Conjugates of Fields, 6 of 6; Other Conjugate Fields

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)

Ergo:

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!