mfpic-doc.pdf.

4.12 FOR ADVANCED USERS. 69
METAFONT has no general command for calculating the lengths of paths; METAPOST does, but it is
quite slow. Thus neither program has an efficient method for finding the described point, so MFPIC
uses METAFONT/METAPOST macros that are faster, but less accurate than they could be. Still, the
results should (except in pathological cases) be accurate to within a couple of percent of the length
of the path. If they are not, adjust the value of the fraction. These remarks about accuracy also hold
for any other command (such as \partpath in subsection 4.4.2) that take the fraction of a path
length as a parameter.
The pathpoint command is not a basic METAFONT command, but is defined by the GRAFBASE
macros that accompany MFPIC.
METAFONT pairs can conveniently be viewed as complex numbers. So grafbase also contains
some functions useful in complex analysis (my research field). In what follows a, z and w denote
pair variables or constants, and each function interprets them as complex numbers. Also t denotes
an angle in radians. There are both numeric and pair valued functions, the type of each is noted after
the description:
Arg z The principle argument of z in radians (numeric).
Log z The principle logarithm of z (pair).
cis t (cost,sint), same as dir degrees(t) (pair).
zexp w The complex exponential, e w (pair).
zsqrt w The (principal) complex square root: that z with −π/2 ≤ Argz ≤ π/2 and
z 2 = w (pair).
sgn z The signum, sgn(0,0) = (0,0) otherwise sgnz = z/|z| (pair).
conj z The complex conjugate, ¯z (pair).
Moebius(a) z The Möbius transformation (z + a)/(1 + āz) (pair)
pshdist(z,w) The pseudohyperbolic distance between z and w: |z−w|/|1− ¯wz| (numeric).
kelvin(z) The Kelvin transform 1/¯z (pair)
METAFONT will happily add and subtract pairs but to multiply and divide complex numbers
requires new operations. These are given by (z zmul w) and (z zdiv w). They operate on pairs
and produce pairs.
4.12.6 MANIPULATING METAFONT PICTURE VARIABLES
\tile{〈tilename〉,〈unit〉,〈wd〉,〈ht〉,〈clip〉}
〈MFPIC drawing commands〉
\endtile
In this environment, all drawing commands contribute to a tile. A tile is a rectangular picture
which may be used to fill the interior of closed paths. Actually, a tile is a composite object. After
\tile{Nick, ... } ... \endtile a picture variable Nick.pic is created as well as numeric
variable Nick.wd and Nick.ht. These are needed by the \tess command, below.
The units of drawing are given by 〈unit〉, which should be an explicit dimension (like 1pt or
.2in). The tile’s horizontal dimensions are 0 to 〈wd〉 · 〈unit〉 and its vertical dimensions 0 to 〈ht〉 ·
〈unit〉, so 〈wd〉 and 〈ht〉 should be pure numbers. If 〈clip〉 is true then the drawing is clipped to be
within the tile’s boundary.
By using this macro, you can design your own fill patterns (to use them, see the \tess macro
below), but see the warning about memory use by the \tess command. The 〈tilename〉 is globally
defined by this command.
\tess{〈tilename〉}...
This rendering macro tiles the interior of a closed path with a tessellation comprised of copies of
the tile specified by 〈tilename〉. The tile must have been previously created by \tile{〈tilename〉,