Equations for Robert Dixon's (Mathographics) Involute Egg-curve

Question

In Mathographics Robert Dixon shows how to draw an Egg-curve (or really half of an egg curve) as an involute:

This allows you to draw a smoother version of eggs created from the 'arcs of a circle method.'

I want to know the formula for idealized involute curves of this type (i.e., 0-D pins, and 0 width string, etc....):

What is the formula for parametric equations for points, $(x_1, y_1), (x_2, y_2), (x_3, y_3)..., (x_n, y_n)$, given a starting point $(x_f, y_f)$ and a point $(x_O, y_O)$ where the string is attached?

Answer 1

Have a look at Fig. 1 modelizing the position of the extremity of a string rolled/unrolled along a convex polygon (here the particular case of a regular octagon) generating a kind of spiral (modelizing rather well a half-egg-shape) ?

Fig. 1.

But one can obtain much general type of curves as one can see on Fig. 2 (see Matlab program below). Here, four different curves ("offsets" of the first one) are displayed just by changing the curve length.

Fig. 2.

Last but not least, being a Frenchman, I don't resist the pleasure to offer you a "croissant" (with eggs entering into its composition :)) :

Fig. 3.

Matlab program (using complex numbers geometry) having generated figure 2 :

 function main;
    a=80;axis([-a,a,-a,a])
    A=[0,pi/4,pi/3,pi/2,2*pi/3,pi,5*pi/4,3*pi/2,7*pi/4,2*pi];% sides polar angles
    Lg=[1,1,3,2,7,1,4,5,20,20];% side lengths
    V=Lg.*exp(A*i);A=[A,A(1)];Lg=[Lg,Lg(1)];L=sum(Lg);
    LA=length(A);
    Pt=cumsum(V);% points, obtained by cumulating vectors
    for k=0:2:6
       C=[]; % curve
       M=L+k;
       for k=1:LA-2
          T=cs(Pt(k),M,A(k),A(k+1));
          C=[C,NaN,T];
          M=M-Lg(k+1);
       end;
       plot(C,'b','linewidth',2);hold on;
    end;
    plot(Pt,'color','r','linewidth',4);hold on
    %%%
 function T=cs(ce,r,t1,t2);% circular sector
    s=sign(t2-t1);
    T=ce+r*[0,exp(i*(t1:s*0.005:t2)),0];