ) r u ( uv, ) r v ( uv, ) Le A, B an C be be he componens of vecor N. To simplyfy he noaion we efine A 0 := A u 0, v 0 B 0 := B u 0, v 0 C 0 := C u 0, v 0 Auv (, ) := N( u, v) Buv (, ) := N( u, v) Cuv (, ) := N( u, v) 3 x 0 := xu 0, v 0 y 0 := yu 0, v 0 z 0 := zu 0, v 0 Then he equaion of he angen plane o he surface a poin P0 is A 0 ZXY (, ) := z 0 X x C 0 0 B 0 Y y C 0 0 P0, r, Z Surfaces_angen_plane.mc Pag. 4 /

16 Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces Since he poin has he coorinaes P 0 T = ( ) we choose in QuickPlo Daa for Plo 3 ha e variable X has values in he inerval [0, 6] an he variable Y has he value in [-5, 5]. Ploing he normal line The equaion of he normal line is N( u 0 v 0 ) NL() := r u 0, v 0 +, NL() P0, r, TP, NL P0, r, Z, NL Surfaces_angen_plane.mc Pag. 5 /

Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces Ellipic Paraboloi ORIGIN The ellipic paraboloi of semiaxis a an b has he equaion or x = u x = a u y = v y = b v z = A simple way o

17 Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces Ellipic Paraboloi ORIGIN The ellipic paraboloi of semiaxis a an b has he equaion or x = u x = a u y = v y = b v z = A simple way o obain parameric equaions for his surface is o pu z = z = u + v x y a + b u v a + b To plo his surface we efine a := b := ruv (, ) := a u b v u v + The graph is r Surfaces_ellipic_paraboloi.mc Pag. /

Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces A beer paramerizaion o plo he ellipic paraboloi is ruv (, ) := a u cos( v) b u sin( v) u r Le us consier he

18 Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces A beer paramerizaion o plo he ellipic paraboloi is ruv (, ) := a u cos( v) b u sin( v) u r Le us consier he following poin on his surface π T u 0 := 3 v 0 := P 4 0 := ru ( 0, v 0 ) P 0 = ( 3 3 9) P ():= P 0 P0 := CreaeSpace( P) P0, r P0, r Surfaces_ellipic_paraboloi.mc Pag. /

19 Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces Ploing he angen plane o he ellipic paraboloi a he given poin Firsly we efine he parial erivaive vecors r u ( uv, ) u ruv (, ) := u ruv (, ) r v ( uv, ) u ruv (, ) 3 := v ruv (, ) v ruv (, ) v ruv (, ) 3 r u ( uv, ) cos( v) sin v) u( r v ( uv, ) u sin( v) u 0 cos( v) The equaion of he angen plane o he surface a he poin P0 is Tab (, ) := r u 0, v 0 ar u ( u 0, v 0 ) + + br v u 0, v 0 P0, r, T Surfaces_ellipic_paraboloi.mc Pag. 3 /

20 Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces Ploing he normal line Define he normal vecor o he surface a any poin Nuv (, ) := r u ( uv, ) r v ( uv, ) Nuv (, ) u cos( v) u sin( v) cos( v) u + sin( v) u Then he equaion of he normal line is N( u 0 v 0 ) L ():= r u 0, v 0 +, L () P0, r, T, L Ploing he coorinae curves on he surface π u 0 := 3 v 0 := 4 ρ( u) := r u, v 0 ρ( v) := r u 0, v Γv := CreaeSpace ρ, 4, 4, 00 Γu := CreaeSpace ρ, 0, 5, 00 Surfaces_ellipic_paraboloi.mc Pag. 4 /

21 Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces P0, Γu, Γv, r P0, Γu, Γv, r, T P0, Γu, Γv, L, r, T Surfaces_ellipic_paraboloi.mc Pag. 5 /

22 Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces Ploing a curve on he surface Le us consier he following curve in he planar omain of efiniion of he surface u := 4 sin() v ():= 5 v () u () We consier he porion of plane curve for which varies in inerval [, 4] :=, v u Then ρ () := r( u(), v() ) ρ () 4 sin() cos() 4 sin() 6 sin() is a curve on he ellipic paraboloi. To plo his space we use CreaeSpace funcion in which we inicae ha belongs o he inerval [, 4]. C := CreaeSpace ρ,, 4, 00 Surfaces_ellipic_paraboloi.mc Pag. 6 /

23 Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces C, r C, r Lengh of a curve on a surface To compue he lengh of a curve we efine he erivaive of he vecor ρ an compue is norm on he given inerval. ρ' () ρ () := ρ () ρ' () ρ () 3 4 cos() 4 sin() 8 sin() cos() 3 sin() cos() 4 ρ' () = 5.74 Surfaces_ellipic_paraboloi.mc Pag. 7 /

24 Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces Ploing wo curves on he surface Le us consier he following wo curves which pass hrough he given poin in he planar omain efiniion of he surface. u() := u 0 v() := v u() := u v() := v 0 4 u 0 v() v() 0 v u( ), u() Then we efine he wo curves on he ellipic paraboloi ρ() := r( u( ), v() ) ρ() := r( u( ), v() ) ρ() ( 3 ) cos 4 π + 3 ( 3 ) sin 4 π + 3 ( 3 ) ρ() sin 4 π + cos 4 π For ploing hese curves we use CreaeSpace funcion. Γ := CreaeSpace ρ, 0.7, 0.7 Γ := CreaeSpace ρ, 0.7, 0.7 Surfaces_ellipic_paraboloi.mc Pag. 8 /

25 Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces P0, Γ, Γ, r Now we compue he angle beween hese wo curves a he given poin. Firs we efine an compue he erivaives of hese vecor funcions. ρ' () ρ () := ρ () ρ' () ρ () 3 := ρ () ρ () ρ () 3 ρ' () ρ' () cos 4 π ( 3 ) sin 4 π + 3 sin 4 π ( 3 ) cos 4 π sin 4 π cos 4 π cos 4 π + sin 4 π + Surfaces_ellipic_paraboloi.mc Pag. 9 /

Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces The angen lines a hese curves a he given poin have he equaions () := ρ( 0) + ρ' ( 0) () s := ρ( 0) + s ρ' ( 0) The value of he angle

26 Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces The angen lines a hese curves a he given poin have he equaions () := ρ( 0) + ρ' ( 0) () s := ρ( 0) + s ρ' ( 0) The value of he angle beween hese curves a he given poin is θ ρ' ( 0) ρ' ( 0) := acos θ ρ' ( 0) ρ' ( 0) π P0, Γ, Γ,,, r Lengh of hese curves L := ρ' () L = 9.4 L := ρ' () L = Surfaces_ellipic_paraboloi.mc Pag. 0 /

27 Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces The perimeer an he angles of a curvilinear riangle on an ellipic paraboloi We consier he ellipic paraboloi given by he following equaions: ORIGIN := a := b := ruv (, ) := a u cos( v) b u sin( v) u r On his surface we consier a curvilinear riangle given by he following hree curves: The firs curve u() := v() := 0 ρ() := r( u( ), v() ) C := CreaeSpace ρ, 0, 3, 00 ρ() 0 ρ' () := ρ () ρ () ρ () 3 ρ' () 0 Surfaces_curvilinear_riangle.mc Pag. /

28 Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces The secon curve u() := v() := π 4 ρ() := r( u( ), v() ) C := CreaeSpace ρ, 0, 4., 00 ρ() ρ' () := ρ () ρ () ρ () 3 ρ' () ρ' () The hir curve u3() := 3 v3() := ρ3() := r( u3( ), v3() ) π C3 := CreaeSpaceρ3, 0,, 50 4 ρ3() 6 cos() 6 sin() 8 ρ3' () := ρ3 () ρ3 () ρ3 () 3 ρ3' ( ) simplify 6 sin() 6 cos() 0 The ploing of his riangle C, C, C3, r Surfaces_curvilinear_riangle.mc Pag. /

Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces The perimeer of curvilinear riangle: L := 0 3 ρ' () L = 9.494 L := 0 3 ρ' () L = 9.494 π 4 L3 := 0 ρ3' () L3 = 4.

29 Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces The perimeer of curvilinear riangle: L := 0 3 ρ' () L = L := 0 3 ρ' () L = π 4 L3 := 0 ρ3' () L3 = 4.7 P := L + L + L3 P = The angen lines a origin a he firs wo curves: () s := ρ( 0) + ρ' ( 0) s () s := ρ( 0) + ρ' ( 0) s := CreaeSpace(, 0, 3) := CreaeSpace(, 0, 4) C, C, C3,,, r The value of he angle beween hese curves: θ acos ρ' ( 0) ρ' ( 0) := ρ' ( 0) ρ' ( 0) θ 4 π = 45eg Surfaces_curvilinear_riangle.mc Pag. 3 /

30 Nicolae Dane DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES Surfaces In a similar manner we compue he angle beween he firs an he hir an he secon an he hir curves. ρ3' 0 ρ3' 0 ρ' 3 θ3 := acos θ3 ρ' 3 π = 90 eg ρ3' π ρ' 3 4 θ3 := acos θ3 π ρ' ( 3 ) ρ3' 4 π = 90 eg Surfaces_curvilinear_riangle.mc Pag. 4 /

Project #1 Math 285 Name:

Project #1 Math 285 Name: Projec #1 Mah 85 Name: Solving Orinary Differenial Equaions by Maple: Sep 1: Iniialize he program: wih(deools): wih(pdeools): Sep : Define an ODE: (There are several ways of efining equaions, we sar wih