Mfcttrf

I am trying to reproduce this figure but I have the impression that alpha does not correspond to the length of the path subpath (0,1.3) of p and that the value -pi/2 is not correct.

 documentclass[border=5mm]standalone
 usepackageluatex85
 usepackageluamplib
 begindocument
 mplibtextextlabelenable
 beginmplibcode

 beginfig(1);

 numeric u, pi;
 u = 10mm;
 pi = 3.141592654;

 path C, T, xx, yy;

 C = fullcircle scaled 5u;

 xx = (point 4 of C -- point 0 of C) scaled 1.1;
 yy = (point 6 of C -- point 2 of C) scaled 1.1;

 T = ((xpart point 0 of C,200) -- (xpart point 0 of C,-200));

 numeric l;
 l = length subpath(0,1.3) of C;

 draw xx;
 draw yy;
 draw C withpen pencircle scaled 3/4;
 draw origin -- (point 1.2 of C) withcolor blue;
 draw subpath(0,1.2) of C withcolor blue;
 draw T withcolor red;

 label.urt(btex $pi$ etex,(xpart point 0 of C,u*(pi)));
 label.urt(btex $1$ etex,(xpart point 0 of C,u*1));
 label.urt(btex $2pi$ etex,(xpart point 0 of C,u*(2pi)));
 label.urt(btex $fracpi2$ etex,(xpart point 0 of C,u*(pi/2)));
 label.urt(btex $-fracpi2$ etex,(xpart point 0 of C,u*(-pi/2)));
 label.urt("$M$", point 1.3 of C);
 label.llft("$O$", origin);
 label.lrt("$I$", point 0 of C);
 label.ulft("$J$", point 2 of C);
 label.urt(btex $fracpi2$ etex, point 2 of C);

 fill fullcircle scaled dotlabeldiam
 shifted (xpart point 1.2 of C, ypart point 1.2 of C)
 withcolor blue;

 endfig;
 endmplibcode
 enddocument

For the labels on your y-axis, you use

label.urt(btex $pi$ etex,(xpart point 0 of C,u*(pi)));
label.urt(btex $1$ etex,(xpart point 0 of C,u*1));
label.urt(btex $2pi$ etex,(xpart point 0 of C,u*(2pi)));
label.urt(btex $fracpi2$ etex,(xpart point 0 of C,u*(pi/2)));
label.urt(btex $-fracpi2$ etex,(xpart point 0 of C,u*(-pi/2)));
label.urt("$M$", point 1.3 of C);`

First, this has a lot of repetitions which makes it hard to adjust, so let's move this into a macro:

vardef labeled (expr t, y) =
 save p; pair p;
 p = (xpart point 0 of C, y*u);
 label.urt(t, p);
enddef;

labeled(btex $pi$ etex,pi);
labeled(btex $1$ etex,1);
labeled(btex $2pi$ etex,2pi);
labeled(btex $fracpi2$ etex,pi/2);
labeled(btex $-fracpi2$ etex,-pi/2);

Now we can add little red lines to indicate the exact positions: Add

draw p shifted (-0.5mm, 0) -- p shifted (0.5mm, 0) withcolor red;

to labeled. This gives

Now we see why -pi/2 seemed to be in a odd position and not symmetric to pi/2:
label.urt places the label in an upper right position and therefore moves all labels a bit up. We can use label.rt in labeled instead to avoid this:

Now the axis still looks odd: The drawing seems to illustrate the correspondence between the length of the circular arc and the angle in radians, so it only works if the radius of the circle is 1 unit. So we change C = fullcircle scaled 5u; to C = fullcircle scaled 2u; (The diameter should be 2u) This make the entire diagram quite small, so we could also increase u. Our new figure is

Now we want to add alpha. First we calculate alpha by asking for the length of the arc (arclength) of the path you are interested in (I use 1.2 instead of 1.3 because that is the path you actually used in your code):

alpha*u = arclength subpath(0,1.2) of C;

Then we can add

labeled(btex $alpha$ etex, alpha);

It turns out that this arc has a length slightly below 1, so let's use subpath(0, 1.4) instead to make it look a bit more like your original:

Now wwe can make everything a bit bigger and add some alignment to the labels to get:

documentclass[border=5mm]standalone
usepackageluatex85
usepackageluamplib
begindocument
mplibtextextlabelenable
beginmplibcode

beginfig(1);

numeric u, pi;
u = 20mm;
pi = 3.141592654;

path C, T, xx, yy;

C = fullcircle scaled 2u;

xx = (point 4 of C -- point 0 of C) scaled 1.1;
yy = (point 6 of C -- point 2 of C) scaled 1.1;

T = ((xpart point 0 of C,6.5u) -- (xpart point 0 of C,-2u));

alpha*u = arclength subpath(0,1.4) of C;

draw xx;
draw yy;
draw C withpen pencircle scaled 3/4;
draw origin -- (point 1.4 of C) withcolor blue;
draw subpath(0,1.4) of C withcolor blue;
draw T withcolor red;

vardef labeled (expr t, y) =
 save p; pair p;
 p = (xpart point 0 of C, y*u);
 label.rt(t, p);
 draw p shifted (-0.5mm, 0) -- p shifted (0.5mm, 0) withcolor red;
enddef;

labeled(btex $alpha$ etex,alpha);
labeled(btex hbox to 1.5emhfill$pi$ etex,pi);
labeled(btex hbox to 1.5emhfill$1$ etex,1);
labeled(btex hbox to 1.5emhfill$2pi$ etex,2pi);
labeled(btex hbox to 1.5emhfill$fracpi2$ etex,pi/2);
labeled(btex hbox to 1.5emhfill$-fracpi2$ etex,-pi/2);
label.urt("$M$", point 1.4 of C);
label.llft("$O$", origin);
label.lrt("$I$", point 0 of C);
label.ulft("$J$", point 2 of C);
label.urt(btex $fracpi2$ etex, point 2 of C);

fill fullcircle scaled dotlabeldiam
shifted point 1.4 of C
withcolor blue;

endfig;
endmplibcode
enddocument