ARCTAN equasion brackets

[Camerart]

Hi,
I have an equasion for use in a project : arctn= x/y/( 1+0.28125*( x/y )^2)*180/pi

My system doesn't support '^'
Will someone rearange the brackets with '*' instead of '^'  Please?
I've tried a few versions, but the results aren't correct.

Camerart.

[tenochtitlanuk]

Don't understand what you mean by 'my system does not support', but the 'exponential' operator ^2 simply means multiply the number by itself, ie 3^2 is the same as 3*3. Run the following code and see both give the same result?


x =2
y =3

pi =3.14159265

arctn= x /y /( 1+0.28125 *( x /y )^2) *180 /pi
print arctn

arctn= x /y /( 1+0.28125 *( x /y ) *( x /y)) *180 /pi
print arctn

[tsh73]

I don't know what are you actually doing
but if this means some ting like 'arctangent'
en.wikipedia.org/wiki/Inverse_trigonometric_functions
then your formula looks highly suspicious to me.
Here tested JB user function to get arc tangent:

function atan2(y,x)
   'pi = asn(1) * 2    'global
   if x <> 0 then arctan = atn(y/x)

   select case
       case x > 0
       atan2 = arctan

       case y>=0 and x<0
       atan2 = pi + arctan

       case y<0 and x<0
       atan2 = arctan - pi

       case y>0 and x=0
       atan2 = pi / 2

       case y<0 and x=0
       atan2 = pi / -2
   end select
end function


[Camerart]

Hi T and T,

My system is a PIC programmer, and simply doesn't support carets.

First I'll try the offered example: arctn= x /y /( 1+0.28125 *( x /y ) *( x /y)) *180 /pi

Thanks.
C.

[B+]

Reversing x and y, Camerart's Forumula works as approach 45 degrees and then goes wrong way!

pi = acs(-1)
print "check pi = ";pi
print
'edit: comment from old code removed from this line
print pstr$("Degrees", 10);", ";pstr$("Radians", 10);", ";pstr$("TAN", 10);", ";pstr$("ATN", 10);", ";"Camerart Formula"
for degree = 0 to 90 step 5
   radian = degree * pi / 180
   y = cos(radian) 'reversed from normal x, y
   x = sin(radian)
   if x <> 0 then
       t = tan(radian)
       at = atn(t) * 180 / pi
       CamerartArcTan = x /y /( 1+0.28125 *( x /y ) * ( x /y)) * 180 /pi
       print pnum$(degree, 10);", "; pnum$(radian, 10);", ";pnum$(t, 10);", ";pnum$(at, 10);", ";pnum$(CamerartArcTan, 10)
   end if
next

function pnum$(x,lenstring)
   pnum$=right$(space$(lenstring)+str$(x),lenstring)
end function

function pstr$(x$,lenstring)
   pstr$=right$(space$(lenstring)+x$,lenstring)
end function


Output:
check pi = 3.14159265

  Degrees,    Radians,        TAN,        ATN, Camerart Formula
        5, 7266463e-1, 7488664e-1,          5, 5.00196314
       10, 0.17453293, 0.17632698,         10, 10.0152148
       15, 0.26179939, 0.26794919,       15.0, 15.0484862
       20, 0.34906585, 0.36397023,         20, 20.1048824
       25, 0.43633231, 0.46630766,       25.0, 25.1776986
       30, 0.52359878, 0.57735027,       30.0,  30.244328
       35, 0.61086524, 0.70020754,         35, 35.2571756
       40,  0.6981317, 0.83909963,         40, 40.1301103
       45, 0.78539816,        1.0,       45.0, 44.7186572
       50, 0.87266463, 1.19175359,       50.0, 48.7922501
       55, 0.95993109, 1.42814801,       55.0,  51.998478
       60, 1.04719755, 1.73205081,       60.0, 53.8246515
       65, 1.13446401, 2.14450692,         65, 53.5749845
       70, 1.22173048, 2.74747742,         70, 50.4054432
       75, 1.30899694, 3.73205081,         75, 43.4853373
       80,  1.3962634, 5.67128182,         80,   32.34537
       85, 1.48352986, 11.4300523,       85.0, 17.3508382
       90, 1.57079633, 1192836e15,      -90.0, 995373e-13




[Camerart]

Hi B,

That's excellent work.

This is only for 0-45

I'll look closer tomorrow.

Will you explain your comment: reversing X and Y please?

Thanks,

C.

[tsh73]

I tried that formula too
y = x/(1+0.28125*x*x)
and got that it is indeed very close to ATN(x) for x in [0..1]

But I cannot see how that could be Taylor series
Aren't they all written in a form
SUM_{i=0} (c_i*x^i)?

[B+]

Apr 9, 2018 20:37:39 GMT tsh73 said:

I tried that formula too
y = x/(1+0.28125*x*x)
and got that it is indeed very close to ATN(x) for x in [0..1]

But I cannot see how that could be Taylor series
Aren't they all written in a form
SUM_{i=0} (c_i*x^i)?

As a matter of fact, before checking Camerart's formula, I did try Taylor's Series!

I got very good results approaching 45 degrees and then things went badly, worse than Camerart's formula.

Perhaps you can see the error of my way?

'https://math.stackexchange.com/questions/29649/why-is-arctanx-x-x3-3x5-5-x7-7-dots
'arctan(x)=x-x^3/3+x^5/5-x^7/7+…?

' x probably in radians

'n th term denominator = 2n - 1
'n th term numerator = x ^ (2n - 1)
'note: 2n -1 series is just odd numbers!

'if n is even subtract term
'if n is odd add term

pi = acs(-1)
print "check pi ";pi
print
'test formula for arctan with 10 terms from Taylor Series
for degree = 0 to 90 step 5
   rad = degree * pi/180
   print "degree, radian ";degree;", ";rad
   x = tan(rad)
   print "x = tan(radian) = ";x
   'now the arc tangent of x should be = radian and radian * 180/pi = angle in degree
   sum = 0
   for n = 1 to 10
       oddN = 2*n - 1
       numerator = x
       for power = 2 to oddN  'without using ^ for power
           numerator = numerator * x
       next
       'add or subtract
       if n mod 2 = 0 then mult = -1 else mult = 1
       sum = sum + mult*(numerator / oddN)
   next
   print "For angle ";degree
   print "The formula of arctan(tan(angle)) got ";sum * 180 / pi 'convert to degrees
   print
   print
next

'something goes wrong after 40 degrees!



[B+]

Apr 9, 2018 19:54:40 GMT Camerart said:

Hi B,

That's excellent work.

This is only for 0-45

I'll look closer tomorrow.

Will you explain your comment: reversing X and Y please?

Thanks,

C.

Where you are using x and y, I would use y and x.

tangent is well known as = to sin / cos.  Sin is always associated to y height on graph and cos x the horizontal width on graph, so tan(angle) = y/x = "rise over run" AKA slope of a line.

But your formula works better if I change x and y around, or you have an arc cotangent formula ( = x/y = the reciprocal of arc tangent).

[B+]

Does this look familiar to Camerart's Forumla? It is for inverse of arc tangent!
www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/arctanproof1.html


Append: On 2nd thought, this isn't it at all! (Back to drawing board.)

[tsh73]

As a matter of fact, before checking Camerart's formula, I did try Taylor's Series!



I got very good results approaching 45 degrees and then things went badly, worse than Camerart's formula.



Perhaps you can see the error of my way?

Nothing wrong. It's all math.
Sin cos exp has a member of series of a form x^n/n!
n! rises really fast, so member goes to 0 and series converge real fast.

Now arctan has members in the form x^n/n
1/n really doesn't do much good with convergence (SUM 1/n is not converge at all)
So the only hope that x^n will go down.
But it will - only for abs(x)<1.
So our series behaves only for x<1

(btw your code - Taylor formula - changed up to 20 gives 44.284249 for 45 degrees.
A bit better then 43.571143  for 10 members, but still not very good.
But this is really the border.)

[tsh73]

I believe I found source of the formula
R. Lyons, “Another contender in the arctangent race,”
www.researchgate.net/publication/3321505_Another_Contender_in_the_Arctangent_Race

BTW 0.28125 happens to be 9/32

[B+]

Yes! and it shows how the error escalates after 40 degrees.

Thanks tsh73, my faith in Taylor Series restored (by restricting my application of it to less than 45 degrees for arc tan). 

[Camerart]

Hi All,

All impressive stuff, but I am using a PIC programmer, that has quirks, which means I have to be careful leaving the suggested formula.

e,g, Oshonsoft (My simulator/programmer) has ARCTAN, but it doesn't work. This is why I'm using my word 'arctn'

As the original formula was suggested by someone who is working through the Idiosyncrasies of the programmer and the compass module I'm using, I ought to stick closely to it.

Here it is again, after conversion by T : arctn= x /y /( 1+0.28125 *( x /y ) *( x /y)) *180 /pi

Are you suggesting that I change it to : arctn= y/x /( 1+0.28125 *( y/x ) *( y/x)) *180 /pi ?

NOTE: This is only for the 0 to 45º section.
NOTE 2: I've got an 'O' level in Maths and Physics, learned navigation, but still don't properly get SIN, TAN, ARCTAN etc and never will, Sorry.

C.

[tsh73]

Are you suggesting that I change it to ...

You really have to check it by yourself. Try and see if it makes sense.
Here how (x, y) usually drawn. Result of arctan is an angle.
Compared are buildt-in arctg() (SMath Studio - free math package) and your formula.
Second part is using your formula to get angles in 45..90.
_that_ actually changes y/x to x/y.
(if z is y/x then 1/z is naturally x/y).