Principal unit normal vector for a sin curve

sin_unit_normal.txt

An expression for the sin curve's principal unit normal vector.

The principal unit normal vector is different from a unit normal vector in that it always points from a point on the curve
towards the centre of the circle that most closely approximates the curvature of the curve at that point. Thus, the principal
unit normal vector is undefined at inflection points since the radius of the circle is infinite at these points.

Consider the parametric form of a sin curve. Let R be a general position vector on a sin curve and let 

c = cos(t)
s = sin(t)

then

               R = |t|
                   |s|

                 = i(t) + j(s)
  
Now

               ^   dR   ||dR||
               T = -- / ||--||
                   dt   ||dt||

                 = (i + j(c)) / √(1+c²)
    
and

    d(1/√(1+c²))
    ------------ = (-1/2)(1+c²)^(-3/2)(-2sc)
         dt
                 = sc(1+c²)^(-3/2)

Thus

               ^
              dT
              -- = i(sc(1+c²)^(-3/2)) + j(-s/√(1+c²) + c(sc(1+c²)^(-3/2)))
              dt 
                 
Now, let the positive scalar k be the curvature, then
                 
                    ^      
               ^   dT   ||dR||
             k*N = -- / ||--||
                   dt   ||dt||

                 = [i(sc(1+c²)^(-3/2)) + j(-s/√(1+c²)+c(sc(1+c²)^(-3/2)))] / √(1+c²)
                  
                 = [i(c)-j] * s/(1+c²)²
                  
                                           ^
That is, the principal unit normal vector, N, of the sin curve is the unit vector of | c| * s/(1+c²)²
                                                                                     |-1|

Also 

               k = || | c| || * || s/(1+c²)² ||
                   || |-1| ||
                   
                 = √(1+c²) * ||s||/(1+c²)²
                 
                 = ||s|| / (√(1+c²))³