62 CNC Machining
Angles can be added by aligning the degrees, minutes, and seconds
and adding each column separately. When totals for the minutes or seconds
columns add up to 60 or more, subtract 60 (or 120, if appropriate) from that
column, then add 1 (or 2, if appropriate) to the next column to the left (the
higher column).
16° 33′ 14″
5° 17′ 16″
38° 55′ 49″
59° 105′ 79″
-60 -60
45′ 19″
+1 +1
60° 46′ 19″
In the example, the total is 59°105′79″ when the angles are added. Since
79″ equals 1′19″ and 105′ equals 1°45′ , the final answer is 60°46′19″.
When subtracting angles, place the degrees, minutes, and seconds
under each other and subtract the separate columns. If not enough minutes
or seconds exist in the upper number of a column, then borrow 60 from the
next column to the left of it and add it to the insufficient number.
55° 14′ 11″ borrow 60″→ 55° 13′ 71″
–15° 11′ 45″ –15° 11′ 45″
40° 2′ 26″
Since 11″ is smaller than 45″, 60″ must be borrowed from 14′. When
15°11′45″ is subtracted from 55°13′71″, the final answer is 40°2′26″.
Using Trigonometry
Trigonometry is the most valuable mathematical tool used by a
programmer for calculating cutter or tool nose locations. Trigonometric
functions are absolute values derived from the relationships existing
between angles and sides of a right triangle. A function is a magnitude
(size or dimension) that depends upon another magnitude. For example, a
circle’s circumference is a function of its radius, since the circle size depends
on the extent of its radius value.
In the triangle shown in Figure 3-19, A/B is the ratio of two sides and
therefore a function of Angle d. As Angle d increases to the dashed line,
the function will change from A/B to E/B. This shows that the ratio of two
sides of a triangle depends on the size of the angles of the triangle.