/*
 * This file:
 *   http://anggtwu.net/MAXIMA/2025-1-C3-P1-Q1.mac.html
 *   http://anggtwu.net/MAXIMA/2025-1-C3-P1-Q1.mac
 *          (find-angg "MAXIMA/2025-1-C3-P1-Q1.mac")
 * Author: Eduardo Ochs <eduardoochs@gmail.com>
 * See: (find-angg "GNUPLOT/2025-1-C3-P1.dem")
 *  (c3m251p1p 1 "title")
 *  (c3m251p1a   "title")
 *
 * «.item-a»	(to "item-a")
 * «.item-b»	(to "item-b")
 * «.item-c»	(to "item-c")
 * «.item-e»	(to "item-e")
 * «.item-f»	(to "item-f")
 * «.item-g»	(to "item-g")
 * «.gab»	(to "gab")
 *
 * (defun e () (interactive) (find-angg "MAXIMA/2025-1-C3-P1-Q1.mac"))
 * (find-myqdraw "myqdraw3.mac")
 * (find-angg "MAXIMA/2025-cases.mac")
*/

load_myqdraw();
load("~/myqdraw/mydraw3d.mac");
load("~/MAXIMA/2025-cases.mac");
[topdf_a,topdf_b,topdf_c,topdf_n] : ["~/LATEX/", "2025-1-C3/", "P1-Q1", 0];

[xmin,ymin, xmax,ymax] : [0,0, 10,6];

z_NW:2*x-4;   z_NE:x;
z_SW:x-2;     z_SE:2*x-8;

z_W:0;  z_C:x+y-4;  z_E:8;

z_M   : max(min(z_NW,z_NE,z_C),z_SW,z_SE);
z_all : max(z_W,min(z_M,z_E));

define(F_NW(x,y), z_NW);
define(F_NE(x,y), z_NE);
define(F_SW(x,y), z_SW);
define(F_SE(x,y), z_SE);
define(F_W (x,y), z_W);
define(F_E (x,y), z_E);
define(F_C (x,y), z_C);
define(F   (x,y), z_all);

mydraw3d_z(x,y) := F(x,y);
item_a() := mydraw3d(mydraw3d_squares(), noaxes(), prop(), view=[63,287]);

/* «item-a»  (to ".item-a")
** (c3m251p1p 3 "questao-1a")
** (c3m251p1a   "questao-1a")
* (eepitch-maxima)
* (eepitch-kill)
* (eepitch-maxima)
load("2025-1-C3-P1-Q1.mac");
linel : 110;
numerozinhos([x,y]);
numerozinhos(F(x,y));
item_a()$

*/

M_b : matrix(['F_NW(x,y), "=", F_NW(x,y)],
             ['F_NE(x,y), "=", F_NE(x,y)],
             ['F_SW(x,y), "=", F_SW(x,y)],
             ['F_SE(x,y), "=", F_SE(x,y)],
             ['F_W (x,y), "=", F_W (x,y)],
             ['F_E (x,y), "=", F_E (x,y)],
             ['F_C (x,y), "=", F_C (x,y)]);

item_b : M_b;


/* «item-b»  (to ".item-b")
** (c3m251p1p 3 "questao-1b")
** (c3m251p1a   "questao-1b")
* (eepitch-maxima)
* (eepitch-kill)
* (eepitch-maxima)
load("2025-1-C3-P1-Q1.mac");
item_b;

*/


eqs  : [z=F_C(x,y), z=F_NW(x,y), z=2*t];
eqs2 : solve(eqs, [x,y,z])[1];
             subst(eqs2, [x,y,z]);
define(P(t), subst(eqs2, [x,y,z]));

item_c : apply('matrix, makelist([t,P(t)], t,-2,4));
item_d : P(t);

/* «item-c»  (to ".item-c")
* (eepitch-maxima)
* (eepitch-kill)
* (eepitch-maxima)
load("2025-1-C3-P1-Q1.mac");
item_c;
item_d;

[P(0),      diff(P(t),t)];
 P(0) + t * diff(P(t),t);

*/


/* «item-e»  (to ".item-e")
 * (find-es "maxima" "2024.2-C3-P1-Q1" "gradz(x,y) :=")
*/
define(z(x,y), F(x,y));
eps : 1/4;
z_xr (x,y) := (z(x+eps,y)-z(x,y))/ eps;
z_xl (x,y) := (z(x-eps,y)-z(x,y))/-eps;
z_yu (x,y) := (z(x,y+eps)-z(x,y))/ eps;
z_yd (x,y) := (z(x,y-eps)-z(x,y))/-eps;
gradz(x,y) := if (z_xr(x,y) = z_xl(x,y)) and
                 (z_yu(x,y) = z_yd(x,y))
              then [z_xr(x,y), z_yu(x,y)]
              else "X";

item_e : numerozinhos(gradz(x,y));

/*
* (eepitch-maxima)
* (eepitch-kill)
* (eepitch-maxima)
load("2025-1-C3-P1-Q1.mac");
linel : 110;
numerozinhos(gradz(x,y));

*/



Q_L(t) := [1,5] +    t*[1,-2];
Q_R(t) := [5,2] + (t-3)*[2,1];
Q(t)   := if t < 3 then Q_L(t) else Q_R(t);
x(t)   := Q(t)[1];
y(t)   := Q(t)[2];
h(t)   := F(x(t),y(t));


/* «item-f»  (to ".item-f")
** (c3m251p1p 3 "questao-1f")
** (c3m251p1a   "questao-1f")
* (eepitch-maxima)
* (eepitch-kill)
* (eepitch-maxima)
load("2025-1-C3-P1-Q1.mac");
[xmin,ymin, xmax,ymax] : [0,0, 6,8];
ts   : seqby(0,6, 1/16);
ts   : seqby(0,6, 1/32);
ts   : seqby(1,2, 1/32);
ts   : seqby(0,3, 1/16);
ptss : makelist([t,h(t)], t,ts);

myqdraw(pts(ptss));
myqdraw(xyrange0(), pts(ptss));
myqdraw(xyrange(), pts(ptss));

*/


hg(t) := a*t + b;

gethn(t1,t2) := block([eqs,sol],
  eqs : [hg(t1)=h(t1), hg(t2)=h(t2)],
  sol : solve(eqs, [a,b]),
  subst(sol, hg(t)));

intersect(f1,f2) := rhs(solve(f1(t)=f2(t),t)[1]);

eps : 1/100;
define(h1(t), gethn(1/2,   1/2  +eps));
define(h2(t), gethn(1,     1    +eps));
define(h3(t), gethn(14/10, 14/10-eps));
define(h4(t), gethn(2,     2    +eps));
define(h5(t), gethn(3,     3    +eps));
define(h6(t), gethn(4,     4    +eps));
define(h7(t), gethn(5,     5    +eps));

h12 : intersect(h1, h2);
h23 : intersect(h2, h3);
h34 : intersect(h3, h4);
h45 : 3;
h56 : intersect(h5, h6);
h67 : intersect(h6, h7);

myc  : Cases([t<h12, t<h23, t<h34, t<h45, t<h56, t<h67, true],
             [h1(t), h2(t), h3(t), h4(t), h5(t), h6(t), h7(t)])$
mycx : Cases([x<h12, x<h23, x<h34, x<h45, x<h56, x<h67, true],
             [h1(x), h2(x), h3(x), h4(x), h5(x), h6(x), h7(x)])$

item_f() := ([xmin,ymin, xmax,ymax] : [0,0, 6,8],
             myqdraw(xyrange(), mycx@@myex(lc(orange)), mycx@@pts(pc(red))));

item_g : myc@@matrix();

/* like h(t), but defined by cases */
define(hc(t), myc@@makeif());

/* «item-g»  (to ".item-g")
** (c3m251p1p 3 "questao-1g")
** (c3m251p1a   "questao-1g")
* (eepitch-maxima)
* (eepitch-kill)
* (eepitch-maxima)
load("2025-1-C3-P1-Q1.mac");

myc @@pairs();
myc @@matrix();
myc @@makeif();

mycx@@matrix();
item_f();

item_g;

foo(i) := [mycx@@bi(i), mycx@@fLi(i), mycx@@fRi(i)];
mycx@@n();
apply('matrix, makelist(foo(i), i,1,6));

[xmin,ymin, xmax,ymax] : [0,0, 6,8];
ts    : seqby(0,6, 1/16);
ts    : seqby(0,6, 1/32);
ts    : seqby(1,2, 1/32);
ts    : seqby(0,3, 1/16);
pts_h   : makelist([t,h(t)], t,ts);
pts_hc  : makelist([t,hc(t)+1/10], t,ts);
pts_all : [pts(pts_h), pts(pts_hc, pc(orange))];

myqdraw(pts_all);
myqdraw(xyrange0(), pts_all);
myqdraw(xyrange(),  pts_all);

*/


/* «gab»  (to ".gab")
** (c3m251p1p 6 "gab-1")
** (c3m251p1a   "gab-1")
** (find-Maxima2-links "P1-Q1")
* (eepitch-maxima)
* (eepitch-kill)
* (eepitch-maxima)
load("~/MAXIMA/2025-1-C3-P1-Q1.mac");
linel : 110;
item_a()$
item_b;
item_c;
item_d;
item_e;
item_f();
item_g;

*/