Curious about a math problem I can't put my head around.

[Flash]

Give a horizontal view of 90 degrees and a vertical view of 60 degrees one can see about 23% of the surface area of a sphere while sitting in the center. Given the universal orientation a ship can have and factoring in both forward or rearward velocity does that mean the possiblity of having the FPM in view is half of that or 11.5%.

[DaveK]

I'm not very mathsy but if you sit in the centre and look out . . .

1: the 90 degree horizontal view (if you had 180 degree vertical view) would let you see 1/4 of the of the sphere's surface (360/90) which is 25%

2: a 60 degree vertical view (again if you had 180 degree horizontal view) should let you see 1/6th of the surface (360/60) which is about 17%

combining the two should let you see 1/6th of 1/4 (or 17% of 25%) of the surface which is 1/24th (or about 4%)

. . . but as I said, maths and spherical geometry aren't my strong points!

[Capt_Caveman]

I dont think you can break it down to a single percentage and be accurate on both planes.

you will still have to specify vert or horz

example 25% hor and 17% vert.


[Edited on 3-18-2013 by capt_cronic]

[DaveK]

Did the theory (first) and then did the practical!

If you have a 90 degree view horizontally you are looking at a wedge of the sphere that covers 90 degrees out of 360, ie one quarter of the sphere surface - or should I go back to doing science?

If you have a 90 degree view vertically you are also looking at a wedge of the sphere that covers 90 degrees out of 360, ie one quarter of the sphere surface

If you have a view that is 90 degrees horizontally and 90 degrees vertically, the horizontal view cuts a "slot" out of the vertical "wedge" and you are left with a square pyramid (with a slightly convex base!) The length of the sides if the base are one quarter of the circumference of the sphere.

The practical - I've just done this with an apple and a knife.

The apple had a radius of 35mm and the base of the "view" as it cut the surface had sides 54mm long (pi x D/4 = 220/4): surface area of the apple is (well was because I've eaten it ) 15396 square mm (4 x pi x r^2) and the area of the "view" as it intersected the surface was 2916 square mm so it covered 20% of the surface of the apple! And 1/4 x 1/4 does NOT = 1/5!!

Hmmmm . . . So much for having a doctorate in science. This Non-Euclidean geometry is too tricky to get my head round! Yep, I should go back to doing the science and leave the sums to the brighter ones!

Oh how I miss my brain!

[Flash]

During lunch I asked one of our CAD guys here at work who uses Solid Works and this is what he came up with using a sphere with 1 as its radius.

Surface area of the sphere 12.56637061
Surface area of the slice 1.4454614

1.44545614/1.44545614 = 8.6937

so 1 to 8.6937 or about 11.5% of the slice over the whole sphere

So the next question:
Does that mean if a pilot travels equal time in all orientations with any given velocity(except 0) will that be how often the FPM is in view or am I missing something.

I don't know why I obsess over these kind of things. I was once at a friends house who had a pool with two hexagon island pillars at the shallow end. Guess who had to figure out how much water was in the pool

Geez




[Edited on 3-18-2013 by Flash]

[Maarschalk]

I'm curious!, What is FPM?

I tried Google-ing it!, Maybe I should try Bing instead!....
What I got does not make sense!

1. FPM=Fast Process Manager
2. FPM=Fellowship of Postgraduate Medicine
3. FPM=Family Practice Management
4. FPM=Feet Per Minute
5. FPM=Fantastic Plastic Machine
6. FPM=Frames Per Minute
7. FPM=Floral Park Memorial
8. FPM=Figaro's Password Manager
9. FPM=Federal Program Monitoring
10. FPM=Financial Planning Ministry
11. FPM=Fan Powered Modules
12. FPM=Facilities Planning Management
13. FPM=Financial Projection Model
14. FPM=Flexible Packet Matching
15. FPM=Flexible Packet Manager
16. FPM=Fast Page Mode
17. FPM=Field Procedures Manual
18. FPM=Field Plotting Module
19. FPM=Fundamentals of Project Management
20. FPM=Fiber Optic Power Meter
21. FPM=Filling Program Manager
........etc.....etc....Here is where I gave up!

I guess it is all relative to your interpretation of FPM!......:P

[Edited on 3-19-2013 by Maarschalk]

[Dingo]

I'm curious!, What is FPM?

I tried Google-ing it!, Maybe I should try Bing instead!....
What I got does not make sense!

1. FPM=Fast Process Manager
2. FPM=Fellowship of Postgraduate Medicine
3. FPM=Family Practice Management
4. FPM=Feet Per Minute
5. FPM=Fantastic Plastic Machine
6. FPM=Frames Per Minute
7. FPM=Floral Park Memorial
8. FPM=Figaro's Password Manager
9. FPM=Federal Program Monitoring
10. FPM=Financial Planning Ministry
11. FPM=Fan Powered Modules
12. FPM=Facilities Planning Management
13. FPM=Financial Projection Model
14. FPM=Flexible Packet Matching
15. FPM=Flexible Packet Matching
16. FPM=Fast Page Mode
17. FPM=Field Procedures Manual
18. FPM=Flexible Packet Matching
19. FPM=Fundamentals of Project Management
20. FPM=Fiber Optic Power Meter
21. FPM=Filling Program Manager
........etc.....etc....Here is where I gave up!

I guess it is all relative to your interpretation of FPM!......:P

Imma go with number 7. Floral Park Memorial. Nice work Maars.

[Maarschalk]

I did not get the answer to my question though and I'm Still curious about what FPM is so I googled it some more and got a more extensive list:

1. FPM Feet Per Minute
2. FPM Fast Page Mode (RAM)
3. FPM Family Practice Management
4. FPM Fantastic Plastic Machine (band)
5. FPM Free Patriotic Movement (Lebanon)
6. FPM Federal Personnel Manual
7. FPM Flexible Packet Matching (Cisco)
8. FPM Financial Performance Management
9. FPM Free People's Movement (Marxist party)
10. FPM Fluoroelastomer
11. FPM Foot Per Minute
12. FPM Fine Particulate Matter (diameter less than or equal to 2.5 micrometers)
13. FPM Faculty of Pharmaceutical Medicine
14. FPM Feet Per Mile (elevation)
15. FPM Financial Planning Manager (job title)
16. FPM Forest Pest Management (various locations)
17. FPM Flood Plain Management (USACE)
18. FPM Frames Per Minute
19. FPM Flight Path Marker
20. FPM Faculté Polytechnique de Mons (Faculty of Engineering, Mons, Belgium)
21. FPM Front Populaire Mauritanien (Mauritanian Popular Front)
22. FPM Frequency Per Million
23. FPM Family Protection Ministries (Lincoln, CA)
24. FPM Front Panel Module
25. FPM Field Project Manager
26. FPM Fragile Porcelain Mice (band)
27. FPM Frags Per Minute (gaming)
28. FPM Floral Park Memorial (High School; Floral Park, NY)
29. FPM Fast Packet Multiplexing
30. FPM Fuel Pump Module
31. FPM Foro Parlamentario Mundial (Spanish: World Parliamentary Forum)
32. FPM Flicker Phase Modulation (ANSI)
33. FPM Fiber Placement Machine (NASA)
34. FPM Flight Performance Model
35. FPM Fine Point Mode
36. FPM Fault and Performance Manager (Avaya Telephony Product used for monitoring voice systems)
37. FPM Frequency Position Modulation
38. FPM Filarmónica Portuguesa de Montreal (Portugese)
39. FPM Ford Partnership Management
40. FPM Field Production Management
41. FPM Forward Path Matrix
42. FPM Fan Power Module
43. FPM Fast Parallel Multiplier
44. FPM Fabry-Perot Modulator
45. FPM Force Packaging Methodology
46. FPM Fookwit Pink Minkis (gaming)
47. FPM Full-Population Model
48. FPM Field Point Module
49. FPM FIFO and Priority Manager (US Postal Service)
50. FPM Fault-Prone Module
51. FPM Freaks Per Minute (film profanity measure; polite form)
52. FPM Flexible Path Manager
53. FPM Fluoric-Peroxide Mixture
54. FPM Fill-Pattern Monitor (physics)
55. FPM Foro de Poblaciones Migrantes (Spanish: Migrant Populations Forum)
56. FPM Fund Portfolio Manager
57. FPM Fuzzy Pattern Matching (classification method)
58. FPM Friendly Psychics Music (Ohio and Colorado)
59. FPM FastCGI (Common Gateway Interface) Process Manager (software)

I'm sure the answer must be relatively in here some where!.......

But if the pilot uses Track IR and looks in the direction of the FPM equal the time for each velocity direction then would not the FPM be visible for 50% of the time?....:P

[Edited on 3-19-2013 by Maarschalk]

[Rubber Chicken]

In this case FPM = Flight Path Marker. And Flash's math problem is assuming that the pilot would be strictly using the 'forward only' view of the cockpit / third person view.

Hope this helps.

[Markspace]

Having been away for a few years now, but this caught my eye...

All in all the Surface of the sphere is given by a double integral:
multiplied by r^2 where r is the radius.

The surface for an angle that is 90 (=pi/2 or π/2) horizontally and 60 degrees vertically (or pi/3 rad) should be:

, again multiplied by r^2.


If you divide the full surface of a sphere which is 4*pi*r^2 by the surface of your solid angle which should be (pi/4)*r^2 you 'd get 1/16.

So the percentage of this solid angle to the whole sphere surface is 6.25 %.


Always assuming you move where you look at, your possibility of "accidentaly" heading towards the right flight path is 12.5%. Unless my english fail me and I can't tell what exactly you need. Hope it helps!

Hm something feels wrong still.... Percentage seems too small, but take into account the following. Notice how one of the two parts of the integral goes from 0 to pi/3. That means we were taking the angle from the top of the sphere to pi/3, which results in a trianle shaped surface. (Integrals work like scanning beams that scan from one angle to another, starting vertically from the top, horizontally it matters not where you start.)

So depending on whether you start counting the vertical angle, from the top of the sphere or somewhere in between, the surface would be different. The above calculations would be correct for a slice that goes from the top to 60 degrees vertically and 90 degrees horizontally.

If we 'd calculate for vertical angle starting from pi/3 to 2pi/3 we 'd have a different surface ratio, since the integral now would be:


which is pi/2. So by dividing again we get 1/8th = 12,5% which is the maximum possible ratio. So in this case we would indeed have a better probability (the real probability since the shape of the ship view is this case and not the previous one which would result in a triangular shaped view).

To conclude, using the original shape, at first we calculated Surface 1 (triangular), but in second case we calculated 2, which more resembles a ship-human view. What was calculated by the OP's CAD people was the angles between shapes 1 and 2, which is somewhere in between, which is why the surface ratio is a bit smaller. I am guessing vertically they went from 30 to 90 degrees... not wrong since ship view could be of that shape.






[Edited on 23-5-2013 by Markspace]