Chapter 4: Geometry
Chapter 4: Geometry
Chapter 4: Geometry
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2. Line through ´Ü ¼ Ý ¼ Þ ¼ µ and ´Ü ½ Ý ½ Þ ½ µ:<br />
Ü Ü Ý Ý ¼<br />
¼ Þ Þ ¼<br />
(4.13.3)<br />
Ü ½ Ü ¼ Ý ½ Ý ¼ Þ ½ Þ ¼<br />
This line is parallel to the vector ´Ü ½ Ü ¼ Ý ½ Ý ¼ Þ ½ Þ ¼ µ.<br />
4.13.1 DISTANCES<br />
1. The distance between two points in space is the length of the line segment<br />
joining them. The distance between the points ´Ü ¼ Ý ¼ Þ ¼ µ and ´Ü ½ Ý ½ Þ ½ µ is<br />
Ô<br />
´Ü ½ Ü ¼ µ ¾ ·´Ý ½ Ý ¼ µ ¾ ·´Þ ½ Þ ¼ µ ¾ (4.13.4)<br />
2. The point ± of the way from È ¼ ´Ü ¼ Ý ¼ Þ ¼ µ to È ½ ´Ü ½ Ý ½ Þ ½ µ is<br />
ܽ ·´½¼¼ µÜ ¼<br />
Ý ½ ·´½¼¼ µÝ ¼<br />
Þ ½ · ´½¼¼ µÞ ¼<br />
<br />
(4.13.5)<br />
½¼¼<br />
½¼¼<br />
½¼¼<br />
(The same formula also applies in oblique coordinates.) This point divides the<br />
segment È ¼ È ½ in the ratio ´½¼¼ µ. As a particular case, the midpoint of<br />
È ¼ È ½ is given by <br />
ܽ · Ü ¼<br />
(4.13.6)<br />
¾<br />
Ý ½ · Ý ¼<br />
¾<br />
Þ ½ · Þ ¼<br />
¾<br />
3. The distance between the point ´Ü ¼ Ý ¼ Þ ¼ µ and the line through ´Ü ½ Ý ½ Þ ½ µ in<br />
direction ´ µ:<br />
ÚÙ<br />
Ù<br />
Ø<br />
¬<br />
¬ Ý ¼ Ý ½ Þ ¼ Þ ½<br />
<br />
¬<br />
¾<br />
¬<br />
¬<br />
¾<br />
·<br />
¬ Þ ¼ Þ ½ Ü ¼ Ü ½<br />
¬<br />
·<br />
¬ Ü ¼ Ü ½ Ý ¼ Ý ½<br />
¬<br />
¾ · ¾ · ¾ (4.13.7)<br />
¬<br />
¬<br />
¾<br />
4. The distance between the line through ´Ü ¼ Ý ¼ Þ ¼ µ in direction ´ ¼ ¼ ¼ µ and<br />
the line through ´Ü ½ Ý ½ Þ ½ µ in direction ´ ½ ½ ½ µ:<br />
¬ Ü ½ Ü ¼ Ý ½ Ý ¼ Þ ½ Þ ¼¬¬¬¬¬<br />
¼ ¼ ¼<br />
¬<br />
½ ½ ½ × ¬¬¬¬ ¬ ¼ ¼¬¬¬<br />
¾<br />
¬ ·<br />
¬ ½ ½<br />
¬ ¼ ¼¬¬¬<br />
¾<br />
¬ ·<br />
½ ½<br />
¬ ¼ ¼¬¬¬<br />
¾<br />
½ ½ ¬<br />
(4.13.8)<br />
4.13.2 ANGLES<br />
Angle between lines with directions ´Ü ¼ Ý ¼ Þ ¼ µ and ´Ü ½ Ý ½ Þ ½ µ:<br />
<br />
Ó× ½<br />
¼ ½ · ¼ ½ · ¼ <br />
Ô Ô<br />
½<br />
<br />
<br />
¾<br />
¼ · ¾ ¼ · ¾ ¼ <br />
¾<br />
½ · ¾ ½ · ¾ ½<br />
(4.13.9)<br />
In particular, the two lines are parallel when ¼ ¼ ¼ ½ ½ ½ , and<br />
perpendicular when ¼ ½ · ¼ ½ · ¼ ½ ¼.<br />
© 2003 by CRC Press LLC