Question

Intrinsic equation / curvature question...?

Prove if `|psi|` is small then `kappa approx (d^2y)/(dx^2) ( 1 - 3/2 tan^2 psi )`

Answer 1

For `y = y(x)`, curvature `kappa` is:

`kappa := (dpsi)/(ds) = ( y'' )/ (1+(y')^2 )^(3/2)`

`= y'' (1 + (y')^2)^(-3/2)`

By Binomial Expansion :

`= y'' (1 - 3/2(y')^2 + bbbO((y^')^3 ))`

Because:

• `y' = tan psi`

Then:

• `tan psi ~~psi " for " abspsi " << "1 color(blue)(implies) abs(y') " << " 1 " if " abspsi "<< "1`

So ignoring `bbbO((y^')^3 )` and higher:

• `kappa ~~ y'' (1 - 3/2tan^2psi ) " for " abspsi " << "1`