RumgeometriDTU Matematik21.08.2009Initaliseringrestart:with(plots):Vi definerer til senere brug et prik-produkt vha. VectorCalculus-pakken :prik:=(x,y)->VectorCalculus[DotProduct](x,y):\303\230velser om kugler, planer og linjer (nem)Vi vil i denne \303\270velse lave en 3d-illustration vha. kommandoer fra plots-pakken,
idet vi udnytter kuglers og planers ligninger samt linjers parameterfremstillinger.\303\230velse:
Indskriv i stedet for XX en ligning for den kugle som har centrum i LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYkLUYjNigtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGMEY3RjBGNEY0LUY4Ni1RIn5GJ0Y0RjsvRj9GPUZBRkNGRUZHRklGSy9GT0ZNRjQ=og radius 3, og afslut med enter: kugleLign:=XX;L\303\270sningkugleLign:=(x-1)^2+(y-1)^2+(z-1)^2=9;
Vi indtaster centrum C:=<1,1,1>:I punktetP:=<0,3,3>:har kuglen en normalvektor givet vedn:=P-C;\303\230velse:
Indskriv i stedet for XX en ligning for tangentplanen for kuglen i P, og afslut med enter: planLign:=XX;L\303\270sningplanLign:=prik(n,<x,y,z>-P)=0;Normalen i P kan parametriseres s\303\245ledesnormalen:=P+t*n;Vi kan nu plotte hver del for sig, og derefter fremkalde hele scenariet vha. "display": kugleplot:=implicitplot3d(kugleLign,x=-2..4,y=-2..4,z=-2..4,axes=normal):planplot:=implicitplot3d(planLign,x=-2..4,y=-2..4,z=-2..4,color=grey,axes=normal):Nplot:=spacecurve(normalen,t=-2.5..1,thickness=2,color=black):Pplot:=plot3d(P,x=-1..1, y=2..4,style=point,symbol=circle,thickness=5,color=red):display(kugleplot,planplot,Nplot,Pplot,scaling=constrained);\303\230velse:Drej p\303\245 figuren til den tager sig bedst mulig ud , og eksperiment\303\251r med andre forbedringer i den menu-bj\303\246lke der i den nye Maple 13 version fremkommer n\303\245r man klikker p\303\245 figuren. NB: \303\206ndringerne bevares n\303\245r man gemmer arket og lukker Maple. Men ikke hvis man eksekverer kommandoerne p\303\245ny efter genstart!\303\230velser om parametrisering af flader (sv\303\246rere)En cirkelskive som har flg. parameterfremstillingr:=<u*cos(v),u*sin(v),0>;hvor 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 kan plottes direkte s\303\245ledes:plot3d(r,u=0..1,v=0..2*Pi,axes=normal);\303\230velse:En lukket cylinder, som st\303\245r p\303\245 (x,y)-planen med z-aksen som symmetriakse, har radius 1 og h\303\270jde 1. 1. Giv en parameterfremstilling for cylinderens tre dele (bund, l\303\245g og den krumme flade).2. Plot de tre parameterfremstillinger og fremkald den samlet vha "display".L\303\270sningr1:=<u*cos(v),u*sin(v),0>:r2:=<u*cos(v),u*sin(v),1>:r3:=<cos(v),sin(v),u>:r1Plot:=plot3d(r1,u=0..1,v=0..2*Pi):r2Plot:=plot3d(r2,u=0..1,v=0..2*Pi):r3Plot:=plot3d(r3,u=0..1,v=0..2*Pi):display(r1Plot,r2Plot,r3Plot,scaling=constrained,axes=normal,view=[-1.5..1.5,-1.5..1.5,-0.5..2]);Enhedskuglen kan parametriseres og plottes s\303\245ledes:e:= <cos(s)*sin(t),sin(s)*sin(t),cos(t)>;plot3d(e,s=0..Pi,t=0..2*Pi,axes=normal,view=[-1.5..1.5,-1.5..1.5,-1.5..1.5]);For et konkret valg af v\303\246rdier for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0YrRi8= og LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= repr\303\246senterer enhedsvektoren LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= en bestemt retning i rummet, og alle retninger i rummet kan opn\303\245s ved passende valg af 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Vi kan fastl\303\245se de to variable, afpr\303\270ve dem - og evt. "frig\303\270re" dem bagefter s\303\245ledes:s:=Pi/6;
t:=Pi/4;
e;
s:='s';
t:='t';
Vi h\303\246ver nu vores lukkede cylinder med 2 enheder og betragter dens projektion (skygge) langs en bestemt retning LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY1Rjk=ned p\303\245 en plan \316\261 , som g\303\245r gennem Origo og som er vinkelret p\303\245 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= .\303\230velse:G\303\270r rede for at hvis LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY1Rjk=er et vilk\303\245rligt punkt i rummet, s\303\245 er dets projektion B = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRIkFGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSJ+RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZURj0=langs LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY1Rjk=ned p\303\245 \316\261 givet 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L\303\270sningOverlades trygt til l\303\246seren ..
Vi indtaster til senere brug projektionen p :p:=v-> v-prik(v,e)*e;\303\230velse:1. Bestem en parameterfremstilling for projektionen af hver af cylinderens tre dele.2. Plot med retningen LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJ0RidGNEY3Rj5GPi1GOzYtUSI9RidGPkZAL0ZERkJGRUZHRklGS0ZNL0ZQUSwwLjI3Nzc3NzhlbUYnL0ZTRmduLUYsNiQtRiM2Ji1JJm1mcmFjR0YkNigtRjE2JVEnJiM5NjA7RicvRjVGQkY+LUYjNiUtSSNtbkdGJDYkUSIyRidGPkY0RjcvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRl9wLyUpYmV2ZWxsZWRHRkJGOi1GXm82KEZgby1GIzYlLUZnbzYkUSI0RidGPkY0RjdGam9GXXBGYHBGYnBGPkY+LUY7Ni1RIn5GJ0Y+RkBGZW5GRUZHRklGS0ZNRk8vRlNGUUY+som eksempel, cylinderen sammen med dens skygge p\303\245 \316\261 . Drej figuren.L\303\270sningVi h\303\246ver cylinderen med 2 enheder og plotter den:r1:=<u*cos(v),u*sin(v),0+2>:r2:=<u*cos(v),u*sin(v),1+2>:r3:=<cos(v),sin(v),u+2>:r1Plot:=plot3d(r1,u=0..1,v=0..2*Pi):r2Plot:=plot3d(r2,u=0..1,v=0..2*Pi):r3Plot:=plot3d(r3,u=0..1,v=0..2*Pi):Vi fastl\303\246gger projektionsretningen:s:=Pi/2:t:=Pi/4:Herefter de tre projektioner og deres plotning:p1:=p(r1);
p2:=p(r2);
p3:=p(r3);p1Plot:=plot3d(p1,u=0..1,v=0..2*Pi,color=grey):p2Plot:=plot3d(p2,u=0..1,v=0..2*Pi,color=grey):p3Plot:=plot3d(p3,u=0..1,v=0..2*Pi,color=grey):Vi kan evt. tilf\303\270je projektionssk\303\246rmenskaerm:=prik(e,<x,y,z>)=0;sPlot:=implicitplot3d(skaerm,x=-1.3..1.3,y=-2.2..0,z=0..2.5,style=patchnogrid,transparency=0.8):display(r1Plot,r2Plot,r3Plot,p1Plot,p2Plot,p3Plot,sPlot,axes=normal,scaling=constrained,orientation=[20,80]);