@mmcdara The unicity (uniqueness) of the linear combination makes the family free. So, the definitions are the same.

@Carl Love Vote up (especially for your comeback)!

@tomleslie The .eps is correct (with a few artifacts though). ghostscript converts it properly into a vector graphics pdf.
Maybe GIMP has an option to do that too.

@tomleslie Your file test2.pdf obviously contains a low resolution bitmap. Have you zoomed into it?

Both your graphics are bitmaps (not vectorized).
To obtain true vector graphics, convert the eps to pdf using e.g. ghostscript or latex.
Compare the results by zooming in:

from-eps.pdf

@Anthrazit Unlike sin or arctan, arctand does not check the (number of) arguments. You may use:

arctandeg := () -> 180/Pi*arctan(args);

@mmcdara Using changes of variables it is easy to integrate f (just like K above). This actually means that the user must know to compute it by hand in order to use Maple (!).

@max125 In a vector space, you can add two vectors and multiply a scalar by a vector. But adding a vector and a scalar is a nonsense (however, in Maple, for matrices, a scalar is identified with a scalar matrix).

3 + <1,2,3>;
Error, a constant cannot be added to a Vector; use +~ for elementwise addition instead of +
The same convention was implemented for lists, but here instead of an error, the expression is returned unsimplified (just like 3 + "abc")

Note that map(f, 3, L)  ==> f(3, L), and for f = `+`, the above remark applies.

@jeffreyrdavis75 It is not clear (at least for me) whether you have some mathematical problems, or you just want some nice pictures.

@mmcdara  Sorry for the typo  :-)

Why don't you continue your previous similar question?
Anyway, this one is not well formulated. For x = 2, ==> f(2) +2* f(1/2) + 3*f(1) = 2, but f(1) is not defined!

@Dr Jean-Michel Collard  Yes, the functions u:=x->1/x; v:=x->x/(x-1);  generate o subgroup of order 6 (isomorphic to S_3) in the Moebius group. 

@maple fan 

1. For older versions, try (I don't have Maple 2015):

restart;
V:=<0,0,0>, <1,2,0>, <0,2,0>, <1,0,0>, <0,0,1>, <1,2,1>, <0,2,1>, <1,0,1>:
T:=<-1,0,0>, <0,-1,0>, <0,0,-1>: 
PM:=LinearAlgebra:-ProjectionMatrix([T[2]-T[1], T[3]-T[1]]):
Pr := v -> T[1]+PM.v:
Pr2:= v -> (PM.v)[[1,2]]:  # convexhull works in dimension 2.
P:=Pr~([V]):
P2:=convert~(Pr2~([V]),  list):
ch:=simplex:-convexhull(P2):
ind:=map(proc(u) local i; member(u, P2,'i'); i end,  ch):
C:=P[ind]: n:=nops(C):  # the vertices of the projection
Tri:=(v1,v2,v3) -> Surface( v1 + t*(v2+s*(v3-v2)-v1), s=0..1, t=0..1):
# the parametrization of the interior of a triangle
add(VectorCalculus:-SurfaceInt( x^2+y^2, [x,y,z] = Tri(C[1],C[i],C[i+1])), i=2..n-1);

2. The method works for polyhedra (the projections will be convex polygons). The projection of a cylinder is more complicated, with infinitely many extreme points in general.

3. A paremetrization of the triangle (v1,v2,v3) is   t*v1 + u*v2 + (1-t-u)*v3,  0<=t<=1, 0<=u<=1-t.
To have both parameters in [0,1], we may take  u = (1-t)*s.

@MathStudent0807 Otherwise, the function would not have existed.