6. Other Conjugate Fields
Here are some useful conjugate field operators: 2 arctan[+], arccos[*] , tanh[+], tan[+].
2 arctan[+] ; for subtended angles.
If a line segment of length L is held with midpoint at distance D away from an observer, at right angles to the observer's line of sight, then it will subtend an angle of
θ = 2 arctan ( L/(2D) ) .
Therefore two lengths, which subtend angles of (respectively) θ1 and θ2, when joined together will subtend an angle of
θ12 = θ1 (2 arctan)[+] θ2
arccos[*] ; for spherical right triangles.
If a spherical right triangle has sides A, B and hypotenuse C, (as measured in radians), then we have the equation:
cos(A) * cos(B) = cos(C)
C = A arccos[*] B
tanh[+] ; for relativistic velocity addition.
The relativistic velocity addition law can be written with tanh[+]:
V12 = ( V1 + V2 ) / ( 1 + V1V2/c2) = c ( V1/c tanh[+] V2/c )
tan[+] ; for slope rotation.
If two lines have slopes x and y, and a third line is tilted from the horizontal at an angle equal to the sum of the other two line’s tilts, then the third line has this slope:
tan( arctan(x) + arctan(y)) = ( x + y ) / ( 1 – x y )
Note that ( x tan[+] y ) = i ( x/i + y/i ) / ( 1 + (x/i)(y/i) ) = i ( (x/i) tanh[+] (y/i))
Therefore : tan[+] = (i*tanh)[+]
And likewise: tanh[+] = (i*tan)[+]
Slope rotation and relativistic velocity addition are imaginary conjugates of each other!