Let point A be on a diagonal of one side of a cube. Let point B be on a diagonal of an adjacent side of the cube such that the two diagonals do not meet at a vertex.
Minimize the distance AB.
First a program to explore the territory:
range=0:.05:1 ;
z1=0; y2=1;
for t1=range
x1=t1; y1=t1;
fprintf('%5.3f ',t1)
for t2=range
z2=t2; x2=t2;
dist=sqrt((x2-x1)^2+(y2-y1)^2+(z2-z1)^2);
fprintf(' %5.3f',dist);
end
fprintf('\n');
end
where t1 and t1 are the parameters for the respective parametric equations of the two lines.
0.000 1.000 1.002 1.010 1.022 1.039 1.061 1.086 1.116 1.149 1.185 1.225 1.267 1.311 1.358 1.407 1.458 1.510 1.564 1.619 1.675 1.732
0.050 0.951 0.951 0.957 0.967 0.982 1.002 1.027 1.056 1.089 1.125 1.164 1.206 1.251 1.298 1.347 1.398 1.451 1.505 1.560 1.617 1.675
0.100 0.906 0.903 0.906 0.914 0.927 0.946 0.970 0.997 1.030 1.065 1.105 1.147 1.192 1.239 1.288 1.340 1.393 1.447 1.503 1.560 1.619
0.150 0.863 0.857 0.857 0.863 0.875 0.892 0.914 0.941 0.972 1.007 1.046 1.089 1.134 1.181 1.231 1.283 1.336 1.391 1.447 1.505 1.564
0.200 0.825 0.815 0.812 0.815 0.825 0.840 0.860 0.886 0.917 0.951 0.990 1.032 1.077 1.125 1.175 1.227 1.281 1.336 1.393 1.451 1.510
0.250 0.791 0.778 0.771 0.771 0.778 0.791 0.809 0.834 0.863 0.897 0.935 0.977 1.022 1.070 1.120 1.173 1.227 1.283 1.340 1.398 1.458
0.300 0.762 0.745 0.735 0.731 0.735 0.745 0.762 0.784 0.812 0.846 0.883 0.925 0.970 1.017 1.068 1.120 1.175 1.231 1.288 1.347 1.407
0.350 0.738 0.718 0.704 0.696 0.696 0.704 0.718 0.738 0.765 0.797 0.834 0.875 0.919 0.967 1.017 1.070 1.125 1.181 1.239 1.298 1.358
0.400 0.721 0.696 0.678 0.667 0.663 0.667 0.678 0.696 0.721 0.752 0.787 0.828 0.872 0.919 0.970 1.022 1.077 1.134 1.192 1.251 1.311
0.450 0.711 0.682 0.660 0.644 0.636 0.636 0.644 0.660 0.682 0.711 0.745 0.784 0.828 0.875 0.925 0.977 1.032 1.089 1.147 1.206 1.267
0.500 0.707 0.675 0.648 0.628 0.616 0.612 0.616 0.628 0.648 0.675 0.707 0.745 0.787 0.834 0.883 0.935 0.990 1.046 1.105 1.164 1.225
0.550 0.711 0.675 0.644 0.620 0.604 0.596 0.596 0.604 0.620 0.644 0.675 0.711 0.752 0.797 0.846 0.897 0.951 1.007 1.065 1.125 1.185
0.600 0.721 0.682 0.648 0.620 0.600 0.587 0.583 0.587 0.600 0.620 0.648 0.682 0.721 0.765 0.812 0.863 0.917 0.972 1.030 1.089 1.149
0.650 0.738 0.696 0.660 0.628 0.604 0.587 0.579 0.579 0.587 0.604 0.628 0.660 0.696 0.738 0.784 0.834 0.886 0.941 0.997 1.056 1.116
0.700 0.762 0.718 0.678 0.644 0.616 0.596 0.583 0.579 0.583 0.596 0.616 0.644 0.678 0.718 0.762 0.809 0.860 0.914 0.970 1.027 1.086
0.750 0.791 0.745 0.704 0.667 0.636 0.612 0.596 0.587 0.587 0.596 0.612 0.636 0.667 0.704 0.745 0.791 0.840 0.892 0.946 1.002 1.061
0.800 0.825 0.778 0.735 0.696 0.663 0.636 0.616 0.604 0.600 0.604 0.616 0.636 0.663 0.696 0.735 0.778 0.825 0.875 0.927 0.982 1.039
0.850 0.863 0.815 0.771 0.731 0.696 0.667 0.644 0.628 0.620 0.620 0.628 0.644 0.667 0.696 0.731 0.771 0.815 0.863 0.914 0.967 1.022
0.900 0.906 0.857 0.812 0.771 0.735 0.704 0.678 0.660 0.648 0.644 0.648 0.660 0.678 0.704 0.735 0.771 0.812 0.857 0.906 0.957 1.010
0.950 0.951 0.903 0.857 0.815 0.778 0.745 0.718 0.696 0.682 0.675 0.675 0.682 0.696 0.718 0.745 0.778 0.815 0.857 0.903 0.951 1.002
1.000 1.000 0.951 0.906 0.863 0.825 0.791 0.762 0.738 0.721 0.711 0.707 0.711 0.721 0.738 0.762 0.791 0.825 0.863 0.906 0.951 1.000
range=.6666664:.00000001:.6666668 ;
z1=0; y2=1;
for t1=range
x1=t1; y1=t1;
fprintf('%13.8f ',t1)
prev=99;
for t2=.33333:.00000000001:.4
z2=t2; x2=t2;
dist=sqrt((x2-x1)^2+(y2-y1)^2+(z2-z1)^2);
if dist>prev
fprintf('%17.14f',prev,prevt2,t2);
break
end
prev=dist; prevt2=t2;
end
fprintf('\n');
end
is a final stage in narrowing down where the solution lies.
The t1 parameter identifies the fraction of the way along one line, and the t2 parameter for the other line.
The output shows
fractional fractional
position minimal position
along one along other next sample
line distance line point
0.66666640 0.57735026919019 0.33333267658000 0.33333267659000
0.66666641 0.57735026919046 0.33333254643000 0.33333254644000
0.66666642 0.57735026919052 0.33333252294000 0.33333252295000
0.66666643 0.57735026919020 0.33333267926000 0.33333267927000
0.66666644 0.57735026919036 0.33333260144000 0.33333260145000
0.66666645 0.57735026919067 0.33333247128000 0.33333247129000
0.66666646 0.57735026919028 0.33333263968000 0.33333263969000
0.66666647 0.57735026919074 0.33333245183000 0.33333245184000
0.66666648 0.57735026919022 0.33333267926000 0.33333267927000
0.66666649 0.57735026919029 0.33333264505000 0.33333264506000
0.66666650 0.57735026919034 0.33333262626000 0.33333262627000
0.66666651 0.57735026919015 0.33333272019000 0.33333272020000
0.66666652 0.57735026919035 0.33333262626000 0.33333262627000
0.66666653 0.57735026919065 0.33333250684000 0.33333250685000
0.66666654 0.57735026919075 0.33333247128000 0.33333247129000
0.66666655 0.57735026919072 0.33333248806000 0.33333248807000
0.66666656 0.57735026919048 0.33333258400000 0.33333258401000
0.66666657 0.57735026919057 0.33333255179000 0.33333255180000
0.66666658 0.57735026919083 0.33333246055000 0.33333246056000
0.66666659 0.57735026919051 0.33333258400000 0.33333258401000
0.66666660 0.57735026919073 0.33333250282000 0.33333250283000
0.66666661 0.57735026919026 0.33333270073000 0.33333270074000
0.66666662 0.57735026919059 0.33333256320000 0.33333256321000
0.66666663 0.57735026919068 0.33333253569000 0.33333253570000
0.66666664 0.57735026919065 0.33333255179000 0.33333255180000
0.66666665 0.57735026919058 0.33333258400000 0.33333258401000
0.66666666 0.57735026919069 0.33333254643000 0.33333254644000
0.66666667 0.57735026919026 0.33333273160000 0.33333273161000
0.66666668 0.57735026919037 0.33333268597000 0.33333268598000
0.66666669 0.57735026919012 0.33333281009000 0.33333281010000
0.66666670 0.57735026919020 0.33333277588000 0.33333277589000
0.66666671 0.57735026919053 0.33333263230000 0.33333263231000
0.66666672 0.57735026919038 0.33333270208000 0.33333270209000
0.66666673 0.57735026919058 0.33333262559000 0.33333262560000
0.66666674 0.57735026919039 0.33333270744000 0.33333270745000
0.66666675 0.57735026919051 0.33333266383000 0.33333266384000
0.66666676 0.57735026919048 0.33333268396000 0.33333268397000
0.66666677 0.57735026919062 0.33333263096000 0.33333263097000
0.66666678 0.57735026919084 0.33333255716000 0.33333255717000
0.66666679 0.57735026919088 0.33333255179000 0.33333255180000
0.66666680 0.57735026919089 0.33333255179000 0.33333255180000
where the minimum distance seems to be 2/3 of the way from one end to the other of one of the diagonals, and 1/3 of the way from the other. Of course this depends on which end your starting from, but if you start from the ends that together are the ends of a space diagonal, its 2/3 of the way from the end of each.
The distance apart is shown as about 0.5773502691906 which Wolfram Alpha identifies as 1/sqrt(3), as we're using coordinates in the distance calculation. If the cube is a unit cube, this is the actual distance, and the location of the endpoints of the location, being referenced by it's coordinates, is sqrt(2)/3 from one end of each face diagonal and 2*sqrt(2)/3 from the other, being in a face diagonals of length sqrt(2).
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Posted by Charlie
on 2022-10-20 11:12:32 |
computer solution
