## 375 Reputation

6 years, 206 days

## Looking for focal points of an ellipse...

Maple 2021
```_local(D);
f := (x, y) -> 3*x^2 - 3*y*x + 6*y^2 - 6*x + 7*y - 9;
coeffs(f(x, y));
A, B, C, D, E, F := %;
theta := 1/2*arctan(B/(A - C));
solve({-2*A*xc - B*yc = D, -B*xc - 2*C*yc = E});
assign(%);
x := xcan*cos(theta) - ycan*sin(theta) + xc;
y := xcan*sin(theta) + ycan*cos(theta) + yc;
Eq := simplify(expand(f(x, y)));
xcan^2/simplify(sqrt(-tcoeff(Eq)/coeff(Eq, xcan^2)))^`2` + ycan^2/simplify(sqrt(-tcoeff(Eq)/coeff(Eq, ycan^2)))^`2` = 1;
a := sqrt(-tcoeff(Eq)/coeff(Eq, xcan^2));
b := sqrt(-tcoeff(Eq)/coeff(Eq, ycan^2));
c := sqrt(a^2 - b^2);
F1 := [xc + c*cos(theta), yc + c*sin(theta)];
evalf(%);
F1 := [xc - c*cos(theta), yc - c*sin(theta)];
evalf(%);
Points := pointplot([F1[], F2[]], symbol = solidcircle, color = [red], symbolsize = 6);
xcan := plot(yc + tan(theta)*('x' - xc), 'x' = -2 .. 3.5, color = black);
ycan := plot(yc - ('x' - xc)/tan(theta), 'x' = 0.1 .. 1.5, color = black);
Ellipse := plots[implicitplot](f('x', 'y'), 'x' = -2 .. 3.5, 'y' = -2 .. 1.5, color = red, thickness = 2, gridrefine = 5);
labels := plots[textplot]([[0.4, 1.3, "ycan"], [3.2, 0.75, "xcan"]], font = [TIMES, ROMAN, 14]);
plots[display](xcan, ycan, Points, Ellipse, labels, scaling = constrained);
```

I do not why I get this message:

Error, (in plots:-display) expecting plot structure but received: pointplot([17/21-(2/21)*(1407/(-(3/2)*2^(1/2)+9/2)-1407/((3/2)*2^(1/2)+9/2))^(1/2)*cos((1/8)*Pi), -8/21-(2/21)*(1407/(-(3/2)*2^(1/2)+9/2)-1407/((3/2)*2^(1/2)+9/2))^(1/2)*sin((1/8)*Pi), F2[]], symbol = solidcircle, color = [red], symbolsize = 6) NULL;

## This warning message is incomprehensible...

Maple 2021

Fig := proc(t)
local xD, yD, D, C, Ii, Points, tex,sol;
global A, B, b, Omega1, EL1, EL2;
xD := Omega1[1] + aa*cos(t);
yD := bb*sin(t);
D := [xD, yD];
C := [xD + b, yD];
sol:=solve({EQ(A,D),EQ(C,B)},{x,y});
Ii:=[subs(sol,x),subs(sol,y)]:
Points := pointplot([A[], B[], C[], C[], D[], E[], Omega1[]], symbol = solidcircle, color = [red], symbolsize = 6);
tex := textplot([[A[], "A"], [B[], "B"], [C[], "C"], [D[], "D"], [E[], "E"], [Omega1[], "&Omega;1"]], align = ["above", "right"]);
display([polygonplot([A, B, C, D], color = blue, filled = true, transparency = 0.9), Points, tex, EL1, EL2,plot([D,Ii]),plot([Ii,C])], axes = normal, scaling = constrained); end proc:
Fig((3*Pi)/4):
display([seq(Fig((2*Pi*i)/40), i = 1 .. 80)], insequence = true);
Warning, data could not be converted to float Matrix
Warning, data could not be converted to float Matrix
Warning, data could not be converted to float Matrix
Warning, data could not be converted to float Matrix
Warning, data could not be converted to float Matrix
Warning, data could not be converted to float Matrix
Warning, data could not be converted to float Matrix
Warning, data could not be converted to float Matrix
I am sorry; How to manage with such a message. Thank you very much.

## Animation of a trapeze...

Maple 2021

restart;
with(plots):
_local(D):

EQ := proc(M, N) local eq; eq := (y - M[2])/(x - M[1]) = (N[2] - M[2])/(N[1] - M[1]); end proc;
EQ := proc (M, N) local eq; eq := (y-M[2])/(x-M[1]) =

(N[2]-M[2])/(N[1]-M[1]) end proc

On considère un trapèze dans lequel une base est fixe l'autre base a une longueur constante et la somme des 2 autres côtés est constaante.
Trouver :
1-. le lieu des sommets mobiles.
A := [xA, 0]:
B := [xA + a, 0]:
D := [xD, yD]:
C := [xD + b, yD]:
EQ(B, C);
E := [xA + a - b, 0]:
Omega1 := (A + E)/2;
Application numérique :
Lieux des sommets C et D

xA := -5:
a := 13:#a>=b
b := 7:
c := -3:
xD := -6:
xC := xD + c:

A:
B:
C:
D:
Ll:=11:aa:= Ll/2:
cc := (a - b)/2:
bb := sqrt(aa^2 - cc^2):
el1 := (x - Omega1[1])^2/aa^2 + y^2/bb^2 = 1:
sol := solve(subs(x = xD, (x - Omega1[1])^2/aa^2 + y^2/bb^2 = 1), y):
yD := sol[1]:
el2 := (x - Omega1[1] - b)^2/aa^2 + y^2/bb^2 = 1:
EL1 := implicitplot(el1, x = -9 .. 4, y = -6 .. 6, color = blue):
EL2 := implicitplot(el2, x = -9 .. 12, y = -6 .. 12, color = blue):
Trap := polygonplot([A, B, C, D], color = blue, filled = true, transparency = 0.9):
Points := pointplot([A[], B[], C[], C[], D[], E[], Omega1[]], symbol = solidcircle, color = [red], symbolsize = 6):
tex := textplot([[A[], "A"], [B[], "B"], [C[], "C"], [D[], "D"], [E[], "E"], [Omega1[], "&Omega;1"]], align = ["above", "right"]):
display([Trap, EL1, EL2, tex, Points], axes = normal, scaling = constrained):
Fig := proc(xD)
local yD, D, C,Points,tex;
global A, B, b, Omega, xA, xB, EL1, EL2;
solve(subs(x = xD, (x - Omega1[1])^2/aa^2 + y^2/bb^2 = 1), y);
yD := %[1]; D:= [xD, yD]; C := [xD + b, yD];
Points := pointplot([A[], B[], C[], C[], D[], E[], Omega1[]], symbol = solidcircle, color = [red], symbolsize = 6):
tex := textplot([[A[], "A"], [B[], "B"], [C[], "C"], [D[], "D"], [E[], "E"], [Omega1[], "&Omega;1"]], align = ["above", "right"]):
display([polygonplot([A, B, C, D], color = blue, filled = true, transparency = 0.9), Points,tex,EL1, EL2], axes = normal, scaling = constrained);
end proc:

Fig(2):Fig(-4):
Fig([seq(-6 + 3*i/10), i = 1.20], insequence = true);
Error, (in Engine:-Dispatch) badly formed input to solve: not fully algebraic
;I don't understand this error message. Thank you gfor your help.

## calculation of Major Axis and Minor Axis...

Maple 2021

How to improve this program ? Thank you.

restart;
Equation de la conique
eqcon := (45 - 27*cos(alpha))*x^2 - 54*sin(alpha)*x*y + (45 + 27*cos(alpha))*y^2 - 8;
Delta := (-54*sin(alpha))^2 - 4*(45 - 27*cos(alpha))*(45 + 27*cos(alpha));
expand(%);
simplify(%);
Discriminant : Δ<0 ce qui correxpond à une ellipse
Eq := simplify(expand(subs(x = cos(alpha/2)*X - sin(alpha/2)*Y, y = sin(alpha/2)*X + cos(alpha/2)*Y, eqcon)));
kx := coeff(Eq, X, 2);
ky := coeff(Eq, Y, 2);
k := -tcoeff(Eq);

EQ := X^2/(sqrt(1/kx^2)*k) + Y^2/(sqrt(1/ky^2)*k) = 1;
Calcul du grand et du petit axe
a := 1/sqrt(coeff(lhs(EQ), X, 2));
b := 1/sqrt(coeff(lhs(EQ), Y, 2));
print(X^2/('a^2') + Y^2/('b^2') = 1);

## how to put color inside circles...

Maple 2021

L’éventail de la Geisha
restart:with(plots):with(geometry):
NULL;
_EnvHorizontalName := 'x':
_EnvVerticalName := 'y':

NULL;
EqBIS := proc(P, U, V)
local a, eq1, M1, t, PU, PV, bissec1;
description "P est le sommet de l'angle dont on chercche la bissectrice" ;
a := (P - U)/LinearAlgebra:-Norm(P - U, 2) + (P - V)/LinearAlgebra:-Norm(P - V, 2);
M1 := P + a*t; eq1 := op(eliminate({x = M1[1], y = M1[2]}, t));
RETURN(op(eq1[2])); end proc:

with(plottools);
with(plots);

r1 := 1/2;
r2 := r1/2;
R := r1*(21 - 12*sqrt(3));
21      (1/2)
R := -- - 6 3
2

a := arc([0, 0], 2*r1, Pi/6 .. (5*Pi)/6);
b := arc([0, 0], r1, Pi/6 .. (5*Pi)/6);

with(geometry);
eq := EqBIS(<sqrt(3)/2, 1/2>, <0, 0>, <0, 1/2>);
line(bis, eq);
(1/2)
eq := 2 3      y - 2 x + 4 y - 2

bis

OpT := 2*sqrt(r1*R);
line(lv, x = OpT);
intersection(Omega, bis, lv);
coordinates(Omega);
evalf(%);
(1/2)
OpT := 2 3      - 3

lv

Omega

[                / (1/2)    \]
[   (1/2)      2 \3      - 1/]
[2 3      - 3, --------------]
[                     (1/2)  ]
[                2 + 3       ]

[0.464101616, 0.3923048456]

retarrt;
with(plots);
with(plottools);
[cos((5*Pi)/6), sin((5*Pi)/6)];
[  1  (1/2)  1]
[- - 3     , -]
[  2         2]

a := arc([0, 0], 2*r1, Pi/6 .. (5*Pi)/6);
b := arc([0, 0], r1, Pi/6 .. (5*Pi)/6);
NULL;
A:=[cos(Pi/6), sin(Pi/6)];
B:=[cos(5*Pi/6), sin(5*Pi/6)];
Oo:=[0,0];
Op:=[0,1/2];
poly:=[A,B,Oo];
R := r1*(21 - 12*sqrt(3))
[1  (1/2)  1]
A := [- 3     , -]
[2         2]

[  1  (1/2)  1]
B := [- - 3     , -]
[  2         2]

Oo := [0, 0]

[   1]
Op := [0, -]
[   2]

[[1  (1/2)  1]  [  1  (1/2)  1]        ]
poly := [[- 3     , -], [- - 3     , -], [0, 0]]
[[2         2]  [  2         2]        ]

21      (1/2)
R := -- - 6 3
2

Omega := [2*sqrt(3) - 3, 2*(sqrt(3) - 1)/(2 + sqrt(3))];
Omega1 := [3 - 2*sqrt(3), 2*(sqrt(3) - 1)/(2 + sqrt(3))];

[                / (1/2)    \]
[   (1/2)      2 \3      - 1/]
Omega := [2 3      - 3, --------------]
[                     (1/2)  ]
[                2 + 3       ]

[                 / (1/2)    \]
[    (1/2)      2 \3      - 1/]
Omega1 := [-2 3      + 3, --------------]
[                      (1/2)  ]
[                 2 + 3       ]

r3 := 3/16;
EF := sqrt(r3);

3
r3 := --
16

1  (1/2)
EF := - 3
4

r := (150 - 72*sqrt(3))/193*1/2;
alpha := -5/3*r + 1/2*1/2;
p := sqrt(3)/3*1/2 - sqrt(3)/18*r;
75    36   (1/2)
r := --- - --- 3
193   193

307   60   (1/2)
alpha := - --- + --- 3
772   193

1  (1/2)   1   (1/2) /75    36   (1/2)\
p := - 3      - -- 3      |--- - --- 3     |
6          18        \193   193       /

p2 := textplot([[A[], "A"], [B[], "B"], [Oo[], "O"]], align = ["above", "right"]);
display(a, b, p2, polygonplot(poly, thickness = 3, color = blue, transparency = 0.3), circle(Omega, R, color = blue, filled = true), circle(Omega1, R, color = blue, filled = true), circle([0, 3/4], 1/4, color = yellow, filled = true), circle([EF, 1/2 + r3], r3, color = green, filled = true), circle([-EF, 1/2 + r3], r3, color = green, thickness = 5), circle([p, 3/4 + alpha], r, color = red, thickness = 5), circle([-p, 3/4 + alpha], r, color = red, thickness = 5), axes = none, scaling = constrained, size = [500, 500]);
how to put color inside circles ? Thabk you.

 5 6 7 8 9 10 11 Last Page 7 of 27
﻿