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In designing transfer curves to connect sections of straight railroad tracks, it's important to realize that the acceleration of the train should be continuous so that the reactive force exerted by the train on the track is also continuous. This will be the case if the curvature varies continuously.
A logical candidate for a transfer curve to join existing train tracks given by y = 1, for x<= 0, and y = sqrt(2) - x, for x >= 1/sqrt(2) might be the function f(x) = sqrt(1 - x^2), 0 < x < 1/sqrt(2).
Show that the function:
F(x) = 1 if x<= 0
sqrt(1 - x^2 if 0 < x < 1/sqrt(2)
sqrt(2) - x if x >= 1/sqrt(2)is continuous and has continuous slope, but does not have continuous curvature. Therefore f is not an appropriate transfer curve.
A logical candidate for a transfer curve to join existing train tracks given by y = 1, for x<= 0, and y = sqrt(2) - x, for x >= 1/sqrt(2) might be the function f(x) = sqrt(1 - x^2), 0 < x < 1/sqrt(2).
Show that the function:
F(x) = 1 if x<= 0
sqrt(1 - x^2 if 0 < x < 1/sqrt(2)
sqrt(2) - x if x >= 1/sqrt(2)is continuous and has continuous slope, but does not have continuous curvature. Therefore f is not an appropriate transfer curve.