Ggtcf

Pg 65 of the PGF Manual, shows different operations with the intersections library in TikZ. But I am not able to understand the code. What is the meaning of the option by here.

The explanation given is:

The name intersections takes an optional argument
by, which lets you specify names for the coordinates and options for them. This creates more compact code.

This is not clear.

The relevant line is

path [name intersections=of=D and E, by=[label=above:$C$]C, [label=below:$C’$]C’];

Compare with this simpler version:

path [name intersections=of=D and E, by=C, C’];

Here the intersection points are computed and named C and C' ("name the intersection points of D and E by the names C and C'").

It is shortcut for

coordinate (C) at ...;
coordinate (C') at ...;

for some computed coordinates.

Adding the optional styling [label=above:$C$]C is equivalent to

coordinate[label=above:$C$] (C) at ...;

and allows you to style the intersection point directly. It would be equivalent, though longer, to write

path [name intersections=of=D and E, by=C, C’];
node[above] at (C) $C$;
node[below] at (C') $C'$;

By default, intersections are named (intersection-1), (intersection-2), etc.

When you write by=a,b the first two intersections will be called (a) and (b).

Let's look at the example on page 142, slightly modified. It displays the 9 intersections of two curves.
The total number of intersections is given by total.

By writing by=a,b, the first 2 intersections now have two names:

• 
  (a) or (intersection-1)
  


• 
  (b) or (intersection-2)
  

(a) is an alias of (intersection-1), the others do not have aliases and remain accessibles.

documentclass[border=5mm,tikz]standalone

usetikzlibraryintersections

begindocument
begintikzpicture
clip (-2,-2) rectangle (2,2);
draw [name path=curve 1] (-2,-1) .. controls (8,-1) and (-8,1) .. (2,1);
draw [name path=curve 2] (-1,-2) .. controls (-1,8) and (1,-8) .. (1,2);
fill [name intersections=of=curve 1 and curve 2, by=a,b, total=t]
[red, opacity=0.5, every node/.style=above left, black, opacity=1]
foreach s in 1,...,t(intersection-s) circle (2pt) node footnotesizes;
draw[fill=blue!50,opacity=.5] (a) circle (4pt);
endtikzpicture
enddocument

Just for completeness. You can name the intersections by C-1 etc. by just using name=C. What is perhaps also worth pointing out is that, if you want so sort the intersections along a straight line, then you have to draw the straight line pretending it is a curve.

documentclassstandalone
usepackagetikz
usetikzlibraryintersections

begindocument
begintikzpicture
 draw[name path=grid] [xstep=3,ystep=2] (9,8) grid (0,0);
 draw[->, name path=line] (2,1) to[bend left=0] (7,7);
 draw[name intersections=of=grid and line, sort by=line, name=C, total=t]
 foreach s in 1,...,t(C-s) node s;
endtikzpicture
enddocument