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Some expressions have ˜true™ or ˜false™ values, instead of
numbers; we will see later that they can be used to adapt programs
to special conditions.
You type And the result is
0<1 true
0=1 false
a+1>a true
false (with error message)
a>=b
"abc"<="b" true
"B">"a!" false
"b">"a?" true
(1,2)<>(0,4) true
(1,2)<(0,4) false
(1,a)>(0,b) true
Chapter 8: Algebraic Expressions 65


numeric a true not
and
known a false or
comparison
not pen a true ¿=
¡=
known "a" and numeric 1 true
¡¿
(0>1) or (a<a) false relations
greater-than-or-equal-to
a (with error messages)
0>1 or a<a less-than-or-equal-to
unequal-to
The tokens ˜>=™, ˜<=™, and ˜<>™ stand respectively for the relations greater-than- pair
numeric
or-equal-to, less-than-or-equal-to, and unequal-to. When strings are compared, pen
uses the order of words in a dictionary, except that it uses ASCII known
max
code to de¬ne ordering of individual characters; thus, all uppercase letters are min
considered to be less than all lowercase letters. (See Appendix C.) When pairs maximum
minimum
of numbers are compared, considers only the x coordinates, unless integers
the x coordinates are equal; in the latter case it compares the y coordinates. The
type of an expression can be ascertained by an expression like ˜pair a™, which is
true if and only if a is a pair. The expression ˜known a™ is true if and only if the
value of a is fully known.
EXERCISE 8.4
What causes the error messages in ˜0>1 or a<a™ ?
The rest of this chapter is entirely preceded by “dangerous bend” signs, so
you can safely omit it on ¬rst reading (unless you™re hooked and can™t stop).
expressions can include many operations that are less familiar but
still useful. For example, the max and min operations compute the maximum
and minimum of numbers, strings, or pairs:
You type And the result is
max(1,-2,4) 4
min(1,-2,4) -2
max("a","b","ab") "b"
min("a","b","ab") "a"
max((1,5),(0,6),(1,4)) (1,5)
min((1,5),(0,6),(1,4)) (0,6)
max(.5a+1,.5a-1) 0.5a+1
Numbers can be converted to integers in a variety of ways:
You type And the result is
floor 3.14159 3
floor -3.14159 -4
floor -epsilon -1
floor infinity 4095
ceiling 3.14159 4
ceiling -3.14159 -3
66 Chapter 8: Algebraic Expressions


round 3.14159 3 ¬‚oor
greatest integer
round -3.14159 -3 ceiling
least integer
round(1.1,2.8) (1,3)
round
round(3.5,-3.5) (4,-3) remainder
mod
a+0.5 (with error message)
round a abs
length
8 mod 3 2
absolute value
-8 mod 3 1 ++
Pythagorean addition
.8 mod .3 0.2
square root
Pythagorean subtraction
The ˜¬‚oor™ operation computes the greatest integer that is less than or equal to its +-+
operand; this quantity is often denoted by x in mathematics texts. Plain
also includes the analogous ˜ceiling™ operation x , which is the least integer greater
than or equal to x. Furthermore, ˜round x™ is the integer nearest to x; plain
computes this by using the formula x + .5 , and applies it to both components of a
pair if a pair is being rounded. The remainder of x with respect to y, written ˜x mod y™,
is calculated by using the formula x ’ y x/y .

You type And the result is
abs -7 7
abs(3,4) 5
length(3,4) 5
3++4 5
300++400 500
181.01933 (with error messages)
sqrt(300**2 + 400**2)
1++1 1.4142
0 ++ -7 7
5+-+4 3

The ˜++™ operation is called Pythagorean addition; a++b is the same thing as a2 + b2 .
Most of the square root operations in computer programs could probably be avoided
if ++ were more widely available, because people seem to want√ square roots primarily
when they are computing distances. Notice that a ++ b ++ c = a2 + b2 + c2 ; we have
the identity (a ++ b) ++ c = a ++ (b ++ c) as √ well as a ++ b = b ++ a. It is better
to use Pythagorean addition than to calculate a2 + b2 , because the computation of
a2 and b2 might produce numbers that are too large even when a ++ b is rather small.
There™s also an inverse operation, √
Pythagorean subtraction, which is denoted by ˜+-+™;
the quantity a +’+ b is equal to a2 ’ b2 .

EXERCISE 8.5
When the author was preparing these examples he typed ˜0++-7™ and was
surprised to get the answer ˜0™. Why should this not have been a surprise?

EXERCISE 8.6
(For mathematicians.) Although the Pythagorean addition operation is asso-
ciative and commutative, says that 5++4++2++2 = 7 = 2++2++4++5
yet 2 ++ 4 ++ 5 ++ 2 = 6.99998. Why?
Chapter 8: Algebraic Expressions 67


uses the names ˜sind™ and ˜cosd™ for the trigonometric functions sind
cosd
sine and cosine, because ™s operations are designed to deal with
trigonometric
angles expressed in degrees. But it turns out that programmers rarely need to refer sine
to sines and cosines explicitly, because the ˜dir™ and ˜angle™ functions provide most of cosine
dir
what a font designer needs. angle
mlog
You type And the result is mexp

sind 30 0.5
cosd 30 0.86603
sind -30 -0.5
cosd 360 1
sind 10 ++ cosd 10 1
dir 30 (0.86603,0.5)
dir -90 (0,-1)
angle(1,1) 45
angle(1,2) 63.43495
angle(1,-2) -63.43495
sind 63.43495 / cosd 63.43495 1.99997
angle up 90
angle left 180
angle(-1000,-epsilon) -180
angle dir 60 60.00008
0 (with error message)
angle(0,0)

Plain de¬nes ˜dir x™ to be the pair of values (cosd x, sind x); this is a vector,
which points x degrees above the rightward horizon. Conversely, the ˜angle™ operator
determines the angle that corresponds to a given vector.

Logarithms and exponentials are computed with respect to an unusual base,
designed to enhance the accuracy of calculations involving ¬xed-radix numbers
™s range. The values mlog x = 256 ln x and mexp x = ex/256 produce
in
reasonably good results when x —— y is computed by the formula mexp(y — mlog x).

You type And the result is
mlog 2 177.44568
mexp mlog 2 2
mexp 8 mlog 2 256
mexp 256 2.71828
mlog 2.71828 255.99954
mlog 2.71829 256.00098
15 mlog 2 2661.68518
mexp 2661.68518 32767.99998
32767.99998 (with error message)
mexp 2661.68519
mexp-2661.68519 0.00003
68 Chapter 8: Algebraic Expressions


also generates two ¬‚avors of random numbers. It is very unlikely uniformdeviate
normaldeviate
that you will get the particular values shown in the following examples, when
scaled
you do the experiment yourself, because the results come out di¬erent each time the xscaled
computer is asked for a new random number (unless you have speci¬ed a “seed value” yscaled
dir
as explained in Chapter 21).

You type And the result might be
uniformdeviate 100 47.4241
uniformdeviate 100 97.28148
uniformdeviate -100 -36.16279
(normaldeviate,normaldeviate) (0.46236,-1.87648)

The value of ˜uniformdeviate 100™ is a random number between 0 and 100; the value
of ˜normaldeviate™ is a normally distributed random number whose mean value is zero
and whose standard deviation is unity. Chapter 21 explains what this means and gives
several applications.

Besides all of these operations on numbers, has a rich collection
of operations on pairs, some of which are indicated in the following examples:

You type And the result is
right (1,0)
(1,2)+(3,4) (4,6)
1/3(3,10) (1,3.33333)
z2-z1 (-x1+x2,-y1+y2)
.2[z1,z2] (0.8x1+0.2x2,0.8y1+0.2y2)
3z (3x,3y)
z scaled 3 (3x,3y)
z xscaled 2 yscaled 1/2 (2x,0.5y)
z shifted (2,3) (x+2,y+3)
z shifted 3right (x+3,y)
z slanted 1/6 (x+0.16667y,y)
z rotated 90 (-y,x)
z rotated 30 (-0.5y+0.86603x,0.86603y+0.5x)
xpart(z rotated 30) -0.5y+0.86603x
ypart(z rotated 30) 0.86603y+0.5x
(3,4) (with error message)
(1,2)*(3,4)
(1,2)zscaled(3,4) (-5,10)
(a,b)zscaled(3,4) (3a-4b,4a+3b)
(a,b)zscaled dir 30 (0.86603a-0.5b,0.5a+0.86603b)
(1,2)dotprod(3,4) 11
(a,b)dotprod(3,4) 3a+4b
dir 21 dotprod dir 51 0.86603
(3,4)dotprod((30,40)rotated 90) 0
Chapter 8: Algebraic Expressions 69


(Recall that plain converts ˜z$™ into ˜(x$,y$)™ when $ is any su¬x .) The xpart
ypart
operations exhibited here are almost all self-evident. When a point or vector is rotated,
shifted
it is moved counterclockwise about (0, 0) through a given number of degrees. - right
computes the rotated coordinates by using sines and cosines in an appropriate slanted
zscaled
way; you don™t have to remember the formulas! Although you cannot use ˜*™ to multiply dotprod
a pair by a pair, you can use ˜zscaled™ to get the e¬ect of complex number multiplication: z
Since (1 + 2i) times (3 + 4i) is ’5 + 10i, we have (1, 2) zscaled (3, 4) = (’5, 10). There™s rotated
sines
also a multiplication that converts pairs into numbers: (a, b) dotprod (c, d) = ac + bd. cosines
This is the “dot product,” often written ˜(a, b) · (c, d)™ in mathematics texts; it turns zscaled
complex number
out to be equal to a ++ b times c ++ d times the cosine of the angle between the vectors
multiplication
(a, b) and (c, d). Since cosd 90—¦ = 0, two vectors are perpendicular to each other if and dot product
only if their dot product is zero. perpendicular
product
string
There are operations on strings, paths, and the other types too; we shall study
point
such things carefully in later chapters. For now, it will su¬ce to give a few direction
examples, keeping in mind that the ¬le expr.mf de¬nes s with any subscript to be a length
cycle
string, while p with any subscript is a path. Furthermore s1 has been given the value
"abra", while p1 is ˜(0, 0) . . (3, 3)™ and p2 is ˜(0, 0) . . (3, 3) . . cycle ™.
You type And the result is
s2 unknown string s2
s1&"cad"&s1 "abracadabra"
length s1 4
length p1 1
length p2 2
cycle p1 false
cycle p2 true
substring (0,2) of s1 "ab"
substring (2,infinity) of s1 "ra"
point 0 of p1 (0,0)
point 1 of p1 (3,3)
point .5 of p1 (1.5,1.5)
point infinity of p1 (3,3)
point .5 of p2 (3,0)
point 1.5 of p2 (0,3)
point 2 of p2 (0,0)
point 2+epsilon of p2 (0.00009,-0.00009)
point -epsilon of p2 (-0.00009,0.00009)
point -1 of p1 (0,0)
direction 0 of p1 (1,1)
direction 0 of p2 (4,-4)
direction 1 of p2 (-4,4)
The length of a path is the number of ˜. .™ steps that it contains; the construction
˜cycle path ™ can be used to tell whether or not a particular path is cyclic. If you say
70 Chapter 8: Algebraic Expressions


just ˜p1™ you get to see path p1 with its control points: control points
substring
subpath
(0,0)..controls (1,1) and (2,2)
ampersand
..(3,3)

Similarly, ˜p2™ is
(0,0)..controls (2,-2) and (5,1)
..(3,3)..controls (1,5) and (-2,2)
..cycle

and ˜subpath (0,1) of p2™ is analogous to a substring:
(0,0)..controls (2,-2) and (5,1)
..(3,3)

The expression ˜point t of p2 ™ gives the position of a point that moves along path p2 ,
starting with the initial point (0, 0) at t = 0, then reaching point (3, 3) at t = 1,
etc.; the value at t = 1/2 is the third-order midpoint obtained by the construction of
Chapter 3, using intermediate control points (2, ’2) and (5, 1). Since p2 is a cyclic
path of length 2, point (t + 2) of p2 is the same as point t. Path p1 is not cyclic, so its
points turn out to be identical to point 0 when t < 0, and identical to point 1 when
t > 1. The expression ˜direction t of path ™ is similar to ˜point t of path ™; it yields a
vector for the direction of travel at time t.

Paths are not necessarily traversed
at constant speed. For example, the
diagram at the right shows point t of p2 at
twenty equally spaced values of t. -
moves faster in this case at time 1.0
than at time 1.2; but the points are spread (Figure 8a will be inserted here; too bad you
out fairly well, so the concept of fractional can™t see it now.)

time can be useful. The diagram shows, in-
cidentally, that path p2 is not an especially
good approximation to a circle; there is no
left-right symmetry, although the curve from
point 1 to point 2 is a mirror image of the
curve from point 0 to point 1. This lack of
circularity is not surprising, since p2 was de¬ned by simply specifying two points, (0, 0)
and (3, 3); at least four points are needed to get a path that is convincingly round.

The ampersand operation ˜&™ can be used to splice paths together in much the
same way as it concatenates strings. For example, if you type ˜p2 & p1™, you
get the path of length 3 that is obtained by breaking the cyclic connection at the end
of path p2 and attaching p1 :
(0,0)..controls (2,-2) and (5,1)
..(3,3)..controls (1,5) and (-2,2)
..(0,0)..controls (1,1) and (2,2)
..(3,3)

Concatenated paths must have identical endpoints at the junction.
Chapter 8: Algebraic Expressions 71


You can even “slow down the clock” by concatenating subpaths that have precedence
primary
non-integer time speci¬cations. For example, here™s what you get if you ask
secondary
for ˜subpath (0,.5) of p2 & subpath (.5,2) of p2 & cycle™: tertiary
expression
(0,0)..controls (1,-1) and (2.25,-0.75) ± primary
(
..(3,0)..controls (3.75,0.75) and (4,2)
)
..(3,3)..controls (1,5) and (-2,2) ± secondary
..cycle ± tertiary
± expression
When t goes from 0 to 1 in subpath (0, .5) of p2 , you get the same points as when t
goes from 0 to .5 in p2 ; when t goes from 0 to 1 in subpath (.5, 2) of p2 , you get the
same points as when t goes from .5 to 1 in p2 ; but when t goes from 1 to 2 in subpath
(.5, 2) of p2 , it™s the same as the segment from 1 to 2 in p2 .
Let™s conclude this chapter by discussing the exact rules of precedence by
which decides what operations to do ¬rst. The informal notion of
“magnetism” gives a good intuitive picture of what happens, but syntax rules express
things unambiguously in borderline cases.
The four levels of precedence correspond to four kinds of formulas, which
are called primaries, secondaries, tertiaries, and expressions. A primary is
a variable or a constant or a tightly bound unit like ˜2x™ or ˜sqrt 2™; a secondary is a
primary or a sequence of primaries connected by multiplicative operators like ˜*™ or
˜scaled™; a tertiary is a secondary or a sequence of secondaries connected by additive
operators like ˜+™ or ˜++™; an expression is a tertiary or a sequence of tertiaries connected
by external operators like ˜<™ or ˜..™. For example, the expression
a+b/2>3c*sqrt4d
is composed of the primaries ˜a™, ˜b™, ˜2™, ˜3c™, and ˜sqrt4d™; the last of these is a primary
containing ˜4d™ as a primary within itself. The subformulas ˜a™, ˜b/2™, and ˜3c*sqrt4d™
are secondaries; the subformulas ˜a+b/2™ and ˜3c*sqrt4d™ are tertiaries.
If an expression is enclosed in parentheses, it becomes a primary that can be
used to build up larger secondaries, tertiaries, etc.
The full syntax for expressions is quite long, but most of it falls into a simple
pattern. If ±, β, and γ are any “types””numeric, boolean, string, etc.”then
± variable refers to a variable of type ±, β primary refers to a primary of type β,
and so on. Almost all of the syntax rules ¬t into the following general framework:
± primary ’’ ± variable | ± constant | ( ± expression )
| operator that takes type β to type ± β primary
± secondary ’’ ± primary
| β secondary multiplicative op taking types β and γ to ± γ primary
± tertiary ’’ ± secondary
| β tertiary additive op taking types β and γ to ± γ secondary
± expression ’’ ± tertiary
| β expression external op taking types β and γ to ± γ tertiary
These schematic rules don™t give the whole story, but they do give the general structure
of the plot.
72 Chapter 8: Algebraic Expressions


Chapter 25 spells out all of the syntax rules for all types of expressions. We numeric primary
[
shall consider only a portion of the numeric and pair cases here, in order to
,
have a foretaste of the complete menu: ]
length
numeric primary ’’ numeric atom length
| numeric atom [ numeric expression , numeric expression ] length
angle
| length string primary xpart
| length path primary ypart
numeric atom
| length pair primary (
| angle pair primary )

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