MORE THAN YOU EVER WANTED TO KNOW ABOUT SWITCH FROG NUMBERS AND SWITCH FROG ANGLES
-
How does one calculate the frog number using measurements at the track?
-
Given the angle of divergence at the frog: how does one calculate the frog number?
-
Given the frog number: how does one calculate the angle of divergence?
-
Summary
-
Calculate the frog number (“N”)
Visualize a line of symmetry through the center of the frog with the rails diverging on either side of that line. Pick a point x units down the line from the point of the frog and measure, perpendicular to the center line, left or right to the rail; call that measurement y units. The SPREAD is 2y; and N = x/2y. Those of you who don’t have the time to wade through the following detail sections can skip to the Summary. -
Given the angle of divergence at the frog, calculate the frog number
Call the frog angle “alpha”. The cotangent of alpha/2 is x/y (the adjacent side of the right triangle divided by the opposite side), and x/y = 2N:
cotan (alpha/2) = x/y = 2N
N = cotan (alpha/2) / 2.
Or using tangent = 1/cotangent:
tan (alpha/2) = y/x = 1/cotan (alpha/2)
N = cotan (alpha/2) / 2
N = (1 / tan (alpha/2)) / 2
N = 1/(2 tan (alpha/2))
(In case you don’t have cotangent on your hand calculator, it is the inverse of the tangent–1/tan; tan is opposite over adjacent; cotan is adjacent over opposite. The cotan of alpha is also tan (90-alpha). If you have an ancient and honorable analog computer called a “slide rule”, you probably have a cotangent “table” on it.)
Wait! There’s more! Actually, it turns out that the inverse of the sine of alpha
1/sin (alpha)
is very close to the frog number!
N = 1/sin(alpha)
-
Given the frog number, calculate the angle of divergence
From 1) above, N = x/2y so y/x = 1/2N.
The tangent of alpha/2 is also y/x. So
tan (alpha/2) = y/x = 1/2N,
alpha/2 = arctan (1/2N),
alpha = 2 arctan (1/2N).
As a check, solve 2) above for alpha:
N = 1/(2 tan (alpha/2)),
tan (alpha/2) = 1/2N,
alpha/2 = arctan (1/2N),
alpha = 2 arctan (1/2N).
Alternatively, for a close approximation, solve from 2) above:
N = 1/sin (alpha)
for alpha, getting:
alpha = arcsin (1/N)
-
Summary
Exact:
N = x/2y
N = 1/(2 tan (alpha/2))
alpha = 2 arctan (1/2N).
Close Enough:
N = 1/sin (alpha)
alpha = arcsin (1/N)
With a sound module (Conversation between RTC and MoW foreman)
