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|
-- tkz_elements_lines.lua
-- date 2025/01/06
-- version 3.10
-- Copyright 2024 Alain Matthes
-- This work may be distributed and/or modified under the
-- conditions of the LaTeX Project Public License, either version 1.3
-- of this license or (at your option) any later version.
-- The latest version of this license is in
-- http://www.latex-project.org/lppl.txt
-- and version 1.3 or later is part of all distributions of LaTeX
-- version 2005/12/01 or later.
-- This work has the LPPL maintenance status “maintained”.
-- The Current Maintainer of this work is Alain Matthes.
-- -------------------------------------------------------------------------
-- Lines
-- -------------------------------------------------------------------------
line = {}
function line:new(za, zb)
local type = 'line'
local mid = (za+zb)/2
local north_pa = rotation_ (za,math.pi/2,zb)
local south_pa = rotation_ (za,-math.pi/2,zb)
local north_pb = rotation_ (zb,-math.pi/2,za)
local south_pb = rotation_ (zb,math.pi/2,za)
local west = rotation_ (za,math.pi/2,north_pa)
local east = rotation_ (zb,math.pi/2,south_pb)
local slope = angle_normalize_(point.arg(zb-za))
local length = point.mod(zb-za)
local vec = vector : new (za,zb)
local o = {pa = za,
pb = zb,
north_pa = north_pa,
south_pa = south_pa,
west = west,
east = east,
north_pb = north_pb,
south_pb = south_pb,
slope = slope,
mid = mid,
type = type,
vec = vec,
length = length}
setmetatable(o, self)
self.__index = self
return o
end
-------------------
-- Result -> real
-------------------
function line:distance(pt)
return point.mod(projection(self,pt)-pt)
end
function line:slope()
return slope_(self.pa,self.pb)
end
-------------------
-- Result -> boolean
-------------------
function line:in_out(pt)
return math.abs((pt - self.pa) ^ (pt - self.pb)) <= tkz_epsilon
end
function line:in_out_segment(pt)
return point.mod(pt - self.pa) + point.mod(pt - self.pb) - point.mod(self.pb - self.pa) <= tkz_epsilon
end
function line:is_parallel(L)
return math.abs(self.slope - L.slope) < tkz_epsilon
end
function line:is_orthogonal(L)
return math.abs(self.slope * L.slope + 1) < tkz_epsilon
end
function line:is_equidistant(p)
return math.abs( (point.mod(self.pa-p)-(point.mod(self.pb-p)))) < tkz_epsilon
end
-------------------
-- Result -> point
-------------------
function line:barycenter(ka,kb)
return barycenter_({self.pa,ka},{self.pb,kb})
end
function line:point(t) -- t=o A t=1 B t = AM / AB
return barycenter_({self.pa,1-t},{self.pb,(t)})
end
function line:midpoint()
return (self.pa+self.pb)/2
end
function line:harmonic_int(pt)
return div_harmonic_int_(self.pa,self.pb,pt)
end
function line:harmonic_ext(pt)
return div_harmonic_ext_(self.pa,self.pb,pt)
end
function line:harmonic_both(k)
return div_harmonic_both_(self.pa,self.pb,k)
end
function line:gold_ratio()
return self.pa + (self.pb-self.pa)*tkzinvphi
end
function line:normalize()
return self.pa+(self.pb-self.pa)/point.mod(self.pb-self.pa)
end
function line:normalize_inv()
return normalize_ (self.pb,self.pa)
end
function line:_east(d)
local d = d or 1
return self.pb+ d/self.length * (self.pb-self.pa)
end
function line:_west(d)
local d = d or 1
return self.pa+ d/self.length * (self.pa-self.pb)
end
function line:_north_pa(d)
local d = d or 1
return d/self.length * ( self.north_pa - self.pa ) + self.pa
end
function line:_south_pa(d)
local d = d or 1
return d/self.length *( self.south_pa - self.pa ) + self.pa
end
function line:_south_pb(d)
local d = d or 1
return d/self.length *( self.south_pb - self.pb ) + self.pb
end
function line:_north_pb(d)
local d = d or 1
return d/self.length *( self.north_pb - self.pb ) + self.pb
end
function line:report(d, pt)
if not self.length or self.length == 0 then
error("self.length must be non-zero")
end
local t = d / self.length
local result = barycenter_({self.pa, 1 - t}, {self.pb, t})
if pt then
return result + pt - self.pa
else
return result
end
end
function line:colinear_at (pt,k)
if k == nil
then
return colinear_at_ (self.pa,self.pb,pt,1)
else
return colinear_at_ (self.pa,self.pb,pt,k)
end
end
-------------- transformations -------------
function line:translation_pt( pt )
return translation_ ( self.pb-self.pa,pt )
end
function line:translation_C( obj )
local pa,pb,x,y
pa = obj.center
pb = obj.through
x,y = set_translation_( self.pb-self.pa,pa,pb )
return circle : new (x,y)
end
function line: translation_T( obj )
local pa,pb,pc,x,y,z
pa = obj.pa
pb = obj.pb
pc = obj.pc
x,y,z = set_translation_( self.pb-self.pa,pa,pb,pc )
return triangle : new (x,y,z)
end
function line: translation_L( obj )
local pa,pb,x,y
pa = obj.pa
pb = obj.pb
x,y = set_translation_ ( self.pb-self.pa,pa,pb )
return line : new (x,y)
end
function line:translation(...)
local obj, nb, t
local tp = table.pack(...)
obj = tp[1]
nb = tp.n
-- If only one object is passed
if nb == 1 then
if obj.type == "point" then
return translation_(self.pb - self.pa, obj) -- Translate point
elseif obj.type == "line" then
return self:translation_L(obj) -- Translate line
elseif obj.type == "triangle" then
return self:translation_T(obj) -- Translate triangle
elseif obj.type == "circle" then
return self:translation_C(obj) -- Translate circle
else
error("Unsupported object type for translation")
end
else
-- If multiple objects are passed, translate each one
t = {}
for i = 1, nb do
-- Translate each object using the translation vector
table.insert(t, translation_(self.pb - self.pa, tp[i]))
end
return table.unpack(t) -- Return the translated objects
end
end
function line: set_translation ( ...)
return set_translation_ ( self.pb-self.pa,... )
end
function line:projection(...)
local obj,nb,t
local tp = table.pack(...)
obj = tp[1]
nb = tp.n
if nb == 1 then
return projection_ ( self.pa, self.pb, obj )
else
t = {}
for i=1,tp.n do
table.insert( t , projection_ (self.pa, self.pb, tp[i]) )
end
return table.unpack ( t )
end
end
function line:set_projection(...)
local tp = table.pack(...)
local i
local t = {}
for i=1,tp.n do
table.insert( t , projection_ (self.pa,self.pb , tp[i]) )
end
return table.unpack ( t )
end
function line:symmetry_axial_L( obj )
local pa,pb,x,y
pa = obj.pa
pb = obj.pb
x,y = self:set_reflection(pa, pb)
return line : new (x,y)
end
function line:symmetry_axial_T( obj )
local pa,pb,pc,x,y,z
pa = obj.pa
pb = obj.pb
pc = obj.pc
x,y,z = self:set_reflection (pa, pb, pc)
return triangle : new (x,y,z)
end
function line:symmetry_axial_C( obj )
local pa,pb,x,y
pa = obj.center
pb = obj.through
x,y = self:set_reflection( pa,pb )
return circle : new (x,y)
end
function line:reflection(...)
local obj,nb,t
local tp = table.pack(...)
obj = tp[1]
nb = tp.n
if nb == 1 then
if obj.type == "point" then
return symmetry_axial_ ( self.pa,self.pb,obj )
elseif obj.type == "line" then
return self: symmetry_axial_L (obj)
elseif obj.type == "triangle" then
return self: symmetry_axial_T (obj)
else
return self: symmetry_axial_C (obj)
end
else
t = {}
for i=1,tp.n do
table.insert( t , symmetry_axial_ ( self.pa,self.pb , tp[i]) )
end
return table.unpack ( t )
end
end
function line:set_reflection (...)
return set_symmetry_axial_ ( self.pb,self.pa,... )
end
-------------------
-- Result -> line
-------------------
function line:ll_from( pt )
return line : new (pt,pt+self.pb-self.pa)
end
function line:ortho_from( pt )
return line : new (pt+(self.pb-self.pa)*point(0,-1),pt+(self.pb-self.pa)*point(0,1))
end
function line:mediator()
local m = midpoint_ (self.pa,self.pb)
return line : new (rotation_ (m,-math.pi/2,self.pb),rotation_ (m,math.pi/2,self.pb))
end
function line:perpendicular_bisector ()
local m = midpoint_ (self.pa,self.pb)
return line : new (rotation_ (m,-math.pi/2,self.pb),rotation_ (m,math.pi/2,self.pb))
end
-------------------
-- Result -> circle
-------------------
function line:circle(swap)
swap = swap or false
if swap then
return circle:new(self.pb,self.pa)
else
return circle:new(self.pa,self.pb)
end
end
function line:circle_swap()
return circle : new (self.pb,self.pa)
end
function line:diameter()
local c = midpoint_(self.pa,self.pb)
return circle:new (c,self.pb)
end
function line:apollonius(k)
local z1,z2,c
z1 = barycenter_({self.pa,1},{self.pb,k})
z2 = barycenter_({self.pa,1},{self.pb,-k})
c = midpoint_ (z1,z2)
return circle : new (c,z2)
end
function line:test(x,y)
end
-- Circle tangent to a line passing through two points
-- In general, there are two solutions
function line:c_l_pp(a, b) -- a and b on the same side
-- Initialisation
local lab = line:new(a, b) -- Line through a and b
local Cab = circle:diameter(a, b) -- Circle with a and b diameters
local i = intersection(lab, self) -- Intersection with current line
-- One point on the line (a)
if self : in_out(a) and not self:in_out(b) then
local lmed = lab : mediator()
local laperp = self:ortho_from(a)
local o = intersection(lmed,laperp)
return circle:new(o,a),
circle:new(o,a)
end
-- One point on the line (b)
if self:in_out(b) and not self:in_out(a) then
local lmed = lab:mediator()
local laperp = self:ortho_from(b)
local o = intersection(lmed,laperp)
return circle:new(o,b),
circle:new(o,b)
end
-- Check: if the intersection exists and lies on the segment [a, b].
if i and lab:in_out_segment(i) then
return nil, nil -- No circle is possible
end
-- If the current line is orthogonal to lab
if self:is_orthogonal(lab) then
local lmed = lab:mediator()
local m = midpoint(a, b)
local r = length(m, i)
local pt1 = lab:isosceles_s(r)
local pt2 = lab:isosceles_s(r, true)
return circle:new(pt1, a),
circle:new(pt2, a)
end
-- If the two lines are parallel
if lab:is_parallel(self) then
local mid = midpoint(a, b) -- Midpoint of segment [a, b]
local proj = self:projection(mid) -- Mid projection on the running line
return circle:new(circum_center_(a, b, proj), proj),
circle:new(circum_center_(a, b, proj), proj)
end
-- General case
local t = Cab:tangent_from(i).pb
local x, y = intersection(self, circle:new(i, t))
return circle:new(intersection(self:ortho_from(x), lab:mediator()), x),
circle:new(intersection(self:ortho_from(y), lab:mediator()), y)
end
-- Circle tangent to two straight lines passing through a given point
function line:c_ll_p(a, p)
-- Compute the bisector of the triangle formed by self.pa, self.pb, and a
local lbi = bisector(self.pa, self.pb, a)
if lbi:in_out(p) then
-- Orthogonal projection of p onto the bisector
local lp = lbi:ortho_from(p)
-- Intersection of line from p to its projection with self.pa and self.pb
local i = intersection_ll_(p, lp.pb, self.pa, self.pb)
-- Intersection points of the line with the circle defined by (i, p)
local t1, t2 = intersection_lc_(self.pa, self.pb, i, p)
-- Create the main line and find orthogonal projections from t1 and t2
local lab = line:new(self.pa, self.pb)
local x = lab:ortho_from(t1).pb
local y = lab:ortho_from(t2).pb
-- Return two circles based on the orthogonal projections and points t1, t2
return circle:new(intersection_ll_(x, t1, self.pa, p), t1),
circle:new(intersection_ll_(y, t2, self.pa, p), t2)
else
-- Reflection of p across the bisector
local q = lbi : reflection (p)
-- Compute circles from the Wallis construction
local c1, c2 = self:c_l_pp(p, q)
-- Return two circles with centers and points on their circumference
return c1,c2
end
end
----------------------
-- Result -> triangle
----------------------
function line:equilateral(swap)
swap = swap or false
if swap then
return triangle:new(self.pa, self.pb, rotation_(self.pa, -math.pi / 3, self.pb))
else
return triangle:new(self.pa, self.pb, rotation_(self.pa, math.pi / 3, self.pb))
end
end
function line:isosceles(phi,swap)
local pta,ptb
swap = swap or false
if swap then
pta = rotation_(self.pa,-phi,self.pb)
ptb = rotation_(self.pb,phi,self.pa)
return triangle : new (self.pa,self.pb, intersection_ll_(self.pa,pta,self.pb,ptb ))
else
pta = rotation_(self.pa,phi,self.pb)
ptb = rotation_(self.pb,-phi,self.pa)
return triangle : new (self.pa,self.pb, intersection_ll_ (self.pa,pta,self.pb,ptb ))
end
end
line.isosceles_a = line.isosceles
function line:isosceles_s(a,swap)
local c1,c2,pta,ptb,pt1,pt2
c1 = circle : radius (self.pa,a)
c2 = circle : radius (self.pb,a)
pta,ptb = intersection_cc (c1,c2)
if get_angle(self.pa,self.pb,pta) < get_angle(self.pa,self.pb,ptb) then
pt1=pta pt2=ptb
else pt1=ptb pt2=pta end
swap = swap or false
if swap then
return triangle : new (self.pa,self.pb,pt2 )
else
return triangle : new (self.pa,self.pb,pt1)
end
end
function line:two_angles(alpha,beta,swap)
local pta,ptb,pt
swap = swap or false
if swap then
pta = rotation_(self.pa,-alpha,self.pb)
ptb = rotation_(self.pb,beta,self.pa)
else
pta = rotation_(self.pa,alpha,self.pb)
ptb = rotation_(self.pb,-beta,self.pa)
end
pt = intersection_ll_(self.pa,pta,self.pb,ptb)
return triangle:new(self.pa,self.pb,pt)
end
function line:school(swap)
local pta,ptb,pt
swap = swap or false
if swap then
pta = rotation_(self.pa,-math.pi/6,self.pb)
ptb = rotation_(self.pb,math.pi/3,self.pa)
else
pta = rotation_(self.pa,math.pi/6,self.pb)
ptb = rotation_(self.pb,-math.pi/3,self.pa)
end
pt = intersection_ll_(self.pa,pta,self.pb,ptb)
return triangle:new(self.pa,self.pb,pt)
end
function line:half(swap)
local x,pt
x = midpoint_(self.pa,self.pb)
swap = swap or false
if swap then
pt = rotation_(self.pb,math.pi/2,x)
else
pt = rotation_(self.pb,-math.pi/2,x)
end
return triangle:new(self.pa,self.pb,pt)
end
function line:sss(a,b,swap)
local pta,ptb,i,j
swap = swap or false
pta = self.pa + point ( a, 0 )
ptb = self.pb + point ( -b , 0)
i,j = intersection_cc_ (self.pa,pta,self.pb,ptb)
if swap then
return triangle : new (self.pa,self.pb,j)
else
return triangle : new (self.pa,self.pb,i)
end
end
function line:ssa(a, phi,swap)
local x, y, i, j
swap = swap or false
x = rotation_(self.pb, -phi, self.pa)
y = self.pa + polar_(a, self.slope)
i, j = intersection_lc_(self.pb, x, self.pa, y)
if swap then
return triangle:new(self.pa, self.pb, j)
else
return triangle:new(self.pa, self.pb, i)
end
end
function line:sas(a, phi,swap)
local x, pt
swap = swap or false
x = self.pa + polar_(a, self.slope)
if swap then
pt = rotation_(self.pa, -phi, x)
else
pt = rotation_(self.pa, phi, x)
end
return triangle:new(self.pa, self.pb, pt)
end
function line:asa(alpha,beta,swap)
local pta,ptb,pt
swap = swap or false
if swap then
pta = rotation_ (self.pa,-alpha,self.pb)
ptb = rotation_ (self.pb,beta,self.pa)
else
pta = rotation_ (self.pa,alpha,self.pb)
ptb = rotation_ (self.pb,-beta,self.pa)
end
pt = intersection_ll_ (self.pa,pta,self.pb,ptb)
return triangle : new (self.pa,self.pb,pt)
end
---- sacred triangles ----
function line:gold(swap)
local pt
swap = swap or false
if swap then
pt = rotation_ (self.pa,-math.pi/2,self.pb)
return triangle : new (self.pa,self.pb, self.pa + (pt-self.pa) * tkzinvphi)
else
pt = rotation_ (self.pa,math.pi/2,self.pb)
return triangle : new (self.pa,self.pb, self.pa + (pt-self.pa) * tkzinvphi)
end
end
function line:sublime(swap)
local pta,ptb,pt
swap = swap or false
if swap then
pta = rotation_(self.pa,-2*math.pi/5,self.pb)
ptb = rotation_(self.pb,2*math.pi/5,self.pa)
pt = intersection_ll_(self.pa,pta,self.pb,ptb)
return triangle:new(self.pa,self.pb,pt)
else
pta = rotation_(self.pa,2*math.pi/5,self.pb)
ptb = rotation_(self.pb,-2*math.pi/5,self.pa)
pt = intersection_ll_(self.pa,pta,self.pb,ptb)
return triangle:new(self.pa,self.pb,pt)
end
end
line.euclid = line.sublime
function line:euclide (swap)
swap = swap or false
if swap then
return triangle : new (self.pa,self.pb, rotation_(self.pa,-math.pi/5,self.pb))
else
return triangle : new (self.pa,self.pb, rotation_(self.pa,math.pi/5,self.pb))
end
end
function line:divine(swap)
local pta, ptb, pt
swap = swap or false
if swap then
pta = rotation_(self.pa, -math.pi / 5, self.pb)
ptb = rotation_(self.pb, math.pi / 5, self.pa)
pt = intersection_ll_(self.pa, pta, self.pb, ptb)
return triangle:new(self.pa, self.pb, pt)
else
pta = rotation_(self.pa, math.pi / 5, self.pb)
ptb = rotation_(self.pb, -math.pi / 5, self.pa)
pt = intersection_ll_(self.pa, pta, self.pb, ptb)
return triangle:new(self.pa, self.pb, pt)
end
end
function line:cheops(swap)
local m, n, pt
m = midpoint_(self.pa, self.pb)
swap = swap or false
if swap then
n = rotation_(m, math.pi / 2, self.pa)
pt = m + (n - m) * tkzsqrtphi
else
n = rotation_(m, -math.pi / 2, self.pa)
pt = m + (n - m) * tkzsqrtphi
end
return triangle:new(self.pa, self.pb, pt)
end
function line:egyptian(swap)
local n, pt
swap = swap or false
if swap then
n = rotation_(self.pb, math.pi / 2, self.pa)
pt = self.pb + (n - self.pb) / point.mod(n - self.pb) * self.length * 0.75
else
n = rotation_(self.pb, -math.pi / 2, self.pa)
pt = self.pb + (n - self.pb) / point.mod(n - self.pb) * self.length * 0.75
end
return triangle:new(self.pa, self.pb, pt)
end
line.pythagoras = line.egyptian
line.isis = line.egyptian
line.golden = line.sublime
line.golden_gnomon = line.divine
------------------------------
-- Result -> square
------------------------------
function line:square (swap)
swap = swap or false
if swap
then
return square : side (self.pa,self.pb,indirect)
else
return square : side (self.pa,self.pb)
end
end
return line
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