Understanding Scan Angles

For some applications, it may be necessary to figure out the scan angle, the screen distance, or the image width. The formulas and table below will make this easy. To use the formulas, you must know two of the following three parameters:

A = Scanning angle in degrees, peak-to-peak. This is also known as the "optical" angle.

D = Throw distance from the scanners to the screen.

W = Projected image width (Of course, the image height should also be the same, so both scanners are tuned to the same angle).

The distance and width must be in the same units, such as feet or meters. The units themselves do not matter. There are two ways to find the unknown (third) parameter. One is by calculations involving the tangent of A (actually, A divided by 2, since the half-angle must be used). The other is by using the distance-to-width ratio listing in the table below. Both methods are described below. Either method gives the same result, so use whichever you feel is easiest.

The diagrams below will help to explain how the calculation formulas were derived.

Diagram of calculating scan angle

The scan angle and the distance to the screen determine the width of the projected image.

Diagram of scan angle

Here's where the scan angle formulas come from. In a right triangle, the tangent of the scan angle equals the length of the opposite side divided by the length of the adjacent side.

Diagram and calculation of determining scan angle

A laser scan is two right triangles, back-to-back. So the right triangle formula is modified to divide or multiply by two at the appropriate point.

Caution: Although the formulas are right you first multiply by two and then divide by two (or vice versa), these operations do not cancel each other out. This is because the tangent is involved and it is non-linear. So don't skip any steps in the calculations.

To find the scan angle A, knowing the width W and distance D

By Calculation: tan(A / 2) = W / (D * 2)

For example, W is 109 meters and D is 150 meters. First, multiply D times 2 to get 300. Then, W (109) divided by 2*D (300) is 0.3633. Next, look in the table below to find the closest angle which has a tangent of 0.3633. This is 20 degrees (at 0.3640). We have just found the half-angle of scanning; the actual peak-to-peak angle is twice this, or 40 degrees. Thus, the desired scan angle A is about 40 degrees.

Using the table: tablewidth@A = (W * 100) / D

For example, W is 109 meters and D is 150 meters. First, multiply W (109) by 100 to get 10,900. Divide this number by the distance D (150) to get 72.6. Finally, look down the table at the "Distance-to-width ratio" column until you find angle A where the table width is closest to 72.6. This is at 40 degrees, where the ratio is 100:72.8. Thus, the desired scan angle A is about 40 degrees.

To find the distance D, knowing the scan angle A, the width W

By calculation: D = W / (tan(A / 2) * 2)

For example, A is 40 degrees and W is 109 meters. First, look in the table below to find the tangent of 40, or 20 degrees; this is 0.3640. Next, multiple 0.3640 times 2. The equation is now W (109) divided by 0.7280, or 149.7. Thus, the desired distance D is about 150 meters.

Using the table: D = (W / tablewidth@A) * 100

For example, A is 40 degrees and W is 109 meters. First, look in the table below and find 40 degrees. The distance-to-width ratio is given as 100:72.8. Using the formula, divide W (109) by 72.8 (the table width at A), to get 1.497. Finally, multiply this number by 100 to find 149.7 meters, the desired distance D.

To find the width W, knowing the scan angle A and the distance D

By calculation: W = D * (tan(A / 2) * 2)

For example, A is 40 degrees and D is 150 meters. First, look in the table below to find the tangent of half of 40, or 20 degrees; this is 0.3640. Next, multiple 0.3640 times 2. The equation is now D (150) times 0.7280, or 109.2. Thus, the desired width W is about 109 degrees.

Using the table: tablewidth@A*W = (D/100)

For example, A is 40 degrees and D is 150 meters. First, look in the table below and find 40 degrees. The distance-to-width ratio is given as 100:72.8. Therefore, the table width at A is 72.8. Next, divide D (150) by 100 to get 1.5. Finally, multiple 72.8 by 1.5 to find 109 meters, the desired width W.

Angle data table

Angle  Tangent Distance-to-width ratio (if distance is 100 units, width is...) Notes
1 0.0175 100 : 1.7
2 0.0349 100 : 3.5
3 0.0524 100 : 5.2
4 0.0699 100 : 7.0
5 0.0875 100 : 8.7
6 0.1051 100 : 10.5
7 0.1228 100 : 12.2
8 0.1405 100 : 14.0 ILDA 30K test pattern size
9 0.1584 100 : 15.7
10 0.1763 100 : 17.5
11 0.1944 100 : 19.3
12 0.2126 100 : 21.0
13 0.2309 100 : 22.8
14 0.2493 100 : 24.6 Distance is 4 times width
15 0.2679 100 : 26.3
16 0.2869 100 : 28.1
17 0.3057 100 : 29.9
18 0.3249 100 : 31.6
19 0.3443 100 : 33.5 Distance is 3 times width
20 0.3640 100 : 35.3
21 0.3839 100 : 37.1
22 0.4040 100 : 38.9
23 0.4245 100 : 40.7
24 0.4452 100 : 42.5
25 0.4663 100 : 44.3
26 0.4877 100 : 46.2
27 0.5095 100 : 48.0
28 0.5317 100 : 49.9 Distance is 2 times width
29 0.5543 100 : 51.7
30 0.5774 100 : 53.6
31 0.6009 100 : 55.5
32 0.6249 100 : 57.3
33 0.6494 100 : 59.2
34 0.6745 100 : 61.1
35 0.7002 100 : 63.1
36 0.7265 100 : 65.0
37 0.7536 100 : 66.9
38 0.7813 100 : 68.9
39 0.8098 100 : 70.8
40 0.8391 100 : 72.8 This is the nominal limit of G-120 scanners
41 0.8693 100 : 74.8
42 0.9004 100 : 76.8
43 0.9325 100 : 78.8
44 0.9657 100 : 80.8
45 1.0000 100 : 82.8
46 1.0355 100 : 84.9
47 1.0724 100 : 87.0
48 1.1106 100 : 89.0
49 1.1504 100 : 91.1
50 1.1918 100 : 93.3
51 1.2349 100 : 95.4
52 1.2799 100 : 97.5
53 1.3270 100 : 99.7 1:1 ratio -- distance equals width
54 1.3764 100 : 101.9
55 1.4281 100 : 104.1
56 1.4826 100 : 106.3
57 1.5299 100 : 108.6
58 1.6003 100 : 110.9
59 1.6643 100 : 113.2
60 1.7321 100 : 115.5 Normal limit of CTI 6800/6210 scanners
61 1.8040 100 : 117.9
62 1.8807 100 : 120.2
63 1.9626 100 : 122.6
64 2.0503 100 : 125.0
65 2.1445 100 : 127.4
66 2.2460 100 : 129.9
67 2.3559 100 : 132.4
68 2.4751 100 : 134.9
69 2.6051 100 : 137.5
70 2.7475 100 : 140.0
71 2.9042 100 : 142.7
72 3.0777 100 : 145.3
73 3.2799 100 : 148.0
74 3.4874 100 : 150.7
75 3.7321 100 : 153.5
76 4.0108 100 : 156.3
77 4.3315 100 : 159.1
78 4.7046 100 : 162.0
79 5.1446 100 : 164.9
80 5.6713 100 : 167.8 Absolute limit of CTI 6800/6210 scanners...
81 6.3138 100 : 170.8 ...remaining entries are for completeness
82 7.1154 100 : 173.9
83 8.1443 100 : 176.9
84 9.5144 100 : 180.1
85 11.430 100 : 183.3
86 14.300 100 : 186.5
87 19.081 100 : 189.8
88 28.636 100 : 193.1
89 57.290 100 : 196.5
90 --- 100 : 200.0
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