CALCULATOR
Ideal Catenary Height Above Lowest Point Calculator
Ideal Catenary Height Above Lowest Point. Example: Scale: 2 m, Horizontal offset: 3 m gives 2.70481923049 m. Supplied mathematical scale a, ideal catenary centered at lowest point; height above that point only. No load, tension, material or structural safety model. Only this original stated job is proposed, not full historical source coverage. Input domains and output unit must be visible.
Supplied mathematical scale a, ideal catenary centered at lowest point; height above that point only. No load, tension, material or structural safety model. Only this original stated job is proposed, not full historical source coverage. Input domains and output unit must be visible. Derivation: cosh(u)-1=2sinh²(u/2). With u=x/a and E=expm1(u/2)-expm1(-u/2)=2sinh(u/2), height=(a/2)EE. expm1 preserves tiny exponent differences; opposite-signed expm1 terms are subtracted without cancellation. Existing engine mul reduces left to right: multiply0.5 by scale, then E, then E, rather than squaring E first. This preserves representable subnormal heights that premature squaring could erase.
Accepted inputs: Scale: 1 to 1,000 m; Horizontal offset: -10 to 10 m.
How the calculation works
height=2a sinh²(x/(2a)); E=expm1(x/(2a))-expm1(-x/(2a)); evaluate (a/2) E E
Worked example
With scale 2 m, horizontal offset 3 m, ideal catenary height above lowest point is 2.70481923 m.
Assumptions and limits
Supplied mathematical scale a, ideal catenary centered at lowest point; height above that point only. No load, tension, material or structural safety model. Only this original stated job is proposed, not full historical source coverage. Input domains and output unit must be visible. Derivation: cosh(u)-1=2sinh²(u/2). With u=x/a and E=expm1(u/2)-expm1(-u/2)=2sinh(u/2), height=(a/2)EE. expm1 preserves tiny exponent differences; opposite-signed expm1 terms are subtracted without cancellation. Existing engine mul reduces left to right: multiply0.5 by scale, then E, then E, rather than squaring E first. This preserves representable subnormal heights that premature squaring could erase.