Chapter 4: Geometry

4.12 PLANES
The (Cartesian) equation of a plane is linear in the coordinates Ü, Ý, and Þ:
Ü · Ý · Þ · ¼ (4.12.1)
The normal direction to this plane is ´ µ. The intersection of this plane with the
Ü-axis, or Ü-intercept,isÜ , the Ý-intercept is Ý , and the Þ-intercept
is Þ . The plane is vertical (perpendicular to the ÜÝ-plane) if ¼. It is
perpendicular to the Ü-axis if ¼, and likewise for the other coordinates.
When ¾ · ¾ · ¾ ½and ¼ in the equation Ü · Ý · Þ · ¼, the
equation is said to be in normal form. In this case is the distance of the plane to
the origin, and ´ µ are the direction cosines of the normal.
To reduce an arbitrary equation Ü · Ý · Þ · ¼to normal form, divide by
¦ Ô ¾ · ¾ · ¾ , where the sign of the radical is chosen opposite the sign of when
¼, the same as the sign of when ¼and ¼, and the same as the sign of
otherwise.
4.12.1 PLANES WITH PRESCRIBED PROPERTIES
1. Plane through ´Ü ¼ Ý ¼ Þ ¼ µ and perpendicular to the direction ´ µ:
´Ü Ü ¼ µ·´Ý Ý ¼ µ·´Þ Þ ¼ µ¼ (4.12.2)
2. Plane through ´Ü ¼ Ý ¼ Þ ¼ µ and parallel to the two directions ´ ½ ½ ½ µ and
´ ¾ ¾ ¾ µ: ¬ ¬¬¬¬¬
¬
Ü Ü ¼ Ý Ý ¼ Þ Þ ¼¬¬¬¬¬
½ ½ ½ ¼ (4.12.3)
¾ ¾ ¾
3. Plane through ´Ü ¼ Ý ¼ Þ ¼ µ and ´Ü ½ Ý ½ Þ ½ µ and parallel to the direction ´ µ:
¬
Ü Ü ¼ Ý Ý ¼ Þ Þ ¼
Ü ½ Ü ¼ Ý ½ Ý ¼ Þ ½ Þ ¼ ¼ (4.12.4)
4. Plane going through ´Ü ¼ Ý ¼ Þ ¼ µ, ´Ü ½ Ý ½ Þ ½ µ, and ´Ü ¾ Ý ¾ Þ ¾ µ:
¬
¬
Ü Ý Þ ½
Ü ¼ Ý ¼ Þ ¼ ½
¼
Ü ½ Ý ½ Þ ½ ½
Ü ¾ Ý ¾ Þ ¾ ½
or
¬
¬ Ü Ü ¼ Ý Ý ¼ Þ Þ ¼ ¬¬¬¬¬ Ü ½ Ü ¼ Ý ½ Ý ¼ Þ ½ Þ ¼ ¼
¬ Ü ¾ Ü ¼ Ý ¾ Ý ¼ Þ ¾ Þ ¼
(The last three formulae remain true in oblique coordinates.)
(4.12.5)
5. The distance from the point ´Ü ¼ Ý ¼ Þ ¼ µ to the plane Ü · Ý · Þ · ¼is
¬ ¬ Ü ¼ · Ý ¼ · Þ ¼ · ¬¬¬
Ô (4.12.6)
¾ · ¾ · ¾
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