Chapter 4: Geometry

(In an oblique coordinate system, everything in the preceding paragraph remains
true, except for the value of the slope.)
When ¾ · ¾ ½and ¼ in the equation Ü · Ý · ¼, the equation
is said to be in normal form. In this case is the distance of the line to the origin,
and (with Ó× and ×Ò ) is the angle that the perpendicular dropped to
the line from the origin makes with the positive Ü-axis (Figure 4.8, with Ô ).
To reduce an arbitrary equation Ü · Ý · ¼ to normal form, divide by
¦ Ô ¾ · ¾ , where the sign of the radical is chosen opposite the sign of when
¼and the same as the sign of when ¼.
FIGURE 4.8
The normal form of the line Ä is Ü Ó× · Ý ×Ò Ô.
Ý
Ä
Ô
Ü
4.4.1 LINES WITH PRESCRIBED PROPERTIES
1. Line of slope Ñ intersecting the Ü-axis at Ü Ü ¼ : Ý Ñ´Ü Ü ¼ µ
2. Line of slope Ñ intersecting the Ý-axis at Ý Ý ¼ : Ý ÑÜ · Ý ¼
3. Line intersecting the Ü-axis at Ü Ü ¼ and the Ý-axis at Ý Ý ¼ :
Ü Ý · ½ (4.4.2)
Ü ¼ Ý ¼
(This formula remains true in oblique coordinates.)
4. Line of slope Ñ passing though ´Ü ¼ Ý ¼ µ: Ý Ý ¼ Ñ´Ü Ü ¼ µ
5. Line passing through points ´Ü ¼ Ý ¼ µ and ´Ü ½ Ý ½ µ:
Ý Ý Ý ½ ¼ Ý ½
Ü Ü ½ Ü ¼ Ü ½
(These formulae remain true in oblique coordinates.)
or
¬
¬
Ü Ý ½
Ü ¼ Ý ¼ ½ ¼ (4.4.3)
Ü ½ Ý ½ ½
6. Slope of line going through points ´Ü ¼ Ý ¼ µ and ´Ü ½ Ý ½ µ: Ý ½ Ý ¼
Ü ½ Ü ¼
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