for the Silent Hunter Series

How to Calculate the Angle to Intercept

How to Calculate the Angle to Intercept 1

Introduction

The purpose of this document is to assist Silent Hunter (I/II/III/IV/V) or other naval simulation game players with the mathematics required to calculate the angle to intercept. In other words, the following calculations will yield how many degrees port or starboard to steer in order to intercept the target. I use it to figure out what course I must travel to reach a position to set up an ambush on that big juicy freighter. I am presenting this with minimal proofs and background theory save for the trigonometric law and unit conversions. If any one finds any mistakes/typos in this, please contact me at via the forums. One final introductory note: The results of these calculations are to find the angle to intercept, or, an angle that gives you a ramming course for lack of a better term. If you follow this course exactly and the target does not change course at all (in the ideal case) then you will run into your target at the end of the course. It is up to you, the captain, to keep track of the range to your target. One could change the calculations to account for some desired distance from the target, but, I find that I would rather just keep an eye on the target. If there is significant request, I’ll rework this sheet to show how to make the calculations for a desired distance to target at the end of the run. DISCLAIMER: I take no responsibility for what the reader of this document does with the information. The information contained herein is provided as is and it is assumed that the reader will NOT use this information for real sailing situations or in any situation that anything or anyone could come to harm. Should someone use this information for any situation in which someone becomes injured or something becomes damaged in any way, the author (Adam A. Tyler) assumes no responsibility. This information is derived by the author and the mathematical formulas are taken from www.wikipedia.com, the images contained within this document are the author’s work. The author releases this work under the GNU license. Basically, don’t use this if your actually sailing a boat or doing anything where someone or something can come to harm. I am not a professional sailor, navigator, torpedoman, mountaineer, etc. Feel free to share this with any one in its entirety and if you find it useful and do share it or use it in some work of your own, drop me a line (my SubSim Radio Room forum id is aatyler), that would be cool. Further, I am not affiliated with the Silent Hunter production company(ies) (Ubisoft / SSI) in any way.

1.1

Acknowledgments

As with all documentation it grows and usually as a result of comments from readers in the community for which the document is written. This document is no different and the following list is my way of thanking the people who have pointed out flaws, typos, and additions that have made it into this document. Pisces (from the SubSim Radio Room forum) Pointed out the need for some extra diagrams and further explanation in the example section, prompting version 1.1.

2

Trigonometry and Units

The only required trigonometric law is the law of sines. The law of sines will yield an unknown angle so long as another angle and two distances (legs of the triangle) are known.

A. A. Tyler

Version: 1.1

1

for the Silent Hunter Series

How to Calculate the Angle to Intercept

Figure 1: Reference Triangle And the law of sines is: sin(A) sin(B) sin(C) = = a b c

(1)

The units of interest are: 1 knot = 1 Kn = .514

3

Nautical Miles km m =1 = 1.852 s hr hr

(2)

Calculating the Intercept Angle

It is important to note that all of these calculations are being made as if the submarine is heading directly towards the target. It is also important to note that this calculation gives you a COURSE CORRECTION, IE an amount of degrees to steer either port or starboard to be heading towards the intercept point. Thus, if you use this formula and your submarine is not heading directly towards the target’s current position, you MUST account for this difference in course. For example, if you are steering 84◦ true and the bearing to the target is 276◦ relative (the course to the target is due north), then once you complete these calculations you must steer however much port or starboard of due north, not your present course. As a final preparatory note, a bit of terminology: Bearing The relative bearing towards the target. The bearing is the angle from your submarine’s present course (true compass bearing) to an imaginary strait directly to the target from your submarine. This can be found from the periscope, crew reports, or by measurement on the navigation map. Course The true compass bearing. The course reported by your course indicator or the course shown on your ship’s compass. For all calculations, I use the visual aid in figure 2. The aid is setup for a starboard angle on the bow and the result of the formula will be a number of degrees to steer to starboard. If you have a port angle on the bow, then the result of the calculation will be the number of degrees that you must steer to port. Again, all of this assumes that your present course is DIRECTLY TOWARDS your target.

A. A. Tyler

Version: 1.1

2

for the Silent Hunter Series

How to Calculate the Angle to Intercept

Figure 2: Positional Reference So, the variables are: AT I ◦ = the course correction, Angle To Intercept AOB◦ = Angle On the Bow Aux◦ = used to calculate time to intercept d0 = initial range to the target in nautical miles

(3)

Star = Speed o f the target in knots Ssm = Speed o f the Submarine in knots t = the time to intercept in hours The time for the intercept to take place (IE target and submarine to arrive at the same spot) is found by:

A. A. Tyler

Version: 1.1

3

for the Silent Hunter Series

How to Calculate the Angle to Intercept

t=

sin(AOB◦ ) · d0 sin(Aux◦ ) · Ssm

(4)

and Aux◦ = 180◦ − AT I ◦ − AOB◦ which is not a necessary part of the angle calculation, but is useful to the player so that he or she doesn’t set the time compression so high that the target is accidentally rammed. Something important to note is that the distance in equation 4 must be expressed in nautical miles as knots are expressed in nautical miles. Though I have listed this equation first, equation 5 must be done prior to equation 4 (in order to know the value of Aux◦ ). The angle to intercept is found by using the information setup in figure 2 and the trigonometric law in equation 1:   sin(AOB◦ ) · Star ◦ (5) AT I = asin Ssm

4

Example calculation

This example will assume that your current course is NOT directly towards the target, and the situation is: Variable Your current course Target’s current course Your speed Target’s speed Angle On the Bow Distance to target

Value 314◦ true 90◦ true 15 knots 3 knots 111◦ starboard a 4,000 meters = 4 kilometers

NOTE: I play Silent Hunter III most often so I am using metric units, if you wish to make the calculation for the time to intercept while playing Silent Hunter IV (or any game with ranges in feet/yards), you will have to convert feet/yards/whatever to nautical miles. This DOES NOT matter for the angle calculation however. a Calculated

on the map using the protractor.

Table 1: Example situation Now, given the situation in table 1 and shown in figure 3, we first convert the range to nautical miles (Silent Hunter III reports ranges in meters & kilometers)

A. A. Tyler

Version: 1.1

4

for the Silent Hunter Series

How to Calculate the Angle to Intercept

Figure 3: Example situation depicted visually 4 = 2.16 nm 1.852 and our angle to intercept is found with equation 5:   sin(111◦ ) · 3 ◦ AT I = asin = 10.76◦ starboard 15 d0,nm =

and since our initial course is not directly to the target, we find the course to the target as 21◦ true. You can find this by using the protractor on the map (if playing Silent Hunter) or by adding the bearing1 to target to your current course and subtracting 360◦ if the result is greater than 360◦ . Notice how the bearing to the target is not given in the above table? I found it by using the protractor on the Navigation Map. The idea is you need to find the angle (in degrees) from your present course (true or compass bearing) to an imaginary strait line from your submarine to the target’s current position. That angle is the relative target bearing. You may also be wondering how to find the angle on the bow if you can’t visually see your target. You can find the estimated angle on the bow by finding the relative bearing from the target’s course to your submarine. Again, on the navigation map (at least in Silent Hunter III/IV/V) use the protractor to find the angle between the target’s current course and an imaginary line from the target’s current position directly to your submarine’s current position2 . That angle is the angle on the bow. Now, AT I ◦ is 10.76 degrees to starboard3 and our Course To Intercept is: Course To Intercept = Course To Target + AT I = 31.76◦ true Now that we know 2 out of the 3 angles shown in figure 2, we find Aux◦ = 180◦ − 111◦ − 10.76◦ = 58.24 and: 1 Relative

bearing, such as from the periscope or found with the protractor on the map. are basically finding the relative bearing to your submarine from the target’s point of view 3 If the Angle On the Bow had been to port, this would have been 10.76 degrees to port instead. IE CourseToTarget − AT I. 2 You

A. A. Tyler

Version: 1.1

5

for the Silent Hunter Series

How to Calculate the Angle to Intercept

t=

5

sin(111) · 2.16 = .16 hours sin(58.24) · 15

Conclusion

On a mathematical note, if you end up with a really wonky value, say infinity for an angle, then you most likely cannot find an intercept course at your present speed. If that is the case, increase your speed (I usually use flank speed) and recalculate. Also, there will be errors in the calculation due to how you round figures and how accurate your measurements are so keep an eye on the target as you are heading to the intercept point. Finally I hope that this document is helpful and provides more enjoyment for you, during your playing time. I am an engineering student (electrical) at USF and enjoy doing more calculations by hand when I play these games as the realism draws me into the time period. However, even though I enjoy realism, I still refuse to give up the exterior camera because when I first started playing the series with Silent Hunter I, I wished that there was an exterior view! Happy hunting to you all!

5.1

Mathematical “checks”

As I said, sometimes, even though a scientific calculator will report an angle correction for your intercept course, it will For those of you interested, it has to do with the domain of the asin function. The asin has a domain of   π beπ invalid. − 2 , 2 , which means that if the absolute value of the argument to the asin function is greater than 1.571(radians)4 , the result (if any) is invalid. In simple terms, given equation 5, the result will be invalid if: sin(AOB◦ ) · Star > 1.571 radians (6) Ssm or sin(AOB◦ ) · Star > 90◦ (7) Ssm

5.2

“doh” checks

I am by no way insinuating that you have an average intelligence equivalent to Homer Simpson (© Fox), but rather that I make “doh” moves all the time. If you perform the calculations and end up with a course that will lead to a position behind your target, you have done something wrong. For example, my seemingly favorite mistake is to convince myself that a starboard angle on the bow is actually a port angle on the bow (dyslexia old good). The gist of this is, if you get something that you can visually see is ridiculous, re-check the math and measurements, if those seem good to go, refer to equations 6 & 7 for further possible mistakes. Another generally good check is that if you are moving faster than your target, you should have a valid intercept course. While this is true, it is also true that if you are already ahead of your target you can be moving slower than your target and still have a valid course to intercept. Basically, if you are behind the target, you need to crank up the speed. I personally like to use these formulas for when I am submerged ahead of my target and I want to find a course to steer to get into torpedo range or a better position.

4 1.571 is a value in radians. Just like a circle has 360 degrees, it also has 2π radians. To convert radians to degrees multiply the radians by 180◦ /π.

A. A. Tyler

Version: 1.1

6