-- tkz_elements_circles.lua
-- date 2025/05/25
-- version 4.00c
-- Copyright 2025  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.
---------------------------------------------------------------------------
--                           circles
---------------------------------------------------------------------------
circle = {}
 circle.__index = circle
function circle:new(c, t) -- c --> center t --> through
	local type = "circle"
	local ct = line:new(c, t)
	local opp = antipode_(c, t)
	local radius = point.abs(c - t)
	local south = c - point(0, radius)
	local east = c + point(radius, 0)
	local north = c + point(0, radius)
	local west = c - point(radius, 0)
	local perimeter = 2 * math.pi * radius
	local area = 4 * math.pi * radius * radius
	local cir = {
		center = c,
		through = t,
		ct = ct,
		opp = opp,
		radius = radius,
		south = south,
		east = east,
		north = north,
		west = west,
		type = type,
		perimeter = perimeter,
		area = area,
	}
	setmetatable(cir, self)
	return cir
end

setmetatable(circle, {
  __call = function(cls, ...)
    return cls:new(...)
  end
})

function circle:get()
	return self.center, self.through
end

-- other definition
function circle:radius(center, radius)
	return circle:new(center, center + point(radius, 0))
end

function circle:diameter(za, zb)
	return circle:new(midpoint_(za, zb), zb)
end
-----------------------
-- boolean --
-----------------------
function circle:in_out(pt)
	return math.abs(point.abs(pt - self.center) - self.radius) < tkz_epsilon
end

function circle:in_out_disk(pt)
	return point.abs(pt - self.center) <= self.radius
end

function circle:is_tangent(l)
	local a, b = intersection(self, l)
	-- Checks whether the intersection produces valid points
	if not a or not b then
		return false
	end
	-- Checks whether the distance between the two intersection points is less than a given tolerance
	return (point.abs(b - a) < tkz_epsilon)
end
------------------------
-- string --------------
------------------------

function circle:circles_position(C)
	return circles_position_(self.center, self.radius, C.center, C.radius)
end

-----------------------
-- real --
-----------------------
function circle:power(pt)
	local d = point.abs(self.center - pt)
	return d * d - self.radius * self.radius
end
-----------------------
-- points --
-----------------------
function circle:antipode(pt)
	return 2 * self.center - pt
end

function circle:midarc(z1, z2)
	local phi = 0.5 * get_angle(self.center, z1, z2)
	return rotation_(self.center, phi, z1)
end

function circle:point(t)
	local phi = 2 * t * math.pi
	return rotation_(self.center, phi, self.through)
end

function circle:random(inside)
	inside = (inside == "inside")
	math.randomseed(os.time())
	if inside then
		local x = self:point(math.random())
		local L = line:new(self.center, x)
		return L:random()
	else
		return self:point(math.random())
	end
end

function circle:internal_similitude(C)
	return internal_similitude_(self.center, self.radius, C.center, C.radius)
end

function circle:external_similitude(C)
	return external_similitude_(self.center, self.radius, C.center, C.radius)
end

function circle:radical_center(C1, C2)
	if C2 == nil then
		if self.radius > C1.radius then
			return radical_center_(self.center, self.through, C1.center, C1.through)
		else
			return radical_center_(C1.center, C1.through, self.center, self.through)
		end
	else
		return radical_center3(self, C1, C2)
	end
end

-----------------------
-- lines --
-----------------------
function circle:tangent_at(pt)
	return line:new(rotation_(pt, math.pi / 2, self.center), rotation_(pt, -math.pi / 2, self.center))
end

function circle:tangent_from(pt)
	local t1, t2 = tangent_from_(self.center, self.through, pt)
	return line:new(pt, t1), line:new(pt, t2)
end

function circle:radical_axis(C)
	local t1, t2
	if self.radius > C.radius then
		t1, t2 = radical_axis_(self.center, self.through, C.center, C.through)
	else
		t1, t2 = radical_axis_(C.center, C.through, self.center, self.through)
	end
	return line:new(t1, t2)
end

function circle:polar(p)
	local q = self:inversion(p)
	local qa = (p - q):orthogonal(1):at(q)
	local qb = (q - p):orthogonal(1):at(q)
	return line:new(qa, qb)
end

function circle:external_tangent(C)
	local i, t1, t2, k, T1, T2
	-- Find the barycenter of the two circles
	i = barycenter_({ C.center, self.radius }, { self.center, -C.radius })

	-- Calculate the tangents from the circle to the point of intersection
	t1, t2 = tangent_from_(self.center, self.through, i)

	-- Calculate the scaling factor for the homothety
	k = point.mod((C.center - i) / (self.center - i))

	-- Apply homothety to the tangents
	T1 = homothety_(i, k, t1)
	T2 = homothety_(i, k, t2)

	-- Return the two tangent lines
	return line:new(t1, T1), line:new(t2, T2)
end

function circle:internal_tangent(C)
	local i, t1, t2, k, T1, T2
	-- Find the barycenter of the two circles with opposite signs for radii
	i = barycenter_({ C.center, self.radius }, { self.center, C.radius })

	-- Calculate the tangents from the circle to the point of intersection
	t1, t2 = tangent_from_(self.center, self.through, i)

	-- Calculate the scaling factor for the homothety
	k = -point.mod((C.center - i) / (self.center - i))

	-- Apply homothety to the tangents
	T1 = homothety_(i, k, t1)
	T2 = homothety_(i, k, t2)

	-- Return the two tangent lines
	return line:new(t1, T1), line:new(t2, T2)
end

function circle:common_tangent(C)
	local o, s1, s2, t1, t2

	-- Calculating the external similarity between the two circles
	o = external_similitude_(self.center, self.radius, C.center, C.radius)

	if self.radius < C.radius then
		-- Si le rayon de 'self' est plus petit que celui de 'C'
		t1, t2 = tangent_from_(C.center, C.through, o) -- Tangentes depuis le cercle C
		s1, s2 = tangent_from_(self.center, self.through, o) -- Tangentes depuis 'self'

		-- Retourner les tangentes dans un ordre spécifique
		return s1, t1, t2, s2
	else
		-- Si le rayon de 'self' est plus grand ou égal à celui de 'C'
		s1, s2 = tangent_from_(C.center, C.through, o) -- Tangentes depuis le cercle C
		t1, t2 = tangent_from_(self.center, self.through, o) -- Tangentes depuis 'self'

		-- Retourner les tangentes dans un ordre spécifique
		return s1, t1, t2, s2
	end
end

-----------------------
-- circles --
-----------------------

function circle:orthogonal_from(pt)
	-- Calculate tangents from point ‘pt’.
	local t1, t2 = tangent_from_(self.center, self.through, pt)
	-- Return a circle with the center in 'pt' and one of the tangents as the radius	
  return circle:new(pt, t1)
end

function circle:orthogonal_through(pta, ptb)
	-- Retourne un cercle défini par l'orthogonale passant par 'pta' et 'ptb'
	local o = orthogonal_through_(self.center, self.through, pta, ptb)
	return circle:new(o, pta)
end

function circle:radical_circle(C1, C2)
	local rc = self:radical_center(C1, C2)
	if C2 == nil then
		return self:orthogonal_from(rc)
	else
		return C1:orthogonal_from(rc)
	end
end

function circle:draw()
	-- Récupère les coordonnées du centre et le rayon du cercle
	local x, y = self.center:get()
	local r = self.radius

	-- Format de la commande LaTeX pour dessiner le cercle
	local frmt = "\\draw (%0.3f,%0.3f) circle [radius=%0.3f];"

	-- Affiche la commande LaTeX en formatant les valeurs de x, y et r
	tex.sprint(string.format(frmt, x, y, r))
end

function circle:midcircle(C)
	-- Retourne le cercle médian entre 'self' et 'C'
	return midcircle_(self, C)
end

-- -----------------------------------------------------------
-- Circle tangent to a circle passing through two points
function circle:c_c_pp(a, b)
	-- test If one point is inside the disk and the other is outside, there is no solution.
	if (self:in_out_disk(a) and not self:in_out_disk(b)) or (self:in_out_disk(b) and not self:in_out_disk(a)) then
		tex.error("An error has occurred", { "Bad configuration. Only one point is in the disk" })
		return
	end

	-- Find the mediator of the current line
	local lab = line:new(a, b)
	local lmed = lab:mediator()

	if self:is_tangent(lab) then
		local c = intersection(self, lab)
		local d = self:antipode(c)

		return circle:new(circum_circle_(a, b, d), a), circle:new(circum_circle_(a, b, d), a)
	end

	-- pb are (AB) tgt to circle A and B equidistant of O tgt and equidistant
	if lab:is_equidistant(self.center) then
		local t1, t2 = intersection(lmed, self)
		return circle:new(circum_circle_(a, b, t1), t1), circle:new(circum_circle_(a, b, t2), t2)
	else
		-- Create a circumcircle passing through a, b, and a point on C
		local Cc = circle:new(circum_circle_(a, b, self.center), a)
		-- Find the intersection points of C and Cc
		local c, d = intersection(self, Cc)
		-- Create a line passing through the two intersection points
		local lcd = line:new(c, d)
		-- Find the intersection of the current line (self) with the line lcd
		local i = intersection(lab, lcd)
		-- Create tangents from the intersection point to C
		local lt, ltp = self:tangent_from(i)
		-- Get the tangent points
		local t, tp = lt.pb, ltp.pb
		-- Return two new circles tangent to C and passing through the tangent points
		return circle:new(intersection(lmed, line:new(self.center, t)), t),
			circle:new(intersection(lmed, line:new(self.center, tp)), tp)
	end
end

-- Circle  tangent to two circles passing through a point
function circle:c_cc_p(C, p)
	-- Calcule la similitude externe entre les cercles 'self' et 'C'
	local i = self:external_similitude(C)

	-- Crée la ligne passant par les centres des cercles 'self' et 'C'
	local lofcenters = line:new(self.center, C.center)

	-- Trouve les intersections de la ligne avec 'self' et avec 'C'
	local u1, u2 = intersection(lofcenters, self)
	local v1, v2 = intersection(lofcenters, C)

	-- Trouve les tangentes communes entre 'self' et 'C'
	local u1, v1 = self:common_tangent(C)

	-- Calcule le cercle circonscrit passant par u1, v1 et p
	local o = circum_circle_(u1, v1, p)

	-- Trouve les intersections du cercle 'o' avec la ligne i et le point p
	local a, b = intersection_lc_(i, p, o, p)

	-- Si les deux intersections sont très proches, retourne un cercle défini par la ligne et l'intersection
	if point.abs(a - b) < tkz_epsilon then
		local li = line:new(i, p)
		return C:c_lc_p(li, a)
	else
		local q
		-- Résout le cas où p et q sont égaux
		if point.abs(a - p) < tkz_epsilon then
			q = b
		else
			q = a
		end

		-- Retourne le cercle défini par p et q
		return C:c_c_pp(p, q)
	end
end

-- Circle  tangent to one circle, on line and passing through a point
function circle:c_lc_p(l, p, inside)
	inside = (inside == "inside")

	-- Vérifie si le point p est à l'intérieur ou à l'extérieur du cercle
	if self:in_out(p) then
		-- Trouve les intersections de la ligne avec les deux bords du cercle
		local t1 = intersection_ll_(self.north, p, l.pa, l.pb)
		local t2 = intersection_ll_(self.south, p, l.pa, l.pb)

		-- Trouve les lignes orthogonales aux tangentes
		local l1 = l:ortho_from(t1)
		local l2 = l:ortho_from(t2)

		-- Trouve les intersections de ces lignes avec le cercle
		local o1 = intersection_ll_(self.center, p, l1.pa, l1.pb)
		local o2 = intersection_ll_(self.center, p, l2.pa, l2.pb)

		-- Return circles defined by intersection points
		return circle:new(o1, t1), circle:new(o2, t2)
	else
		-- If point p is outside the circle, check whether p is inside or outside the line
		if l:in_out(p) then
			-- Calculates the projection of 'self.center' onto the line
			local i = l:projection(self.center)

			-- Find the orthogonal lines passing through p
			local lortho = l:ortho_from(p)

			-- Find the ratio points for the orthogonal
			local u = lortho:report(self.radius, p)
			local v = lortho:report(-self.radius, p)

			-- Find the medians between the center of the circle and the ratio points
			local ux, uy = mediator_(self.center, u)
			local vx, vy = mediator_(self.center, v)

			-- Find mediator intersections
			local o1 = intersection_ll_(u, v, ux, uy)
			local o2 = intersection_ll_(u, v, vx, vy)

			-- Returns circles defined by points of intersection and p
			return circle:new(o1, p), circle:new(o2, p)
		else
			-- General case
			local u = self.north
			local v = self.south
			local h = intersection_ll_(u, v, l.pa, l.pb)

			if inside then
				-- Calculate the circumscribed circle between p, u and h
				local o = circum_circle_(p, u, h)

				-- Finds the intersection of the circle and returns the result
				local q = intersection_lc_(p, v, o, p)
				return self:c_c_pp(p, q)
			else
				-- if 'inside' is false, we calculate the other circumscribed circle
				local o = circum_circle_(p, v, h)

				-- Finds the intersection of the circle and returns the result
				local q = intersection_lc_(u, p, o, v)
				return self:c_c_pp(p, q)
			end
		end
	end
end

-----------------------------------------------------------
----  Inversion ----
-----------------------------------------------------------
function circle:set_inversion(...)
	local tp = table.pack(...)
	local i
	local t = {}
	for i = 1, tp.n do
		table.insert(t, inversion_(self.center, self.through, tp[i]))
	end
	return table.unpack(t)
end

function circle:inversion_L(L)
	-- Vérifie si le centre du cercle est à l'intérieur ou à l'extérieur de la ligne L
	if L:in_out(self.center) then
		return L -- Retourne la ligne L inchangée si le centre est du bon côté
	else
		-- Calcul de la projection du centre sur la ligne L
		local p = L:projection(self.center)

		-- Inversion du point projeté par rapport au cercle défini par 'self.center' et 'self.through'
		local q = inversion_(self.center, self.through, p)

		-- Retourne un cercle avec le centre au milieu de 'self.center' et 'q' et comme rayon 'q'
		return circle:new(midpoint_(self.center, q), q)
	end
end

function circle:inversion_C(C)
	local p, q, x, y, X, Y
	if C:in_out(self.center) then
		p = C:antipode(self.center)
		q = inversion_(self.center, self.through, p)
		x = ortho_from_(q, self.center, p)
		y = ortho_from_(q, p, self.center)
		return line:new(x, y)
	else
		x, y = intersection_lc_(self.center, C.center, C.center, C.through)
		X = inversion_(self.center, self.through, x)
		Y = inversion_(self.center, self.through, y)
		return circle:new(midpoint_(X, Y), X)
	end
end

function circle:inversion(...)
	local tp = table.pack(...)
	local obj = tp[1]
	local nb = tp.n
	if nb == 1 then
		if obj.type == "point" then
			return inversion_(self.center, self.through, obj)
		elseif obj.type == "line" then
			return self:inversion_L(obj)
		else
			return self:inversion_C(obj)
		end
	else
		local t = {}
		for i = 1, nb do
			table.insert(t, inversion_(self.center, self.through, tp[i]))
		end
		return table.unpack(t)
	end
end

function circle:path(za,zb,nb)
  local list = {}
  for i = 0, nb do
    local t = i / nb
    local P = self:point_arc(za, zb, t)
     table.insert(list, "(" .. checknumber(P.re) .. "," .. checknumber(P.im) .. ")")
   end
	return path:new(list)  
end

function circle:point_arc(za, zb, t)
  local anglezazb = get_angle(self.center,za,zb)
	local phi =  t * anglezazb
	return rotation_(self.center, phi, za)
end

return circle