Trigonometry Functions
This is a repository of functions used in trigonometry - namely Sine, Cosine, Tangent, Arcsine, Arccosine, and Arctangent.
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.
Formulae
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! + ...
Taylor Series Variant 1
ScriptName SineCosineTangent
; 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
if ( ang < -180 )
set ang to ( ang + 360 )
elseif ( ang > 180 )
set ang to ( ang - 360 )
endif
; approximate sine and cosine of the angle
set rad to ( ang * 0.0174533 )
set cos to 1
set exp to rad ; 1st
set sin to exp
set exp to ( exp * rad ) ; 2nd
set cos to ( cos - exp / 2 )
set exp to ( exp * rad ) ; 3rd
set sin to ( sin - exp / 6 )
set exp to ( exp * rad ) ; 4th
set cos to ( cos + exp / 24 )
set exp to ( exp * rad ) ; 5th
set sin to ( sin + exp / 120 )
set exp to ( exp * rad ) ; 6th
set cos to ( cos - exp / 720 )
set exp to ( exp * rad ) ; 7th
set sin to ( sin - exp / 5040 )
set exp to ( exp * rad ) ; 8th
set cos to ( cos + exp / 40320 )
set exp to ( exp * rad ) ; 9th
set sin to ( sin + exp / 362880 )
; get tangent
set tan to ( sin / cos )
End
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.
This script is more accurate than it needs to be. Galerion suggests that only the first seven steps are necessary.
Taylor Series Variant 2
ScriptName SineCosineTangent
; script originally by JOG
float angle ; Input
float sina ; Output Sin
float cosa ; Output Cos
float cosa ; Output Tan
float t1
float t2
float t5
float t6
Begin {appropriate blocktype}
set angle to player.getangle z
set angle to angle + 360*(angle < -180) - 360*(angle > 180) ; Normalize
set t1 to angle / 57.29577951 ; precalculate powers of "angle"
set t2 to t1*t1
set t5 to t2*t2*t1
set t6 to t5*t1
set sina to t1 - t1*t2/6 + t5/120 - t5*t2/5040 + t6*t2*t1/362880
set cosa to 1 - t2/2 + t2*t2/24 - t6/720 + t6*t2/40320
set tana to sina/cosa
end
Taylor Series Variant 3
The idea here is to approximate sin(z) and cos(z) with taylor. Since taylor works best with values close to 0, I think it's a good idea to use sin and cos periodicity as many times as possible. then I used taylor to approximate the functions. (Scripts by bifmadeinsabbioni)
ScriptName Sin(Z)
set Angolo to AngleZ ; Angolo is the angle of which we want to calculate the sin
set Sign to 1 ; it's the sign adjustment of the result
if (Angolo < 0) ; sin(-z) = -sin(z)
set Angolo to Angolo * (-1)
set Sign to Sign * (-1)
endif
if (Angolo > 180)&& (Angolo <=360) ; if 180<=z<=360 sin(z) = -sin(z-180)
set Sign to Sign * (-1)
set Angolo to (Angolo - 180)
endif
if (Angolo > 90)&&(Angolo <=180) ; if 90<=z<=180 sin (z) = sin (180-z)
set Angolo to (180 - angolo)
endif
if (Angolo >45)&&(Angolo <=90) ; if 45<z<90 sin(z) = cos(90-z)
set Angle to (90 - Angolo)
set Angle to (3.14159265 * Angle / 180) ;now in radiants
set X2 to Angle * Angle
set X4 to X2 * X2
set X6 to X4 * X2
set X8 to X4 * X4
set SinZ to (1 - X2 / 2 + X4 / 24 - X6 / 720 + X8 / 40320)
set SinZ to SinZ*Sign
endif
if (Angolo>=0)&&(Angolo<=45)
set Angle to Angolo
set Angle to (3.14159265 * Angle / 180) ;now in radiants
set X to Angle
set X3 to X*X*X
set X5 to X3*X*X
set X7 to X3*X3*X
set X9 to X3*X3*X3
set SinZ to (X - X3 / 6 + X5 / 120 - X7 / 5040 + X9 / 362880)
set SinZ to SinZ*Sign
endif
ScriptName Cos(Z)
set Angolo to AngleZ ; Angolo is the angle of which we want to calculate the cos
set Sign to 1 ; it's the sign adjustment of the result
if (Angolo < 0) ; cos(-z) = cos(z)
set Angolo to Angolo * (-1)
endif
if (Angolo > 180)&& (Angolo <=360) ; if 180<=z<=360 cos(z) = -cos(z-180)
set Sign to Sign * (-1)
set Angolo to (Angolo - 180)
endif
if (Angolo > 90)&&(Angolo <=180) ; if 90<=z<=180 cos (z) = -cos(180-z)
set Angolo to (180 - Angolo)
set Sign to Sign * (-1)
endif
if (Angolo >45)&&(Angolo <=90) if 45<z<90 cos(z) = sin(90-z)
set Angle to (90 - Angolo)
set Angle to (3.14159265 * Angle / 180) ;now in radiants
set X to Angle
set X3 to X*X*X
set X5 to X3*X*X
set X7 to X3*X3*X
set X9 to X3*X3*X3
set CosZ to (X - X3 / 6 + X5 / 120 - X7 / 5040 + X9 / 362880)
set CosZ to CosZ * Sign
endif
if (Angolo>=0)&&(Angolo<=45)
set Angle to Angolo
set Angle to (3.14159265 * Angle / 180) ;now in radiants
set X2 to Angle * Angle
set X4 to X2 * X2
set X6 to X4 * X2
set X8 to X4 * X4
set CosZ to (1 - X2 / 2 + X4 / 24 - X6 / 720 + X8 / 40320)
set CosZ to CosZ * Sign
endif
Arcsine, Arccosine, and Arctangent
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.
Arcsine - 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))
scriptname Arcsine
; script originally by JOG
float x ; Input (Sine)
float arcsinx ; Output (Angle)
float n
Begin {appropriate blocktype}
set x to (player.getpos z - NPC.getpos z)/player.getdistance NPC
set n to 1 - x ; arcsinx = SquareRoot(1-x)
set arcsinx to n/2
set arcsinx to (arcsinx+(n/arcsinx))/2
set arcsinx to (arcsinx+(n/arcsinx))/2
set arcsinx to (arcsinx+(n/arcsinx))/2
set n to (arcsinx+(n/arcsinx))/2
set arcsinx to 57.2957795*(1.5707963-n*(1.5707288-0.2121144*x+0.0742610*x*x-0.0187293*x*x*x))
End
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.
Arcsine - Approximation by Taylor Series
According to Wikipedia, arcsine can be found with this series:
This can be approximated by Oblivion using the following script:
scriptname Arcsine
;script originally supplied by DragoonWraith
float z
float z3
float z5
float z7
float arcsinz
Begin {appropriate blocktype}
;z can be set depending on implementation
;here it is set to the number of places discovered by the player
;(a completely bizarre thing to find the arcsine of)
set z to GetPCMiscStat 7
set z3 to ( z * z * z )
set z5 to ( z3 * z * z )
set z7 to ( z5 * z * z )
set arcsinz to ( z + (1/2)*(z3/3) + (3/8)*(z5/5) + (15/48)*(z7/7) )
End
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.
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.
scriptname Arctan
;modified from arcsine script supplied by DragoonWraith
float z
float z3
float z5
float z7
float arctanz
Begin {appropriate blocktype}
;z can be set depending on implementation
;here it is set to the number of places discovered by the player
;(a completely bizarre thing to find the arcsine of)
set z to GetPCMiscStat 7
set z3 to ( z * z * z )
set z5 to ( z3 * z * z )
set z7 to ( z5 * z * z )
set arctanz to ( z - (z3/3) + (z5/5) - (z7/7) )
End
Again, more terms equals more accuracy.
See Also
Elder Scrolls Forums - Trigonometry Function Workarounds?
Elder Scrolls Forums - Is there an actor for the target you are attacking currently?