R
Ingenieurmathematik Prüfung 1
8.April2008
Zeit 90 Minuten, Reihenfolge beliebig, 8 Punkte pro Hauptaufgabe,
40 Pt. = N.6.
plot(zv); axis equal
ein auf der
Spitze stehendes Quadrat
mit der Seitenlänge 1 zeichnet.
T = [0 1 0; 0 0 1; 1 0 0]; T^3
E = [1 0 0; 0.5 1 0; 0 0 1]
und
F = [1 0 0; 0 1 0; 0 -0.2 1]
gilt.
G = (E*F)^-1 % =[ 1 0 0; -0.5 1 0; -0.1 0.2 1]
0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 Pl = 0 0 0 0 0 Pr = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
A=[-4 0 0]'; B=[-4 4 0]'; C=[8 4 0]'; D=[8 0 0]'; E=[-4 0 2]'; F=[-4 4 2]'; G=[8 4 2]'; H=[8 0 2]'; u1 = C-A, v1 = G-A, sp1 = u1'*v1, u1b = norm(u1), v1b = norm(v1) co1 = sp1/u1b/v1b % 0.9877 % u1 = [12 4 0]' ; v1 = [12 4 2]'; sp1 = 160; u1b = 12.6491; v1b = 12.8062 w1 = acos(u1'*v1/norm(u1)/norm(v1)) % 0.1568 rad 8.9849 Grad u2 = H-A, v2 = G-A, sp2 = u2'*v2, u2b = norm(u2), v2b = norm(v2) co2 = sp2/u2b/v2b % 0.9500 % u2 = [12 0 2]' ; v2 = [12 4 2]'; sp2 = 148; u2b = 12.1655; v2b = 12.8062 w2 = acos(u2'*v2/norm(u2)/norm(v2)) % 0.3177 rad 18.2008 Grad Q = [ A B C D A E F B F G C G H D H E]; plot3(Q(1,:), Q(2,:), Q(3,:),'k'); hold on UV1 = [G A C]; UV2 = [H A G]; plot3(UV1(1,:), UV1(2,:), UV1(3,:),'r'); axis equal plot3(UV2(1,:), UV2(2,:), UV2(3,:),'m'); hold off
% Punkte auf vertikalen Linien ueber 0/0 und 1/0 % Achse bei 0.5/0, h = 2 Radius = 0.5 2 Umgaenge t = (0:0.01:2)*2*pi; r = 0.5; h = 2; xl = 0.5 - r*cos(t); yl = r*sin(t) ; zl = t*h/2/pi; xr = 0.5 - r*cos(t); yr = -r*sin(t) ; zr = t*h/2/pi; plot3(xl,yl,zl,'k'); hold on; plot3(xr,yr,zr,'r'); axis equal; plot3([0 1 0 1 0], [0 0 0 0 0], [0 1 2 3 4], 'go'); hold off
Rur = [2 10 10 2 2; 3 3 5 5 3; 1 1 1 1 1]; % M = (6/4) (2+10)/2 , (3+5)/2 Tz = [1 0 -6; 0 1 -4; 0 0 1]; R = [0 1 0; -1 0 0; 0 0 1]; Tb = [1 0 6; 0 1 4; 0 0 1]; TT = Tb*R*Tz % = [0 1 2; -1 0 10; 0 0 1]; Rtr = Tb*R*Tz*Rur % Rtr = [5 5 7 7 5; 8 0 0 8 8; 1 1 1 1 1]; plot(Rur(1,:),Rur(2,:),'k') ; axis equal; hold on plot(Rtr(1,:),Rtr(2,:),'r') ; Rzr = Tz*Rur plot(Rzr(1,:),Rzr(2,:),'g') ; Rrr = R*Tz*Rur plot(Rrr(1,:),Rrr(2,:),'g') ;