Y
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 8 zeichnet.
T = [0 1 0; 0 0 1; 1 0 0]; T^3
E = [1 0 0; -0.4 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; 0.4 1 0; -0.08 -0.2 1]
0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 Pl = 0 0 0 0 0 Pr = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0
A=[-2 0 0]'; B=[-2 6 0]'; C=[4 6 0]'; D=[4 0 0]'; E=[-2 0 3]'; F=[-2 6 3]'; G=[4 6 3]'; H=[4 0 3]'; u1 = C-A, v1 = G-A, sp1 = u1'*v1, u1b = norm(u1), v1b = norm(v1) co1 = sp1/u1b/v1b % 0.9428 % u1 = [6 6 0]' ; v1 = [6 6 3]'; sp1 = 72; u1b = 8.4853; v1b = 9 w1 = acos(u1'*v1/norm(u1)/norm(v1)) % 0.3398 rad 19.4712 Grad u2 = H-A, v2 = G-A, sp2 = u2'*v2, u2b = norm(u2), v2b = norm(v2) co2 = sp2/u2b/v2b % 0.7454 % u2 = [6 0 3]' ; v2 = [6 6 3]'; sp2 = 45; u2b = 6.7082; v2b = 9 w2 = acos(u2'*v2/norm(u2)/norm(v2)) % 0.7297 rad 41.8103 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 2/0 % Achse bei 1/0, h = 2 Radius = 1 2 Umgaenge t = (0:0.01:2)*2*pi; r = 1; h = 2; xl = 1 - r*cos(t); yl = r*sin(t) ; zl = t*h/2/pi; xr = 1 - r*cos(t); yr = -r*sin(t) ; zr = t*h/2/pi; plot3(xl,yl,zl,'b'); hold on; plot3(xr,yr,zr,'k'); axis equal; plot3([0 2 0 2 0], [0 0 0 0 0], [0 1 2 3 4], 'rx'); hold off
Rur = [3 3 7 7 3; 1 11 11 1 1; 1 1 1 1 1]; % M = (5/6) (3+7)/2 , (1+11)/2 Tz = [1 0 -5; 0 1 -6; 0 0 1]; R = [0 1 0; -1 0 0; 0 0 1]; Tb = [1 0 5; 0 1 6; 0 0 1]; TT = Tb*R*Tz % = [0 1 -1; -1 0 11; 0 0 1]; Rtr = Tb*R*Tz*Rur % Rtr = [0 10 10 0 0; 8 8 4 4 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') ;