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Trigonometry Functions (edit)

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This is a repository of functions used in trigonometry - namely Sine, Cosine, Tangent, Arcsine, Arccosine, and Arctangent.  
This is a repository of functions used in trigonometry - namely Sine, Cosine, Tangent, Arcsine, Arccosine, and Arctangent.


There are a number of methods for obtaining these, some more accurate and others faster. Here is one for Sine, Cosine, and Tangent.
There are many methods of obtaining any or all of these, and each has its own benefits and drawbacks in terms of length, efficiency, and speed. This page is a work in progress - please add any scripts you have for approximating these functions. Please remember to comment them thoroughly and to use generic variable names.
 
These are [[:Category:Stage_Functions|Stage Functions]], and as such must be created as [[Quest Stages Tab|Quest Stages]] of a [[Global Script]] and called as a function. These examples assume a [[Quest]] called Wiki with the variables declared. Therefore, you must modify this code to include the name of your [[Quest]] (assuming it is not Wiki, which it shouldn't be). Any contributors, please read that article and write your code in that format.
 
= Formulae =
[[Image:Triangle.png|160px|right|Right triangle]]
 
'''''A + B + C = 180°'''''
 
'''''A + B = 90°'''''
 
'''''a² + b² = c²'''''
 
'''''sin(A) = a / c'''''
 
'''''cos(A) = b / c'''''
 
'''''tan(A) = a / b'''''
 
'''''tan(A) = sin(A) / cos(A)'''''
 
 
 
 
 
= Sine, Cosine, and Tangent =
 
You can use a Taylor series to approximate the value of sine, cosine and tangent:
 
 
'''''sin(x) = x^1/1! - x^3/3! + x^5/5! - x^7/7! + ...'''''
 
'''''cos(x) = x^0/0! - x^2/2! + x^4/4! - x^6/6! + ...'''''
 
'''''tan(x) = x^1/1! + x^3/3! + x^5/5! + x^7/7! + ...'''''
 
== Galsiah Version ==
 
Galsiah says that this version of sine/cosine approximation is faster and more accurate than the Taylor series. (For source, see [http://www.convertit.com/Go/EducationPlanet/Reference/AMS55.ASP?Res=150&Page=76 Handbook of Mathematical Functions].)
 
This [[:Category:Stage Functions|function]] requires the following variables to be declared in your [[Global Script]]:
float ang ;(the measure of the angle)
float n ;(used internally)
float t2 ;(used internally)
float sin ;(sine of the angle)
float cos ;(cosine of the angle)
float tan ;(tangent of the angle)
You may rename these if you wish, but that means changing the code to match. All of them must be floats.
 
<pre>;script originally by Galsiah
 
if Wiki.ang < 0
  set Wiki.ang to (Wiki.ang + 360)
elseif Wiki.ang >= 360
  set Wiki.ang to (Wiki.ang - 360)
endif
set Wiki.ang to (Wiki.ang/Wiki.rad)
 
set Wiki.n to 1
if (Wiki.ang > 4.7123)
    Set Wiki.ang to (Wiki.ang - 6.2832)
elseif ( Wiki.ang > 1.5708 )
    Set Wiki.ang to (Wiki.ang - 3.1416)
    Set Wiki.n to -1
endif
 
set Wiki.t2 to (Wiki.ang * Wiki.ang)
set Wiki.sin to Wiki.n*(Wiki.ang*(1 - Wiki.t2*0.16605 + 0.00761*Wiki.t2*Wiki.t2))
set Wiki.cos to Wiki.n*(1 - Wiki.t2*0.4967 + 0.03705*Wiki.t2*Wiki.t2)
set Wiki.tan to (Wiki.sin/Wiki.cos)</pre>
 
For working example (with somewhat simpler code), see [[Summon Object]].
 
== Taylor Series Variant 1 ==
 
This [[:Category:Stage Functions|function]] requires the following variables to be declared in your [[Global Script]]:
float ang ;(the measure of the angle)
float rad ;(used internally)
float exp ;(used internally)
float sin ;(the sine of the angle)
float cos ;(the cosine of the angle)
float tan ;(the tangent of the angle)
You may rename these if you wish, but that means changing the code, which is not advised, as there are a lot of variables here. All of them must be floats.


ScriptName SineCosineTangent
  ; script originally by Galerion
  ; script originally by Galerion
float ang
float rad
float cos
float sin
float tan
float exp
Begin {appropriate blocktype}
; how one sets ang depends on implementation. Here it is the player's Z rotation
  set ang to Player.GetAngle Z
  ; normalize angle
  ; normalize angle
   if ( ang < -180 )
   if ( Wiki.ang < -180 )
     set ang to ( ang + 360 )
     set Wiki.ang to ( Wiki.ang + 360 )
   elseif ( ang > 180 )
   elseif ( Wiki.ang > 180 )
     set ang to ( ang - 360 )
     set Wiki.ang to ( Wiki.ang - 360 )
   endif
   endif
   
   
  ; approximate sine and cosine of the angle
  ; approximate sine and cosine of the angle
   set rad to ( ang * 0.0174533 )
   set Wiki.rad to ( Wiki.ang * 0.0174533 )
   set cos to 1
   set Wiki.cos to 1
   set exp to rad                     ; 1st
   set Wiki.exp to Wiki.rad                         ; 1st
   set sin to exp
   set Wiki.sin to Wiki.exp
   set exp to ( exp * rad )          ; 2nd
   set Wiki.exp to ( Wiki.exp * Wiki.rad )          ; 2nd
   set cos to ( cos - exp / 2 )
   set Wiki.cos to ( Wiki.cos - Wiki.exp / 2 )
   set exp to ( exp * rad )          ; 3rd
   set Wiki.exp to ( Wiki.exp * Wiki.rad )          ; 3rd
   set sin to ( sin - exp / 6 )
   set Wiki.sin to ( Wiki.sin - Wiki.exp / 6 )
   set exp to ( exp * rad )          ; 4th
   set Wiki.exp to ( Wiki.exp * Wiki.rad )          ; 4th
   set cos to ( cos + exp / 24 )
   set Wiki.cos to ( Wiki.cos + Wiki.exp / 24 )
   set exp to ( exp * rad )          ; 5th
   set Wiki.exp to ( Wiki.exp * Wiki.rad )          ; 5th
   set sin to ( sin + exp / 120 )
   set Wiki.sin to ( Wiki.sin + Wiki.exp / 120 )
   set exp to ( exp * rad )          ; 6th
   set Wiki.exp to ( Wiki.exp * Wiki.rad )          ; 6th
   set cos to ( cos - exp / 720 )
   set Wiki.cos to ( Wiki.cos - Wiki.exp / 720 )
   set exp to ( exp * rad )          ; 7th
   set Wiki.exp to ( Wiki.exp * Wiki.rad )          ; 7th
   set sin to ( sin - exp / 5040 )
   set Wiki.sin to ( Wiki.sin - Wiki.exp / 5040 )
   set exp to ( exp * rad )          ; 8th
   set Wiki.exp to ( Wiki.exp * Wiki.rad )          ; 8th
   set cos to ( cos + exp / 40320 )
   set Wiki.cos to ( Wiki.cos + Wiki.exp / 40320 )
   set exp to ( exp * rad )          ; 9th
   set Wiki.exp to ( Wiki.exp * Wiki.rad )          ; 9th
   set sin to ( sin + exp / 362880 )
   set Wiki.sin to ( Wiki.sin + Wiki.exp / 362880 )
   
   
  ; get tangent
  ; get tangent
   set tan to ( sin / cos )
   set Wiki.tan to ( Wiki.sin / Wiki.cos )
   
   
  End
  End
This script is more accurate than it needs to be. Galerion suggests that only the first seven steps are necessary.


This can be placed within your own code or run as a separate script and checked by yours (though the former is probably easier). The sin, cos, and tan variables are your outputs, ang is your input.
== Taylor Series Variant 2 ==


This script is more accurate than it needs to be. Galerion suggests that only the first seven steps are necessary.
This [[:Category:Stage Functions|function]] requires the following variables to be declared in your [[Global Script]]:
float angle ;(measure of the angle)
float t1 ;(used internally)
float t2 ;(used internally)
float t5 ;(used internally)
float t6 ;(used internally)
float sina  ;(sine of the angle)
float cosa  ;(cosine of the angle)
float tana  ;(tangent of the angle)
All must be floats. Though they can be renamed, doing so means changing the function code to match.
; script originally by JOG
; normalize the angle
set Wiki.angle to Wiki.angle + 360*(Wiki.angle < -180) - 360*(Wiki.angle > 180)
; precalculate powers of "angle"
set Wiki.t1 to Wiki.angle / 57.29577951
set Wiki.t2 to Wiki.t1*Wiki.t1
set Wiki.t5 to Wiki.t2*Wiki.t2*Wiki.t1
set Wiki.t6 to Wiki.t5*Wiki.t1
set Wiki.sina to Wiki.t1 - Wiki.t1*Wiki.t2/6 + Wiki.t5/120 - Wiki.t5*t2/5040
set Wiki.sina to Wiki.sina + Wiki.t6*Wiki.t2*Wiki.t1/362880
;note that the previous two lines have been split in order to display properly
;they should be on one line in your code
set Wiki.cosa to 1 - Wiki.t2/2 + Wiki.t2*Wiki.t2/24 - Wiki.t6/720 + Wiki.t6*Wiki.t2/40320
set Wiki.tana to Wiki.sina/Wiki.cosa
 
= Arcsine =
 
== Arcsine ==
 
Sin=Opp/Hyp - The arcsine is the angle between two points, when the distance is the hypotenuse of the triangle and the opposing side is either the x- or the y-difference (whatever is shorter). To get the z-angle (left/right) between two objects in Oblivion you can use the function [[GetHeadingAngle]]. But when you need the x-angle (up/down) you still have to calculate the arcsin.
 
== Abramowitz/Stegun Approximation ==
Abramowitz and Stegun found this polynomal approximation for arcsine:
 
'''''a0=1.5707288 / a1=-0.2121144 / a2=0.0742610 / a3=-0.0187293'''''
 
'''''arcsin(x) = 180/Pi * (pi/2 - sqrt(1 - x) * (a0 + a1*x + a2*x^2 + a3*x^3))'''''
 
This [[:Category:Stage Functions|function]] requires the following variables to be declared in your [[Global Script]]:
float x ;(the sine of the angle)
float n ;(used internally)
float arcsinx ;(the arcsine, that is, the measure of the angle)
These may be renamed, but doing so means changing them in the code.
 
; script originally by JOG
set Wiki.x to (player.getpos z - NPC.getpos z)/player.getdistance NPC
set Wiki.n to 1 - Wiki.x ; arcsinx = SquareRoot(1-x)
set Wiki.arcsinx to Wiki.n/2
set Wiki.arcsinx to (Wiki.arcsinx+(Wiki.n/Wiki.arcsinx))/2
set Wiki.arcsinx to (Wiki.arcsinx+(Wiki.n/Wiki.arcsinx))/2
set Wiki.arcsinx to (Wiki.arcsinx+(Wiki.n/Wiki.arcsinx))/2
set Wiki.n to (Wiki.arcsinx+(Wiki.n/Wiki.arcsinx))/2
set Wiki.arcsinx to 57.2957795*(1.5707963-Wiki.n*(1.5707288-0.2121144*Wiki.x+0.0742610*Wiki.x*Wiki.x-0.0187293*Wiki.x*Wiki.x*Wiki.x))
;note that this does not display properly - be sure to get the entire line with copy and paste
 
This approximation is very accurate (99.9% - 99.9999%) for angles of 45 degrees and higher. For smaller angles you can get a higher precision when you use a taylor series with 15+ iterations.
 
== Approximation by Taylor Series ==
 
According to [http://www.wikipedia.org Wikipedia], arcsine can be found with this series:
 
[[Image:Arcsine.png]]
 
This can be approximated by Oblivion using a [[:Category:Staged Functions|function]]. This function requires the following variables be declared in your [[Global Script]]:
float z ;(the sine of the angle)
float z3 ;(used internally)
float z5 ;(used internally)
float z7 ;(used internally)
float arcsinz ;(the arcsine, that is, the measure of the angle)
These can be renamed, but must also be changed in the code.
;script originally by DragoonWraith
set Wiki.z3 to (Wiki.z*Wiki.z*Wiki.z)
set Wiki.z5 to (Wiki.z3*Wiki.z*Wiki.z)
set Wiki.z7 to (Wiki.z5*Wiki.z*Wiki.z)
set Wiki.arcsinz to (Wiki.z+(1/2)*(Wiki.z3/3)+(3/8)*(Wiki.z5/5)+(15/48)*(Wiki.z7/7))
More terms will increase accuracy. The larger the angle, the more terms are needed to get acceptable accuracy. As a rough guide use 3 terms for every 10° to get an accuracy of at least 1 decimal digit.
 
= Arccosine =
'''Arccos z = (PI / 2) - arcsin(z).'''
 
See above for method of obtaining arcsin. This may be useful when using the Law of Cosines to find the measure of an unknown angle, given three known sides.
 
<b><u>Law of Cosines</u></b>
 
<i>a</i><sup>2</sup> = <i>b</i><sup>2</sup> + <i>c</i><sup>2</sup> - 2<i>bc</i> cos<i>A</i>
 
<i>b</i><sup>2</sup> = <i>a</i><sup>2</sup> + <i>c</i><sup>2</sup> - 2<i>bc</i> cos<i>B</i>
 
j<i>c</i><sup>2</sup> = <i>a</i><sup>2</sup> + <i>b</i><sup>2</sup> - 2<i>ab</i> cos<i>C</i>
 
That is, the square of one side is equal to the sum of the other squares of the other two, minus twice the product of the other two sides with the cosine of the angle opposite that side.
 
Simple algebra gives the following equations for any given angle in a triangle, if you know the lengths of all three sides.
 
[[Image:LawOfCosinesForAngle.png]]
 
= Arctangent =
'''Arctan z = z - (z^3/3) + (z^5/5) - (z^7/7) +...'''
 
This is very similar to the Taylor Expansion method of generating Arcsine, but without the extra multipliers, and with alternating signs.
 
This function requires the following variables be declared in your [[Global Script]]:
float z ;(the tangent of the angle)
float z3 ;(used internally)
float z5 ;(used internally)
float z7 ;(used internally)
float arctanz ;(the arctangent, that is, the measure of the angle)
These can be renamed, but must also be changed in the code.
;modified from arcsine script supplied by DragoonWraith
  set Wiki.z3 to ( Wiki.z * Wiki.z * Wiki.z )
  set Wiki.z5 to ( Wiki.z3 * Wiki.z * Wiki.z )
  set Wiki.z7 to ( Wiki.z5 * Wiki.z * Wiki.z )
  set Wiki.arctanz to ( Wiki.z - (Wiki.z3/3) + (Wiki.z5/5) - (Wiki.z7/7) )
 
Again, more terms equals more accuracy.
 
= See Also =
 
[http://www.elderscrolls.com/forums/index.php?showtopic=374963 Elder Scrolls Forums - Trigonometry Function Workarounds?]
 
[http://www.elderscrolls.com/forums/index.php?showtopic=410923 Elder Scrolls Forums - Is there an actor for the target you are attacking currently?]


'''Note''': this is a work in progress. As stated, there are a number of methods for these. Please add any if you have them. Please make them well-commented and use generically named variables.
[[Ingame Functions|Ingame Functions (example uses simple (3-line) sin/cos script functions: example for player x-hair)]]


[[Category: Extra_Math_Functions]]
[[Category: Stage Functions]]
[[Category: Useful_Code]]
[[Category: Math Functions (CS 1.0)]]
Retrieved from "https://cs.uesp.net/wiki/Special:MobileDiff/32329...32371"