Graphics Reference
In-Depth Information
Table 3.1 Parameter, Arc Length Pairs
Index
Parametric Value (
V
)
Arc Length (
G
)
0
0.00
0.000
1
0.05
0.080
2
0.10
0.150
3
0.15
0.230
4
0.20
0.320
5
0.25
0.400
6
0.30
0.500
7
0.35
0.600
8
0.40
0.720
9
0.45
0.800
10
0.50
0.860
11
0.55
0.900
12
0.60
0.920
13
0.65
0.932
14
0.70
0.944
15
0.75
0.959
16
0.80
0.972
17
0.85
0.984
18
0.90
0.994
19
0.95
0.998
20
1.00
1.000
For example, consider the table of values, V , for u and corresponding values of the function G in
Table 3.1 .
As a simple example of how such a table could be used, consider the case in which the user wants to
know the distance (arc length) from the beginning of the curve to the point on the curve corresponding
to a parametric value of 0.73. The parametric entry closest to 0.73 must be located in the table. Because
the parametric entries are evenly spaced, the location in the table of the closest entry to the given value
can be determined by direct calculation. Using Table 3.1 , the index, i , is determined by Equation 3.8
using the distance between parametric entries, d ¼ 0.05 in this case, and the given parametric value, v ¼
0.73 in this case.
j
k
v
d þ
0
:
73
i ¼
0
:
5
¼
05 þ
0
:
5
¼
15
(3.8)
0
:
A crude approximation to the arc length is obtained by using the arc length entry located in the table
at index 15, that is, 0.959. A better approximation can be obtained by interpolating between the arc
lengths corresponding to entries on either side of the given parametric value. In this case, the index
of the largest parametric entry that is less than the given value is desired ( Eq. 3.9 ) .
¼
jk
¼
v
d
0
:
73
i ¼
14
(3.9)
:
0
05
 
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