Always the same function in function table

Question

Let's say, we have a list of points such as

Points = {{0, 0}, {0, 1}, {1, 1}, {1, 0}, {0, 0}};

This is nothing but the corners of a square. List Plot:

Now I wanted to write a general function that connects the points I came up with this

PointList2Curve[pointlist_] := 
Module[{pointlist0 = pointlist, pathpatchs, Supports, Direcs,  F},
 Direcs = 
 Table[pointlist0[[i + 1]] - pointlist0[[i]], {i, 1, 
  Length[pointlist0] - 1}];
 Supports = pointlist0[[;; -1]];
 pathpatchs = 
 Table[Function[{t}, Supports[[i]] + t*Direcs[[i]]], {i, 1, 
 Length[pointlist0] - 1}];
 F[t_] := Module[{t0 = t, i}, 
 i = IntegerPart[t];
 pathpatchs[[i + 1]][t - i]];

F

 ];

If I then have the function returned as in

X = PointList2Curve[Points];

and then do a parametric plot,

ParametricPlot[X[t], {t, 0, 4}]

This is of course not what it should look like. What should be happening is that that I get a square. I am pretty sure that, as far as I understand everything there is no logical mistake. After some trying I found that if I shorten the function such that the module returns only pathpatchs, (The line segments of the paths) They turn out to be all the same. Now my question is, why is that and does anybody have an idea how to fix that ?

Thanks in advance !

I get

Answer 1

Points = {{0, 0}, {0, 1}, {1, 1}, {1, 0}, {0, 0}};

bsf = BSplineFunction[Points, SplineDegree -> 1];

ParametricPlot[bsf[r], {r, 0, 1}]

In case your points don't already describe a closed curved, you may want to add the SplineClosed -> True option.

Answer 2

Clear["Global`*"]

Points = {{0, 0}, {0, 1}, {1, 1}, {1, 0}, {0, 0}};

FindCurvePath gives an ordering of the {xi, yi} that corresponds to one or more smooth curves.

curve = FindCurvePath[Points]

(* {{2, 1, 4, 3, 2}} *)

Graphics[{Line[Points[[curve[[1]]]]],
  Red, AbsolutePointSize[4], Point[Points]},
 Axes -> True]

Answer 3

Another option would be to use the Graphics commands:

points = {{0, 0}, {0, 1}, {1, 1}, {1, 0}, {0, 0}};
Graphics[Line[points]]

If you want the axes shown:

Show[ListPlot[points], Graphics[Line[points]], AspectRatio -> 1]

Answer 4

I think the OP's key general problem may be the ordering the points... for example if the order happened to be

points = {{0, 0}, {0, 1}, {0, 0}, {1, 1}, {1, 0}};

or indeed any other permutation of the points.

The following, based closely on Bob Hanlon's solution, solves that problem:

Graphics[
 Polygon[
  points[[#]] & /@ FindCurvePath[points]]]

with obvious EdgeForm[] and FaceForm[] directives, if desired.