Vector of Pure Functions

Question

I'm trying to make a function that takes in a vector of pure functions (representing a parametric equation) and returns another vector of pure functions representing an offset curve. Here's what I have as an example of an input curve:

HermiteSpline[{p1x_, p1y_}, {p2x_, p2y_}, {t1x_, t1y_}, {t2x_, t2y_}] :=
{p2x (3 #^2 - 2 #^3) + p1x (1 - 3 #^2 + 2 #^3) + (# - 2 #^2 + #^3) t1x + 
(-#^2 + #^3) t2x , p2y (3 #^2 - 2 #^3) + p1y (1 - 3 #^2 + 2 #^3) + (# - 
   2 #^2 + #^3) t1y + (-#^2 + #^3) t2y } &

Basically, some messy function that returns a vector with two elements, both pure functions. Then, I want to make another function that would take this as an argument and do some operations on the curve and return a different curve, like so:

CurveOffset[P_ , radius_] :=  radius {(P[[2]])'[#], (P[[1]])'[#]}/
Sqrt[((P[[1]])'[#])^2 + ((P[[2]])'[#])^2] &

So P would be the argument containing the function, it gets indexed and had its derivative taken, then returns another vector. But I keep getting the following error:

Part::partw: Part 2 of {0.5 (3 Slot[<<1>>]^2-2 Power[<<2>>])+0.2 (1-3 Power[<<2>>]+2 Slot[<<1>>]^3)+(#1-2 Power[<<2>>]+#1^3) 0+(-Slot[<<1>>]^2+#1^3) 0,0.5 (3 Slot[<<1>>]^2-2 Power[<<2>>])+0 (1-3 Power[<<2>>]+2 Slot[<<1>>]^3)+(#1-2 Power[<<2>>]+#1^3) 1+(-Slot[<<1>>]^2+#1^3) 1}& does not exist.

I've been searching around but I'm just not sure what's going wrong or how to fix it. Any help is appreciated, thanks!

Answer 1

Define hermiteSpline to return a pair of pure functions:

hermiteSpline[{p1x_, p1y_}, {p2x_, p2y_}, {t1x_, t1y_}, {t2x_, t2y_}] :=
Function/@ {p2x (3 #^2 - 2 #^3) + p1x (1 - 3 #^2 + 2 #^3) + (# - 2 #^2 + #^3) t1x + 
 (-#^2 + #^3) t2x , 
 p2y (3 #^2 - 2 #^3) + p1y (1 - 3 #^2 + 2 #^3) + (# -2 #^2 + #^3) t1y + (-#^2 + #^3) t2y } 

or, equivalently,

 hermiteSpline[{p1x_, p1y_}, {p2x_, p2y_}, {t1x_, t1y_}, {t2x_, t2y_}] :=
  {p2x(3 #^2 - 2 #^3) + t1x(# - 2 #^2 + #^3) + t2x(-#^2 + #^3) + p1x (1 - 3 #^2 + 2 #^3)&, 
   p2y(3 #^2 - 2 #^3) + t1y(# - 2 #^2 + #^3) + t2y(-#^2 + #^3) + p1y (1 - 3 #^2 + 2 #^3)&}

Through@hermiteSpline[{0.2, 0}, {0.5, 0.5}, {0, 1}, {0, 1}][0]

{0.2, 0.}

 ParametricPlot[Through@hermiteSpline[{0.2, 0}, {0.5, 0.5}, {0, 1}, {0, 1}][t], 
  {t,  0, 1}]

 CurveOffset[hermiteSpline[{0.2, 0}, {0.5, 0.5}, {0, 1}, {0, 1}], radius] /@ {0, 1, 2}

{{1. radius, 0.}, {1. radius, 0.},{0.889288 radius, -0.457348 radius}}