Equation of a helix parametrized by arc length between two points in space

Question

What is the equation of a helix parametrized by arc length (i.e. a function of arc length) between any two points in space? Is there any function for this ?

Answer 1

To find the arc length parameterization of the helix defined by

    r(t)  =  cos t i + sin t j + t k

Arc Length = s = Integral(a,b){sqrt((dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2) dt}

First find the arc length function

     s(t) = Integral(0,t) { sqrt((sin u)^2 + (cos u)^2 + 1) du }
          = Integral(0,t) { sqrt(2) du } = sqrt(2) * t

Solving for t gives

    t   =  s / sqrt(2)

Now substitute back to get

    r(s)  =  cos(s / sqrt(2)) i + sin(s / sqrt(2)) j + (s / sqrt(2)) k

I'll leave the last bit to you!