-- tkz_elements_conic.lua -- date 2025/03/04 -- version 3.34c -- 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. --------------------------------------------------------------------------- -- conics --------------------------------------------------------------------------- ----------------------------- ----- Conic parameters ----------------------------- -- foci, directrix and eccentricity are the main parameters and are used to define all conics. -------------------- General parameters common to all conics ------------- -- (e) is eccentricity e = 1 for parabola; e = p / h -- (h) distance between the focus and the directrix (sometime called f) ; h = p / e ellipse and hyperbola and h = p for parabola -- (p) semi-latus rectum (The latus rectum is the chord parallel to the directrix and passing through a focus) It's also called, the focal parameter (distance from a focus to the corresponding directrix.) -- vertex We can define the distance between the vertex and the directrix or the distance between the vertex and the focus p / (e + 1) so p / 2 for parabola; (e * h) / (e + 1) = h * e / (e + 1) -- major_axis : The principal axis is the line joining the foci of an ellipse or hyperbola, and its midpoint is the curve's center. A parabola has no center, in this case, the major_axis is the line joining the focus and the vertex. -- parabola has no center and one focus. c/a for hyperbola and ellipse -- (c) is the distance between the center and a focus. (line eccentricity) -- for ellipse -- a semi_major_awis distance between center and vertex -- b semi_mminor_axis distance between center and covertex -- c = math.sqrt(a^2 - b^2) and p = b^2 / a -- h = b^2 / c -- for hyperbola -- a semi_major_awis distance between center and vertex -- b semi_mminor_axis distance between center and covertex b/a slope of asymptote -- c = math.sqrt(a^2 - b^2) and p = b^2 / a -- h = b^2 / c --------------------------------------------------------------------------- conic = {} function conic: new (Fa,Di,ecc) -- focus, directrix, eccentricity local type = 'conic' local e = ecc local subtype = get_subtype(e) local K = projection_(Di.pa,Di.pb,Fa) local h = length(K,Fa) local a = get_a(h, e) local b = get_b(h, e) -- only hy and el local c = get_c(h, e) local p = e * h -- demi latus rectum if hy and el b^2/a local slope = slope_(K, Fa) local major_axis = line : new (K, Fa) local directrix = Di local vertex = get_vertex(Fa, K, e, h) -- pb No center, no covertex, no Fb with parabola local center = conic_center(Fa, K, e, h ) local Fb = next_focus(Fa, K, e, h) local covertex = get_covertex(Fa, K ,e, h) local minor_axis = get_minor_axis(Fa, K ,e, h) local o = { type = type, subtype = subtype, K = K, e = e, h = h , a = a, b = b, c = c, p = p, Rx = a, Ry = b, Fa = Fa, Fb = Fb, center = center, vertex = vertex, covertex = covertex, major_axis = major_axis, minor_axis = minor_axis, directrix = directrix, slope = slope } setmetatable(o, self) self.__index = self return o end function conic:points(ta, tb, nb, swap) -- Retourne les points associés à la conique, selon la valeur de 'swap' if not swap then return get_points_conic_(self, ta, tb, nb) else return get_points_sym_conic_(self, ta, tb, nb) end end function conic:point(t, swap) -- Retourne un seul point sur la conique, selon la valeur de 'swap' if not swap then return get_one_point_conic_(self, t) else return get_one_point_hyperbola_ii(self, t) end end function conic:antipode(pt) -- Calcule l'antipode d'un point sur la conique if self.e ~= 1 then -- Si l'excentricité n'est pas égale à 1, l'antipode est défini comme 2 * center - pt return 2 * self.center - pt else -- Si l'excentricité est égale à 1, un message d'erreur est généré via LuaLaTeX tex.error("Undefined antipode for a parabola.") end end function conic:tangent_at(pt) -- actually only parabola local u, v if self.e == 1 then -- Parabola local h = self.directrix:projection(pt) u = self.vertex:identity(pt) and ll_from_(pt, self.directrix.pa, self.directrix.pb) or in_center_(pt, h, self.Fa) elseif self.e > 1 then -- Hyperbola u = self.vertex:identity(pt) and ll_from_(pt, self.directrix.pa, self.directrix.pb) or in_center_(pt, self.Fb, self.Fa) elseif self.e < 1 then -- Ellipse local zi = in_center_(self.Fa, pt, self.Fb) u = pt + (zi - pt) * point(0, 1) end u = normalize_(pt, u) v = pt:symmetry(u) return line:new(u, v) end -- Intersection entre une parabole et une droite function conic:inter_Pa_line(pa, pb) -- Crée un système de coordonnées basé sur l'axe majeur et le sommet de la parabole local sys = occs:new(self.major_axis, self.vertex) -- Calcule les coordonnées des points pa et pb dans ce système local Xa, Ya = sys:coordinates(pa) local Xb, Yb = sys:coordinates(pb) -- Résout l'intersection entre la parabole et la droite définie par les points pa et pb local r1, r2 = solve_para_line(self.h, param_line(Xa, Ya, Xb, Yb)) -- Si les résultats de l'intersection sont invalides (false), retourne deux faux if r1 == false then local s1 = false local s2 = false return s1, s2 else -- Sinon, calcule les points d'intersection local s1, s2 = self:point(r1), self:point(r2) -- Retourne les points d'intersection dans l'ordre croissant de distance par rapport à pa if length(pa, s1) < length(pa, s2) then return s1, s2 else return s2, s1 end end end function conic:tangent_from(pt) if self.e == 1 then -- Parabola local sys = occs : new (self.major_axis ,self.vertex) local Xb,Yb = sys : coordinates (pt) local p1, p2 = solve_quadratic(self.h, -2 * Xb, 2 * Yb) local s1 = self:point(self.h * p1) local s2 = self:point(self.h * p2) return line:new(pt, s1), line:new(pt, s2) elseif self.e > 1 then -- Hyperbola local C = circle:radius(self.Fb, 2 * self.a) local m, n = intersection_cc_(pt, self.Fa, self.Fb, C.through) local u, v = mediator_(m, self.Fa) local x, y = mediator_(n, self.Fa) local T1, T2 = line:new(u, v), line:new(x, y) local d1, d2 = T1:distance(self.Fa), T2:distance(self.Fa) if d2 < d1 then T2, T1 = T1, T2 end local Fbsym = symmetry_axial_(T1.pa, T1.pb, self.Fb) local t1 = intersection_ll_(Fbsym, self.Fa, T1.pa, T1.pb) local Fasym = symmetry_axial_(T2.pa, T2.pb, self.Fa) local t2 = intersection_ll_(Fasym, self.Fb, T2.pa, T2.pb) return line:new(pt, t1), line:new(pt, t2) elseif self.e < 1 then -- Ellipse local w = report_(self.Fb, self.Fa, 2 * self.a) local s1, s2 = intersection_cc_(pt, self.Fa, self.Fb, w) local u, v = mediator_(s1, self.Fa) local U = intersection_ll_(u, v, self.Fb, s1) u, v = mediator_(s2, self.Fa) local V = intersection_ll_(u, v, self.Fb, s2) return line:new(pt, U), line:new(pt, V) end end -- intersection line hyperbola function conic:inter_Hy_line(pa, pb) local function hyp_fct (x) return self.a * math.sqrt(1 + (x^2) / (self.b)^2) end local sys = occs : new (self.major_axis,self.center) local XA,YA = sys : coordinates (pa) local XB,YB = sys : coordinates (pb) if math.abs(XA - XB) < tkz_epsilon then local xs = XA local fa, c = self.Fa, self.center local s1, s2 = hyp_fct(xs), -hyp_fct(xs) local wx = report_(self.directrix.pa, self.directrix.pb, xs, report_(c, fa, s1)) local wy = report_(self.directrix.pa, self.directrix.pb, xs, report_(c, fa, s2)) return wx, wy else local r, s = param_line(XA, YA, XB, YB) local t1, t2 = solve_hyper_line(self.a, self.b, r, s) if t1 ==false then return pa,pb else local s1, s2 = self : point(t1), self : point(t2) if r * t2 + s < 0 then s2 = self:point(t2,swap) end if r * t1 + s < 0 then s1 = self:point(t1,swap) end if length(pa,s1) < length(pa,s2) then return s1,s2 else return s2,s1 end end end end -- Fonction pour déterminer si un point est à l'intérieur ou à l'extérieur de la conique function conic:in_out(pt) if self.e == 1 then -- Parabole return PA_in_out(self, pt) elseif self.e > 1 then -- Hyperbole return HY_in_out(self, pt) elseif self.e < 1 then -- Ellipse return EL_in_out(self, pt) end end -- Fonction pour obtenir l'orthoptique de la conique function conic:orthoptic() if self.e == 1 then -- Parabole return self.directrix elseif self.e > 1 and self.e < math.sqrt(2) then -- Hyperbole local r = math.sqrt(self.a * self.a - self.b * self.b) local th = report_(self.center, self.vertex, r) return circle:new(self.center, th) elseif self.e < 1 then -- Ellipse local r = math.sqrt(self.a * self.a + self.b * self.b) local th = report_(self.center, self.vertex, r) return circle:new(self.center, th) end end -- Fonction pour calculer les asymptotes de la conique function conic:asymptotes() if self.e > 1 then -- Hyperbole -- Calcul du point sur les foyers et des asymptotes local pa = report_(self.Fa, self.Fb, self.a, self.center) local p1 = (pa - self.center):orthogonal(self.b):at(pa) local p2 = (pa - self.center):orthogonal(-self.b):at(pa) -- Symétrie des points d'asymptotes par rapport au centre local q1 = symmetry_(self.center, p1) local q2 = symmetry_(self.center, p2) -- Retourne les deux asymptotes return line:new(p1, q1), line:new(p2, q2) else -- Si ce n'est pas une hyperbole, renvoie une erreur tex.error("An error has occurred", {"It's not an hyperbola"}) return end end return conic