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The problem of computing the gyro angle setting is a trigonometry problem that is simplified by first considering the calculation of the deflection angle, which ignores torpedo ballistics and parallax.<ref name = Deflection>{{harvnb|COMSUBATL|1950|loc=§ "Definitions", p. 1-2}}</ref>
For small gyro angles, {{math|''θ''<sub>Gyro</sub> ≈ ''θ''<sub>Bearing</sub> − ''θ''<sub>Deflection</sub>}}. A direct application of the [[law of sines]] to Figure 3 produces Equation {{EquationNote|1}}.
{{NumBlk|:|<math>\frac{\left \Vert v_{Target} \right \| }{ \sin(\theta_{Deflection}) } = \frac{\left \Vert v_{Torpedo} \right \| }{ \sin(\theta_{Bow}) } </math>
▲<math>\frac{\left \Vert v_{Target} \right \| }{ \sin(\theta_{Deflection}) } = \frac{\left \Vert v_{Torpedo} \right \| }{ \sin(\theta_{Bow}) } </math> </div>
where
:{{math|''v''<sub>Target</sub>}} is the velocity of the target.
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:{{math|''θ''<sub>Deflection</sub>}} is the angle of the torpedo course relative to the periscope line of sight.
Range plays no role in Equation {{EquationNote|1}}, which is true as long as the three assumptions are met. In fact, Equation {{EquationNote|1}} is the same equation solved by the mechanical sights of [http://www.history.navy.mil/photos/images/h41000/h41761.jpg steerable torpedo tubes] used on surface ships during World War I and World War II. Torpedo launches from steerable torpedo tubes meet the three stated assumptions well. However, an accurate torpedo launch from a submarine requires parallax and torpedo ballistic corrections when gyro angles are large. These corrections require knowing range accurately. When the target range was not known, torpedo launches requiring large gyro angles were not recommended.<ref name = AccurateRange>{{harvnb|COMSUBATL|1950|loc=§ "Theory of Approach and Attack" p. 8-10}}</ref>
Equation 1 is frequently modified to substitute track angle for deflection angle (track angle is defined in Figure 2, {{math|1=''θ''<sub>Track</sub>=''θ''<sub>Bow</sub>+''θ''<sub>Deflection</sub>}}). This modification is illustrated with Equation 2.▼
<math>\frac{\left \Vert v_{Target} \right \| }{ \sin(\theta_{Deflection}) } = \frac{\left \Vert v_{Torpedo} \right \| }{ \sin(\theta_{Track}-\theta_{Deflection})}</math>▼
▲Equation {{EquationNote|1}} is frequently modified to substitute track angle for deflection angle (track angle is defined in Figure 2, {{math|1=''θ''<sub>Track</sub>=''θ''<sub>Bow</sub>+''θ''<sub>Deflection</sub>}}). This modification is illustrated with Equation {{EquationNote|2}}.
▲{{NumBlk|:|<math>\frac{\left \Vert v_{Target} \right \| }{ \sin(\theta_{Deflection}) } = \frac{\left \Vert v_{Torpedo} \right \| }{ \sin(\theta_{Track}-\theta_{Deflection})}</math>|{{EquationRef|2}}}}
where
[[Image:DeflectionAngle.png|noframe|thumb|Figure 4: Deflection angle versus track angle and target speed ({{math|1=''θ''<sub>Gyro</sub> = 0°}}).]]
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A number of publications<ref name = OptimumTrackAngle>{{harvnb|COMSUBATL|1950|loc=§ "Theory of Approach and Attack", p. 8-9}}</ref><ref name="Clear2">{{harvnb|O'Kane|1977|p=303}}</ref> state the optimum torpedo track angle as 110° for a Mk 14 (46 knot weapon). Figure 4 shows a plot of the deflection angle versus track angle when the gyro angle is 0° (''i.e''., {{math|1=''θ''<sub>Deflection</sub>=''θ''<sub>Bearing</sub>}}).<ref name="track">Most work on computing intercept angles is done using track angle as a variable. This is because track angle is a strictly a function of the target's course and speed along with the torpedo's course and speed. It removes the complexities associated with parallax and torpedo ballistics.</ref> Optimum track angle is defined as the point of minimum deflection angle sensitivity to track angle errors for a given target speed. This minimum occurs at the points of zero slope on the curves in Figure 4 (these points are marked by small triangles).
The curves show the solutions of
{{harvnb|COMSUBATL|1950|loc=§ "Duties of the Fire Control Party", p. 5-25}}</ref>
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