{VERSION 6 0 "IBM INTEL LINUX" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "Ebene Kurve" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(plots): with(plottools);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "u:=t->t^2/5+2; v:=t->t*(t-4) *(t+4)/7;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "u1:=D(u); v1:=D(v); d: =t->sqrt(u1(t)^2+v1(t)^2); V:=t->[u1(t)/d(t),v1(t)/d(t)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "bild:=proc(t)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " local A,B,C;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " global u,v,u1,v1,b,d,V;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 " A :=line([u(t),v(t)],[u(t)+u1(t)/d(t),v(t)+v1(t)/d(t)],colour=black, thi ckness=2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 " B:=line([0,0],V(t) ,colour=black, thickness=2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " \+ C:=plot([u(s),v(s),s=-5..5],view=[-1..7,-5..5]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " RETURN( display([A,B,C], scaling=constrained) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "bild(2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "animate( bild, [t], t=-5..4.6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Flaeche - S chritt fuer Schritt" }}{PARA 0 "" 0 "" {TEXT -1 294 "Hier wird eine ei nfache Flaeche betrachtet, der Graph einer Funktion f:=(x,y)->a*x^2+b* y^2. Zu dem Kreis in der xy-Ebene um den Nullpunkt mit dem Radius r ge hoert eine geschlossene Kurve in der Flaeche. Man kann sich anschauen, wie sich der Normaleneinheitsvektor in einem Kurvenpunkt verhaelt." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "a,b und \+ r sind waehlbar. Haben a und b verschiedene Vorzeichen, ergibt sich ei ne Sattelflaeche." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "Hier sind noch recht viele Zwischenergebnisse ausgegeben, das liesse sich abkuerzen." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(plots): with(plottools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "a:=1; \+ b:=2/3; f:=(x,y)->a*x^2+b*y^2+1.5;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Bild der Flaeche:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "plot3d ([x,y,f(x,y)], x=-2..2,y=-2..2, view=[-2..2,-2..2,-1..4], labels=[x,y, z], axes=boxed, scaling=constrained); flaeche:=%:" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 37 "Festsetzung von r und Bild der Kurve:" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 6 "r:=1; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "spacecurve([r*cos(t),r*sin(t),f(r*cos(t),r*sin(t))], t=0..2*Pi, colou r=black, thickness=2); kurve:=%:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Hintergrund: Flaeche und Kurve" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "display3d([kurve,flaeche]); hintergrund:=%:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Normaleneinheitsvektor der Flaeche im Punkt (x,y,f(x ,y))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "N:=proc(x,y)" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 11 " local d;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 " d:=sqrt( (D[1](f)(x,y))^2+(D[2](f)(x,y))^2+1 );" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 55 " RETURN( [ -D[1](f)(x,y)/d, -D[2](f)(x,y)/d, 1/d ] );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "Normaleneinheitsvektor in Flaechenpunkt zu eine m Punkt des Kreises um den Nullpunkt mit Radius r:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 27 "v:=t->N(r*cos(t),r*sin(t));" }}{PARA 0 "" 0 "" {TEXT -1 11 "Kurvenpunkt" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "p:=t->[ r*cos(t),r*sin(t),f(r*cos(t),r*sin(t))]; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Prozedur fuer die Animation:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "bild:=proc(t)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " \+ local L1,L2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " L1:=line([0,0, 0],v(t),colour=blue, thickness=2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " L2:=line(p(t),p(t)+v(t),colour=blue, thickness=2);" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 32 " RETURN( display3d([L1,L2]) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Animation herstellen. (Hilfe: animate)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "animate(bild, [t], t=0..2*Pi, background=hintergrund);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "Hier sieht man nur die Kurve und \+ den Normaleneinheitsvektor, die Flaeche ist weggelassen. Das ist etwas durchsichtiger." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "animate(bild, [ t], t=0..2*Pi, background=kurve, axes = normal);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 33 "Indikatrix auf der Einheitskugel:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "sphere([0,0,0],1): kugel:=display3d(%, style=wir eframe, colour=black): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "animate (spacecurve, [v(t), t=0..A, colour=blue, thickness=2], A=0..2*Pi, back ground=kugel, axes=normal);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Vo n den Normaleneinheitsvektoren beschriebener \"Kegel\" in der Einheits kugel:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "plot3d([u*v(t)[1],u*v(t) [2],u*v(t)[3]],u=0..1,t=0..2*Pi, axes=normal, scaling=constrained, lab els=[x,y,z]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Zum Umlaufsinn: Projektionen des Kurvenpunkts (rot) und des Punktes der Indikatrix (b lau) in die xy-Ebene" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "umlauf:=pro c(t)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " plot([r*cos(u),r*sin(u), u=0..t], colour=red);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 " plot([v (u)[1],v(u)[2],u=0..t], colour=blue);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " RETURN( display([%,%%], axes=normal) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "ani mate( umlauf, [t], t=0..2*Pi, scaling=constrained);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Flaeche - Demoversion" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(plots): with(plottools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "a:=1; b:=-2/3; r:=1; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f:=(x,y)->a*x^2+b*y^2+1.5;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Bild der Flaeche:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "flaech e:=plot3d([x,y,f(x,y)], x=-2..2,y=-2..2, view=[-2..2,-2..2,-1..4], lab els=[x,y,z], axes=boxed, scaling=constrained): " }}{PARA 0 "" 0 "" {TEXT -1 15 "Bild der Kurve:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "kur ve:=spacecurve([r*cos(t),r*sin(t),f(r*cos(t),r*sin(t))], t=0..2*Pi, co lour=black, thickness=2):" }}{PARA 0 "" 0 "" {TEXT -1 30 "Hintergrund: Flaeche und Kurve" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "hintergrund:= display3d([kurve,flaeche]):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Normaleneinheitsvektor der Flaeche im Punkt (x, y,f(x,y))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "N:=proc(x,y)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 " local d;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 " d:=sqrt( (D[1](f)(x,y))^2+(D[2](f)(x,y))^2+1 );" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " RETURN( [ -D[1](f)(x,y)/d, -D[2 ](f)(x,y)/d, 1/d ] );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "Normalene inheitsvektor in Flaechenpunkt zu einem Punkt des Kreises um den Nullp unkt mit Radius r:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "v:=t->N(r*cos (t),r*sin(t)):" }}{PARA 0 "" 0 "" {TEXT -1 12 "Kurvenpunkt:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "p:=t->[r*cos(t),r*sin(t),f(r*cos(t),r*sin (t))]: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Prozedur fuer die Animation:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "bild:=proc(t)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " local L1,L2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " L1:=line([0,0,0],v(t),colour=b lue, thickness=2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " L2:=line(p (t),p(t)+v(t),colour=blue, thickness=2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " RETURN( display3d([L1,L2]) );" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 4 "end:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Einheitskugel" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 " kugel:=display3d( sphere([0,0,0],1), style=wireframe, colour=black):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "Projekt ionen des Kurvenpunkts (rot) und des Punktes der Indikatrix (blau) in \+ die xy-Ebene" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "umlauf:=proc(t)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " plot([r*cos(u),r*sin(u),u=0..t], colour=red);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 " plot([v(u)[1],v (u)[2],u=0..t], colour=blue);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " \+ RETURN( display([%,%%], axes=normal) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG }{EXCHG }{EXCHG }{EXCHG }{EXCHG } {EXCHG }{EXCHG }{EXCHG }{EXCHG }{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Animation herstellen. (Hilfe: animate)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "animate(bild, [t], t=0..2*Pi, background=hintergrund) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "Hier sieht man nur die Kurv e und den Normaleneinheitsvektor, die Flaeche ist weggelassen. Das ist etwas durchsichtiger." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "animate(b ild, [t], t=0..2*Pi, background=kurve, axes = normal);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Indikatrix auf der Einheitskugel:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "animate(spacecurve, [v(t), t=0..A, colour=blue, thickness=2], A=0..2*Pi, background=kugel, axes=normal); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Von den Normaleneinheitsvekto ren beschriebener \"Kegel\" in der Einheitskugel:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 107 "plot3d([u*v(t)[1],u*v(t)[2],u*v(t)[3]],u=0..1,t=0. .2*Pi, axes=normal, scaling=constrained, labels=[x,y,z]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Zum Umlaufsinn der Kurve in der Flaeche u nd der Indikatrix" }}{PARA 0 "" 0 "" {TEXT -1 212 "Gezeichnet wird die Projektion des Kurvenpunktes (rot) und des Punktes der Indikatrix (bl au) auf der Einheitskugel in die xy-Ebene. Man erkennt gut, ob die Kur even den gleichen Umlaufsinn oder verschiedene haben." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "animate( umlauf, [t], t=0..2*Pi, scaling=constra ined);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 69 "Flaeche neu: Quadrat als Referenzgebiet, \+ Zeichnungen und Berechnungen" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(plots): with(pl ottools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "a:=1; b:=2/3; \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f:=(x,y)->a*x^2+b*y^2+1.5;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Bild der Flaeche ueber Quadrat -2* k < x,y < 2*k" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "flaeche:=proc(k)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "RETURN( plot3d(f(x,y), x=-2*k..2 *k,y=-2*k..2*k, view=[-2*k..2*k,-2*k..2*k,f(0,0)-k..f(0,0)+3*k], label s=[x,y,z], axes=boxed, scaling=constrained) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 99 "Bild der Kurve ueber dem Quadrat mit der Kantenlange 2* k, Zentrum Nullpunkt, achsenparallele Seiten" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "kurve:=proc(k)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " local s1,s2,s3,s4,t;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 " s1:=s pacecurve([k,k*t,f(k,k*t)], t=-1..1, colour=black, thickness=3):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 " s2:=spacecurve([-k*t,k,f(-k*t,k) ], t=-1..1, colour=black, thickness=3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 " s3:=spacecurve([-k,-k*t,f(-k,-k*t)], t=-1..1, colour=black, thickness=3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 " s4:=spacecurve ([k*t,-k,f(k*t,-k)], t=-1..1, colour=black, thickness=3):" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 40 " RETURN( display3d( [s1,s2,s3,s4] ) );" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 56 "Normaleneinheitsvektor der Flaeche im Punkt (x,y,f(x,y))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "N:=proc(x,y) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " local dd;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " dd:=sqrt( (D[1](f)(x,y))^2+(D[2](f)(x,y))^2+1 ) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 " RETURN( [ -D[1](f)(x,y)/dd, -D[2](f)(x,y)/dd, 1/dd ] );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Bild \+ des Normaleneinheitsvektors im Punkt P(x,y,f(x,y)), angeheftet in P" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "bildNp:=proc(x,y)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 " local S;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 " S:=line([x,y,f(x,y)],[x,y,f(x,y)]+N(x,y), colour=blue, thicknes s=3);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " RETURN( display3d(S) ); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Bild einiger Normalen auf dem R and des betrachteten Quadrats" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "pn ormalenbild:=proc(k)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " local li ste;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 " liste:=[[-k,-k],[0,-k],[ k,-k],[k,0],[k,k],[0,k],[-k,k],[-k,0]];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " map(X->bildNp( op(X) ), liste);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " RETURN( display3d( % ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 114 "Bild der Indikatrix der Kurve ueber dem Quadrat mit de r Kantenlange 2*k, Zentrum Nullpunkt, achsenparallele Seiten" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "indikatrix:=proc(k)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " local s1,s2,s3,s4,t,opts;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " opts:=(t=-1..1,colour=blue,thickness=3);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " s1:=spacecurve(N(k,k*t), opts):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " s2:=spacecurve(N(-k*t,k), opts): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " s3:=spacecurve(N(-k,-k*t), o pts):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " s4:=spacecurve(N(k*t,-k ), opts):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 " RETURN( display3d( \+ [s1,s2,s3,s4], scaling=constrained, axes=normal ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 23 "Bild der Einheitskugel:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "kugel:=sphere([0,0,0],1):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Inhalt der Flaeche ueber dem Re ferenzquadrat (dezimaler Naeherungswert)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "inhaltflaeche:=proc(k)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " local f1,f2,F;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " f1: =D[1](f)(x,y);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " f2:=D[2](f)(x, y);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 " F:=sqrt( (1+f1^2)*(1+f2^2 )-f1*f1*f2*f2 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " RETURN( eval f( int( int( F, x=-k..k ), y=-k..k ) ) ); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 87 "Naeherungsweise Berech nung des Inhalts der von der Indikatrix eingeschlossenen Flaeche" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fx: =D[1](f): fy:=D[2](f): " }{TEXT -1 27 "partielle Ableitungen von f" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "d:=(x,y)->sqrt(1+fx(x,y)^2+fy(x,y)^ 2): " }{TEXT -1 36 "Laenge des Normalenvektors in P(x,y)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "dxy:=(x,y)->sqrt(fx(x,y)^2+fy(x,y)^2): " } {TEXT -1 42 "Laenge der Proj. des Normvekt. in xy-Ebene" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 51 "Nxy:=(x,y)->(-fx(x,y)/dxy(x,y),-fy(x,y)/dxy(x, y)): " }{TEXT -1 24 "\" auf Laenge 1 gestreckt" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 219 "Der Winkel dphi, den die Projektionen zweier Normalenvektoren in die xy-Ebene bilden, wird fue r die Rechnung benoetigt. Ich nehme als Naeherung die Laenge der Diffe renz der zugehoerigen Einheitsvektoren in der xy-Ebene." }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "dphi:=proc(x1,y1,x2,y2)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " local dNxy;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " dNxy:=Nxy(x2,y2)-Nxy(x1,y1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " RETURN( sqrt( dNxy[1]^2+dNxy[2]^2 ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 195 "Zu zwei Punkten (x1,y1) und (x2,y2) des Randes des Ref erenzquadrats gibt dF(x1,y1,x2,y2) naeherungsweise den Inhalt des Drei ecks auf der Einheitskugel mit den Ecken (0,0,1), N(x1,y1) und N(x2,y2 )" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dF:=proc(x1,y1,x2,y2)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " local xm,ym;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 32 " xm:=(x1+x2)/2; ym:=(y1+y2)/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 " 1/2*(arccos( 1/d(xm,ym) ))^2*dphi(x1,y1,x2,y2); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " RETURN( evalf(%) );" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 105 "Naeherungswert fuer den Inhalt des v on der Indikatrix eingeschlossenen Flaechenstuecks der Einheitskugel. " }}{PARA 0 "" 0 "" {TEXT -1 92 "Dabei ist die Kantenlaenge des Refere nzquadrats 2*k, es wird mit 4*n Teilflaechen gerechnet." }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "Findikatrix:=proc(k,n)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " local S1,S2,S3,S4,dk,i;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " dk:=2*k/n;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 " \+ S1:=sum( dF(k,-k+(i-1)*dk,k,-k+i*dk), i=1..n ); i:='i';" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 58 " S2:=sum( dF(-k+(i-1)*dk,k,-k+i*dk,k), i=1 ..n ); i:='i';" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " S3:=sum( dF(-k ,-k+(i-1)*dk,-k,-k+i*dk), i=1..n ); i:='i';" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " S4:=sum( dF(-k+(i-1)*dk,-k,-k+i*dk,-k), i=1..n ); \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " RETURN( S1+S2+S3+S4 );" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 257 54 "Berechnung der Gausskruemmung (nach \+ Teubner TB S. 820)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "gauss:=proc(u,v)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " local z1,z2,z11,z12,z22;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " z1:=diff( f(x,y), x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " z2:= diff( f(x,y), y);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " z11:=diff( \+ z1, x );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " z22:=diff( z2, y ); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " z12:=diff( z1, y );" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " (z11*z22-z12^2)/((1+z1^2)*(1+z2^ 2)-(z1*z2)^2)^2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " RETURN( subs (\{x=u,y=v\}, % ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 13 "A usprobieren:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG }{EXCHG } {EXCHG }{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "display3d([f laeche(1/2),kurve(1/2),pnormalenbild(1/2)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "indikatrix(1/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "display3d( [kugel,indikatrix(1/2)] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "inhaltflaeche(1/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Findikatrix(1/2,10); %/inhaltflaech e(1/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Findikatrix(1/2, 100); %/inhaltflaeche(1/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Findikatrix(1/2,1000); %/inhaltflaeche(1/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Findikatrix(1/200,30); inhaltflaeche(1/200); \+ %%/%;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "gauss(0,0);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG }{EXCHG }{EXCHG }{EXCHG }}{MARK "3" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }