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Math 251-003 - Lecture Notes #2 - Spring 2008 David Maslanka
Direction Cosines
The direction angles α, β, and γ associated with a nonzero vector v determine three circles on the unit sphere as described
below. Each of these inscribed circles is perpendicular to one of the coordinate axes and their common point of intersection
determines the direction of the vector uniquely.
restart:
with(plots):with(plottools):
s:=sphereplot(1,theta=0..2*Pi,phi=0..Pi,style=wireframe,color=green):
Circle1:=spacecurve([1/2,sqrt(3)*cos(t)/2,sqrt(3)*sin(t)/2],t=0..6*Pi,color=navy):
for n from 0 to 63 do
y||n:=sqrt(3)*cos(n*Pi/32)/2;
z||n:=sqrt(3)*sin(n*Pi/32)/2;
c||n:=line([1/2,y||n,z||n],[0,0,0],color=navy);
end do:
C1:=seq(c||i,i=0..63):
Circle2:=spacecurve([sqrt(13)*cos(t)/6,sqrt(23)/6,sqrt(13)*sin(t)/6],t=0..6*Pi,color=
red):
for n from 0 to 63 do
x||n:=sqrt(13)*cos(n*Pi/32)/6;
z||n:=sqrt(13)*sin(n*Pi/32)/6;
c||n:=line([x||n,sqrt(23)/6,z||n],[0,0,0],color=red);
end do:
C2:=seq(c||i,i=0..63):
Circle3:=spacecurve([sqrt(8)*cos(t)/3,sqrt(8)*sin(t)/3,1/3],t=0..6*Pi,color=magenta):
for n from 0 to 63 do
x||n:=sqrt(8)*cos(n*Pi/32)/3;
y||n:=sqrt(8)*sin(n*Pi/32)/3;
c||n:=line([x||n,y||n,1/3],[0,0,0],color=magenta);
end do:
C3:=seq(c||i,i=0..63):
l1 := line([0,0,0], [1.8,0,0],color=black):
l2 := line([0,0,0], [0,1.8,0],color=black):
l3 := line([0,0,0], [0,0,1.5],color=black):
t1:=textplot3d([1.7,0,-0.15,`x`],font=[TIMES,BOLDITALIC,36],color=black):
t2:=textplot3d([0,1.6,-0.15,`y`],font=[TIMES,BOLDITALIC,36],color=black):
t3:=textplot3d([0,0.15,1.4,`z`],font=[TIMES,BOLDITALIC,36],color=black):
l4 := line([0,0,0], [1/2,sqrt(23)/6,1/3],color=black,thickness=3,linestyle=2):
l5 := line([1,1*sqrt(23)/3,2/3], [1/2,sqrt(23)/6,1/3],color=black,thickness=3):
c:= sphere([1/2,sqrt(23)/6,1/3],0.015,color=black):
t:=0.025:
P:=[0.9+t*1/3,3*sqrt(23)/10+t*2/3,6/10-t*(sqrt(23)/3+1/2)]:
t:=-0.025:
Q:=[0.9+t*1/3,3*sqrt(23)/10+t*2/3,6/10-t*(sqrt(23)/3+1/2)]:
R:=[1,1*sqrt(23)/3,2/3]:
ar:=polygon([P,Q,R],color=black):
t4:=textplot3d([1,1*sqrt(23)/3,3/4,`v`],font=[TIMES,BOLDITALIC,38],color=black):
arx:=polygon([[1.8,0,0],[1.65,0.1,0],[1.65,-0.1,0]],color=black):
ary:=polygon([[0,1.8,0],[0.1,1.65,0],[-0.1,1.65,0]],color=black):
arz:=polygon([[0,0,1.5],[0.1,0,1.35],[-0.1,0,1.35]],color=black):
