Point And Complex 

It is not difficult to redefine the meaning of the usual arithmetic operators for userdefined types. Normally, this will be a runtime error; if I have a table t
then t + 1
will give "attempt to perform arithmetic on global `t' (a table value)". However, if that table has a metatable, then Lua will see if a function __add
is defined, and use it instead.
The main question with operator overloading is whether it will make using your objects easier for other programmers. It is similar to userinterface decisions; it really is a good idea to keep to common userinterface conventions for your platform. In interfaces, dull is good, and the unexpected will cause misunderstanding. For instance, you can make the addition operator concatenate lists, but as Paul Graham observed [1], one tends to misread such expressions as arithmetic.
I'm going to present two cases where redefining operators makes perfect sense, because they are both generalizations of the usual real number arithmetic we use. To add two points with p1 + p2
is not only convenient, but mathematically correct.
Using this Point
class, you will be able to express vector algebra in something like the usual notation. For example,
x = ((p1^p2)..q)*q
p1
and p2
, get its dot product with q
and multiply the resulting scalar with q.
 point.lua  A class representing vectors in 3D  (for class.lua, see SimpleLuaClasses) require 'class' Point = class(function(pt,x,y,z) pt:set(x,y,z) end) local function eq(x,y) return x == y end function Point.__eq(p1,p2) return eq(p1[1],p2[1]) and eq(p1[2],p2[2]) and eq(p1[3],p2[3]) end function Point.get(p) return p[1],p[2],p[3] end  vector addition is '+','' function Point.__add(p1,p2) return Point(p1[1]+p2[1], p1[2]+p2[2], p1[3]+p2[3]) end function Point.__sub(p1,p2) return Point(p1[1]p2[1], p1[2]p2[2], p1[3]p2[3]) end  unitary minus (e.g in the expression f(p)) function Point.__unm(p) return Point(p[1], p[2], p[3]) end  scalar multiplication and division is '*' and '/' respectively function Point.__mul(s,p) return Point( s*p[1], s*p[2], s*p[3] ) end function Point.__div(p,s) return Point( p[1]/s, p[2]/s, p[3]/s ) end  dot product is '..' function Point.__concat(p1,p2) return p1[1]*p2[1] + p1[2]*p2[2] + p1[3]*p2[3] end  cross product is '^' function Point.__pow(p1,p2) return Point( p1[2]*p2[3]  p1[3]*p2[2], p1[3]*p2[1]  p1[1]*p2[3], p1[1]*p2[2]  p1[2]*p2[1] ) end function Point.normalize(p) local l = p:len() p[1] = p[1]/l p[2] = p[2]/l p[3] = p[3]/l end function Point.set(pt,x,y,z) if type(x) == 'table' and getmetatable(x) == Point then local po = x x = po[1] y = po[2] z = po[3] end pt[1] = x pt[2] = y pt[3] = z end function Point.translate(pt,x,y,z) pt[1] = pt[1] + x pt[2] = pt[2] + y pt[3] = pt[3] + z end function Point.__tostring(p) return string.format('(%f,%f,%f)',p[1],p[2],p[3]) end local function sqr(x) return x*x end function Point.len(p) return math.sqrt(sqr(p[1]) + sqr(p[2]) + sqr(p[3])) end
Point
is a simple class which can be constructed using call notation (see SimpleLuaClasses):
> p1 = Point(10,20,30) > p2 = Point(1,2,3) > = p1 (10.000000,20.000000,30.000000) > = p1 + p2 (11.000000,22.000000,33.000000) > = 2*p1 (20.000000,40.000000,60.000000)
Point
defines __tostring
, Lua knows how to print out such objects. It's perhaps not a perfect format, but it's easy to modify the code (one could make precision a class property). The simplified constructor syntax makes it easy for vector operations to return Point
objects. The decision to use indices 1,2 and 3 for x, y and z components is quite arbitrary; it's easy to change to match your preferences. There are some limitations: although p*2
should be valid, it isn't; the scalar must be first.
One can now define higher level operations. For instance, here is a useful function for finding the minimum distance between a point and a line (very useful if you're doing an editable graphics program):
 given a point q, where does the perp cross the line (p1,p2)? function perp_to_line(p1,p2,q) local diff = p2  p1 local x = ((q  p1)..diff)/(diff..diff) return p1 + x*diff end  minimum distance between q and a line (p1,p2) function min_dist_to_line(p1,p2,q) local perp = perp_to_line(p1,p2,q) return Point.len(perpq) end
There is a 'gotcha' which you should keep in mind; Lua objects are always passed by reference, so beware of modifying points passed to functions. Use the copy constructor provided, e.g. say local pc = Point(p)
to make a local copy of a point passed as an argument.
Complex numbers are a generalization of real numbers, so they understand all the usual operations, plus some more. If z
is complex, then 1.5 + z
and z + 1.5
are both complex expressions, so __add
must handle the case where one of the arguments is a plain number.
Please note that this Complex Number class is an example, it should not be used for serious applications. It has some problems (which could, in principle, be fixed): division suffers from loss of precision, modulus overflows on some reasonable values, square root is careless with cuts and does not work for real values, pow only computes positive integer powers (and then slowly).
 complex.lua require 'class' Complex = class(function(c,re,im) if type(re) == 'number' then c.re = re c.im = im else c.re = re.re c.im = re.im end end) Complex.i = Complex(0,1) local sqrt = math.sqrt local cos = math.cos local sin = math.sin local exp = math.exp local function check(z1,z2) if type(z1) == 'number' then return Complex(z1,0),z2 elseif type(z2) == 'number' then return z1,Complex(z2,0) else return z1,z2 end end  redefine arithmetic operators! function Complex.__add(z1,z2) local c1,c2 = check(z1,z2) return Complex(c1.re + c2.re, c1.im + c2.im) end function Complex.__sub(z1,z2) local c1,c2 = check(z1,z2) return Complex(c1.re  c2.re, c1.im  c2.im) end function Complex:__unm() return Complex(self.re, self.im) end function Complex.__mul(z1,z2) local c1,c2 = check(z1,z2) return Complex(c1.re*c2.re  c1.im*c2.im, c1.im*c2.re + c1.re*c2.im) end function Complex.__div(z1,z2) local c1,c2 = check(z1,z2) local a = c1.re local b = c1.im local c = c2.re local d = c2.im local lensq = c*c + d*d local ci = (a*c + b*d)/lensq local cr = (b*c + a*d)/lensq return Complex(cr,ci) end function Complex.__pow(z,n) local res = Complex(z) for i = 1,n1 do res = res*z end return res end  this is how our complex numbers will present themselves! function Complex:__tostring() return self.re..' + '..self.im..'i' end  operations only valid for complex numbers function Complex.conj(z) return Complex(z.re,z.im) end function Complex.mod(z) return sqrt(z.re^2 + z.im^2) end  generalizations of sqrt() and exp() function Complex.sqrt(z) local y = sqrt((Complex.mod(z)z.re)/2) local x = z.im/(2*y) return Complex(x,y) end function Complex.exp(z) return exp(z.re)*Complex(cos(z.im),sin(z.im)) end
Obviously vector and complex arithmetic is going to be fairly slow in Lua, but that's relative. I can create 100,000 points in about two seconds; same time for adding that number. If your program works with a few thousand points, it's going to be fast enough. Programs that need to work with much larger number of points (such as GIS systems) would simply not use this representation, even in C++. Instead, one could write a Lua extension for arrays of points, and manipulate them in groups.
A^(1)Av
. ".." is odd too, though I admit the two dots somewhat resemble dot product. Lua's automatic string conversion might increase the change for errorif one mistakenly writes "(p1..p2)..3" the ".." is silently taken as regular string concatenation. "*" of course could be understood as scalar product, dot product, or cross product. Unfortunately, the number of operators we have to use is limited. These issues detract from the original argument of the article, but I agree that "+", "", and such make perfect sense for vectors. DavidManura
I cannot agree with you, David. Anybody who has done maths up to first year undergraduate level would recognize '^' as vector product for 3dimensional vectors, or as exterior product more generally. The '..' for inner product is unfortunate, I agree. Alas, the historical quirks of mathematical notation really do not fit well with computers. The best programming language I have come across for accommodating mathematical notation is Gofer (not Hugs or Haskell proper  the standardized prelude screws up, because it was designed by people without sufficient mathematical background).  GavinWraith
^
is rightassociative in Lua but crossproducts are not associative. So, a^b^c
would mean a^(b^c)
. DavidManura
p1 + p2
is not only convenient, but mathematically correct." just about made me cry. There is absolutely no geometrical meaning to adding two points. After skimming the article it was fairly clear that the 'Point' class is also intended to be used for vectors, for some reason? I'm not to change anything here, but I thought it should at least be pointed out. anonymous