Chapter 4: Geometry

FIGURE 4.40
Left: a spherical cap. Middle: a spherical zone (of two bases). Right: a spherical segment.
Ô
equation of the sphere is
¬
Ü ¾ · Ý ¾ · Þ ¾ Ü Ý Þ ½
Ü ¾ ½ · ݾ ½ · Þ¾ ½
Ü ½ Ý ½ Þ ½ ½
Ü ¾ ¾ · ݾ ¾ · Þ¾ ¾
Ü ¾ Ý ¾ Þ ¾ ½
Ü ¾ ¿ · ݾ ¿ · Þ¾ ¿
Ü ¿ Ý ¿ Þ ¿ ½
Ü ¾ · ݾ · Þ¾
Ü Ý Þ ½
¬
¼ (4.18.8)
2. Given two points È ½ ´Ü ½ Ý ½ Þ ½ µ and È ¾ ´Ü ¾ Ý ¾ Þ ¾ µ, there is a unique
sphere whose diameter is È ½ È ¾ ; its equation is
´Ü Ü ½ µ´Ü Ü ¾ µ·´Ý Ý ½ µ´Ý Ý ¾ µ·´Þ Þ ½ µ´Þ Þ ¾ µ¼ (4.18.9)
3. The area of a sphere of radius Ö is Ö ¾ , and the volume is ¿ Ö¿ .
4. The area of a spherical polygon (that is, of a polygon on the sphere whose
sides are arcs of great circles) is
Ë
´Ò ¾µ
Ö ¾ (4.18.10)
Ò
½
where Ö is the radius of the sphere, Ò is the number of vertices, and are
the internal angles of the polygons in radians. In particular, the sum of the
angles of a spherical triangle is always greater than ½¼ Æ , and the excess
is proportional to the area.
4.18.1.1 Spherical cap
Let the radius be Ö (Figure 4.40, left). The area of the curved region is ¾Ö Ô ¾ .
The volume of the cap is ½ ¾´¿Ö µ ½ ¿ ´¿¾ · ¾ µ.
4.18.1.2 Spherical zone (of two bases)
Let the radius be Ö (Figure 4.40, middle). The area of the curved region (called a
spherical zone)is¾Ö. The volume of the zone is ½ ´¿¾ ·¿ ¾ · ¾ µ.
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