## 385 Reputation

6 years, 287 days

## Problema with rotation...

restart;
with(geometry);
with(plots);
_EnvHorizomtalName = 'x';
_EnvVerticalName = 'y';
_local(D);
line(delta, y = 1/3*x + 1, [x, y]);
line(deltap, y = (-1)/4*x - 2, [x, y]);
line(D, y = 3*x - 5, [x, y]);
point(S, -1, 0);
intersection(P, delta, D);
intersection(Pp, deltap, D);
intersection(N, delta, deltap);
projection(H, S, delta);
projection(K, S, deltap);
projection(M, S, D);
circle(c1, [H, K, M], 'centername' = O1);
coordinates(O1);
omega := FindAngle(delta, deltap);
line(SP, [S, P]);
line(SPp, [S, Pp]);
alpha := FindAngle(SP, SPp);
segment(sMH, M, H);
segment(sMK, M, K);
line(MH, [M, H]);
line(MK, [M, K]);
omega - alpha;
beta := FindAngle(MH, MK);
is(beta + omega - alpha = 0);
point(S1, 2*HorizontalCoord(O1) + 1, 2*VerticalCoord(O1));
coordinates(S1);
line(SS1, [S, S1]);
c := distance(S, S1)/2;
b := sqrt(a^2 - c^2);
ell := x^2/a^2 + y^2/b^2 = 1;
ellip := implicitplot(ell, x = -2 .. 2, y = -2 .. 2, color = blue);
solve({Equation(SS1, [x, y])}, {y});
rotation(elli2, ellip, arccos(-3283/4253), 'clockwise', O1);
display(textplot([[coordinates(S)[], "S"], [coordinates(P)[], "P"], [coordinates(H)[], "H"], [coordinates(N)[], "N"], [coordinates(K)[], "K"], [coordinates(M)[], "M"], [coordinates(Pp)[], "P'"], [coordinates(S1)[], "S1"], [coordinates(O1)[], "O1"]], font = [times, bold, 16], align = [above, right]), draw([delta(color = green), deltap(color = blue), D(color = red), c1(color = black), MH(color = orange), MK(color = orange), SP(color = magenta), SPp(color = magenta), S(color = black, symbol = solidcircle, symbolsize = 18), H(color = black, symbol = solidcircle, symbolsize = 12), K(color = black, symbol = solidcircle, symbolsize = 12), M(color = black, symbol = solidcircle, symbolsize = 12), P(color = black, symbol = solidcircle, symbolsize = 12), S1(color = black, symbol = solidcircle, symbolsize = 12), O1(color = black, symbol = solidcircle, symbolsize = 12)]), ellip, scaling = constrained, axes = normal);
[-965  -3283]
[----, -----]
[5218  5218 ]

/7 \
omega := arctan|--|
\11/

/276\
alpha := arctan|---|
\61 /

/7 \         /276\
arctan|--| - arctan|---|
\11/         \61 /

/2609\
beta := arctan|----|
\2603/

true

S1

[1644  -3283]
[----, -----]
[2609  2609 ]

SS1

1             (1/2)        (1/2)
c := -------- 28866098      6806881
13613762

1             (1/2)         (1/2)
a := -------- 34595401      13613762
13613762

2          (1/2)
b := ---- 2520294
2609

13613762  2   2609  2
ell := -------- x  + ---- y  = 1
34595401      3864

/      3283   3283  \
{ y = - ---- - ---- x }
\      4253   4253  /

Error, (in geometry:-rotation) wrong type of arguments

NULL;I corrected the scale error of the previous program
I still haven’t understood my problem with display. Thank you very much.

## how to overcome the linearity problem...

restart;
with(geometry);
with(plots);
_EnvHorizontalName := 'x';
_EnvVerticalName := 'y';
cb := color = blue;
t3 := thickness = 3;
l3 := linestyle = 3;
xA := 3;
yA := 0;
xB := -3;
yB := 0;
c := 6;
point(A, xA, yA);
point(B, xB, yB);
R := 5;
alpha := arctan(3/4);
evalf(%*180/Pi);
36.86989765

xH := (xA + xB)/2;
yH := (yA + yB)/2;
point(H, xH, yH);
xO1 := 0;
yO1 := 4;
point(O1, xO1, yO1);
xO2 := 0;
yO2 := -4;
point(O2, xO2, yO2);
segment(sAB, A, B);
segment(sHO, H, O1);
segment(sAO, A, O1);
segment(sBO, B, O1);
sAB

alpha1 := arctan((yO1 - yA)/(xO1 - xA));
beta := Pi + arctan((yO1 - yB)/(xO1 - xB));
AR1 := plottools[arc]([xO1, yO1], R, alpha1 .. beta, l3);
AR2 := plottools[arc]([xO2, yO2], R, -beta .. -alpha1, l3);
N := 80;
dt := (beta - alpha1)/(N - 1);
dr := draw({O1, O2, sAB, sHO, sAO(cb), sBO(cb)});
tex := textplot([[xA, yA - 0.5, "A"], [xB, yB - 0.5, "B"], [xO1, yO1 + 0.5, "O1"], [xO2, yO2 - 0.5, "O2"], [xH, yH - 0.5, "H"]]);
M1 := seq(plottools[disk]([R*cos(dt*t + alpha1) + xO1, R*sin(dt*t + alpha1) + yO1], 0.2, color = orange), t = 0 .. N);
M2 := seq(plottools[disk]([R*cos(dt*t + Pi + alpha1) + xO2, R*sin(dt*t + Pi + alpha1) + yO2], 0.2, color = orange), t = 0 .. N);
point(m2, [R*cos(dt*t + Pi + alpha1) + xO2, R*sin(dt*t + Pi + alpha1) + yO2]);
P1 := seq(plottools[polygon]([[R*cos(dt*t + alpha1) + xO1, R*sin(dt*t + alpha1) + yO1], [xA, yA], [xB, yB]], color = aquamarine, linestyle = dash), t = 0 .. N);
P2 := seq(plottools[polygon]([[R*cos(dt*t + Pi + alpha1) + xO1, R*sin(dt*t + Pi + alpha1) + yO1], [xA, yA], [xB, yB]], color = aquamarine, linestyle = dash), t = 0 .. N);
triangle(ABM, [A, B, m2]);
AreCollinear: hint: could not determine if 24+30*sin(2/79*t*arctan(4/3)+1/79*t*Pi-arctan(4/3)) is zero
Error, (in geometry:-triangle) not enough information: the three points might be AreCollinear
for t to N do
E[t] := display(dr, tex, AR1, AR2, M1[t], P1[t]);
F[t] := display(dr, tex, AR1, AR2, M2[t], ABM);
end do;
Error, (in plots:-display) expecting plot structure but received: ABM
NULL;

display([seq(E[t], t = 1 .. N), seq(F[t], t = 1 .. N)], insequence = true, axes = none, scaling = constrained, view = [-10 .. 10, -10 .. 10]);
Error, (in plots:-display) expecting plot structures but received: [E[2], E[3], E[4], E[5], E[6], E[7], E[8], E[9], E[10], E[11], E[12], E[13], E[14], E[15], E[16], E[17], E[18], E[19], E[20], E[21], E[22], E[23], E[24], E[25], E[26], E[27], E[28], E[29], E[30], E[31], E[32], E[33], E[34], E[35], E[36], E[37], E[38], E[39], E[40], E[41], E[42], E[43], E[44], E[45], E[46], E[47], E[48], E[49], E[50], E[51], E[52], E[53], E[54], E[55], E[56], E[57], E[58], E[59], E[60], E[61], E[62], E[63], E[64], E[65], E[66], E[67], E[68], E[69], E[70], E[71], E[72], E[73], E[74], E[75], E[76], E[77], E[78], E[79], E[80], F[1], F[2], F[3], F[4], F[5], ...  F[71], F[72], F[73], F[74], F[75], F[76], F[77], F[78], F[79], F[80]]
NULL;
how to improve this program ? Thank you.

## Error, (in aProc) illegal use of a forma...

restart;
_EnvHorizomtalName = 'x';
_EnvVerticalName = 'y';
aProc := proc(t) local xA, yA, xB, yB, xH, yH, R, c, cb, O1, O2, xO1, yO1, xO2, yO2, AR1, AR2, M1, M2, P1, P2, N, dt, dr, alpha, alpha1, beta, l3, t3, tex, E, F; cb := color = blue; xA := 3; yA := 0; xB := -3; yB := 0; c := 6; alpha := artan(3/4); l3 := linestyle = 3; t3 := thickness = 3; N := 80; dt := (beta - alpha1)/(N - 1); geometry:-point(A, xA, yA); geometry:-point(B, xB, yB); R := 5; xH := 1/2*xA + 1/2*xB; yH := 1/2*yA + 1/2*yB; geometry:-point(H, xH, yH); xO1 := 0; yO1 := 4; geometry:-point(O1, xO1, yO1); xO2 := 0; yO2 := 4; geometry:-point(O2, xO2, yO2); geometry:-segment(sAB, A, B); geometry:-segment(sHO, H, O1); geometry:-segment(sAO, A, O1); geometry:-segment(sBO, B, O1); alpha1 := arctan((yO1 - yA)/(xO1 - xA)); beta := Pi + arctan((yO1 - yB)/(xO1 - xB)); AR1 := plottools[arc]([xO1, yO1], R, alpha1 .. beta, l3); AR2 := plottools[arc]([xO2, yO2], R, -beta .. -alpha1, l3); dr := geometry:-draw({sAB, sHO, sAO(cb), sBO(cb), O1, O2}); tex := texplot([[xA, yA - 0.5, "A"], [xB, yB - 0.5, "B"]]); M1 := seq(plottools[disk]([R*cos(dt*t + alpha1) + xO1, R*sin(dt*t + alpha1) + yO1], 0.2, color = orange), t = 0 .. N); M2 := seq(plottools[disk]([R*cos(dt*t + alpha1 + Pi) + xO2, R*sin(dt*t + alpha1 + Pi) + yO2], 0.2, color = orange), t = 0 .. N); P1 := seq(plottools[polygon]([[R*cos(dt*t + alpha1) + xO1, R*sin(dt*t + alpha1) + yO1], [xA, yA], [xB, yB]], color = aquamarine, linestyle = dash), t = 0 .. N); P2 := seq(plottools[polygon]([[R*cos(dt*t + alpha1 + Pi) + xO1, R*sin(dt*t + alpha1 + Pi) + yO1], [xA, yA], [xB, yB]], color = aquamarine, linestyle = dash), t = 0 .. N); for t to N do E[t] := plots:-display(dr, tex, AR1, AR2, M1[t], P1[t]); F[t] := plots:-display(dr, tex, AR1, AR2, M2[t], P2[t]); end do; plots:-display([seq(E[t], t = 1 .. N), seq(F[t], t = 1 .. N)], insequence = true, axes = none, scalming = constraioned, view = [-10 .. 10, -10 .. 10]); end proc;
nFig := 60;
plots:-display(seq(aProc(2*Pi*i/nFig), i = 0 .. nFig), insequence = true);
Error, (in aProc) illegal use of a formal parameter

I don't understand this error. Thank you.

## angle that does not turn on the lower ar...

(triangle ABM) the 3 vertices of a triangle are well placed except when the point M is on the bottom arc

## angle that does not turn on the lower ar...

the top of this angle rotates correctly, it is sufficient that these sides pass through A and B ?

## Pythagorean triples...

FromLargestToSmallest(<27304197, 27304196, 38613965>)
Your procedure works very well. Best regards.

## I am trying to say where these triples ...

I am trying to say where these triples come from. With :
R1:=matrix(3,3,[1,-2,2,2,-1,2,2,-2,3]):R2:=matrix(3,3,[1,2,2,2,1,2,2,2,3]):R3:=matrix(3,3,[-1,2,2-,2,-1,2,-2,-2,3]):
Do you understand ? Thank you very much.

## Pythagorean triples from the largest to ...

Best regards,
I wish to write the following in this order (for instance)
<2275,3648,4275>,<975,448,1273>,<275,252,373>,<33,56,65>,<15,8,17>,<3,4,5> Is it possible ?

## Pythagorean triples...

Thank you of your anwer. How to find the sequence of R[i ]to get (3,4,5) from any triplet?

## Zerckendorf's Theorem...

Very good procedure, thank you

## suite of zeckendorf(s numbers...

I misunderstood, I wanted all possible occurences to get 100 and not the first hundred numbers; Thank you.

## Congruent Numbers...

proper procedure; Probleme with conguent(3); false
Have you look for the dimensions of the rectangular triangles ? Thank you;

## Difficulty with display...

restart;
with(LinearAlgebra);
with(geometry);
with(plots);
_EnvHorizontalName := 'x';

_EnvVerticalName := 'y';

xA := -7: yA := 0: xB := 5: yB := 9: xC := 9: yC := 1:
xP:=3:yP:=5:
point(A, xA, yA);
point(B, xB, yB);
point(C, xC, yC);
point(P, xP, yP);
Mat := Matrix(3,3,[xA,xB,xC,yA,yB,yC,1,1,1]);

"v := <x, y, 1>;

1/Mat;

Mcb := MatrixVectorMultiply(1/Mat, v);

Mbc := MatrixVectorMultiply(Mat, Mcb);

Triangle(Tr, [A, B, C]);

line(AP, [A, P]);

line(BP, [B, P]);

line(CP, [C, P]);

line(AB, [A, B]);

line(BC, [B, C]);

line(CA, [C, A]);

alpha := FindAngle(BP, CP);

beta := FindAngle(CP, AP);

eta := FindAngle(BP, AP);

Aa := FindAngle(AB, CA);

Bb := FindAngle(BC, AB);

Cc := FindAngle(BC, CA);

t3 := thickness = 3;

Mcb := <1/(cot(Aa) - cot(alpha)), 1/(cot(Bb) - cot(beta)),

1/(cot(Cc) - cot(eta))>;

Mbc := MatrixVectorMultiply(Mat, Mcb);

L3 := linestyle = 1;

cbl := color = blue;

dr := draw([AB, BC, CA, AP(cbl, L3), BP(cbl, L3), CP(cbl, L3)]);

tp := textplot([[coordinates(A)[], "A"], [coordinates(B)[],

"B"], [coordinates(C)[], "C"], [coordinates(P)[], "P"]],

align = {above, right});

display*[textplot*([[coordinates(A)[], "A"], [coordinates(B)[]\

, "B"], [coordinates(C)[], "C"]], align = {"above",

'right'}), draw*([AP(color = black), BP(color = green),

CP(color = green), Tr(color = blue, filled = true,

transparency = 0.9)], axes = none)];"

Can you tell me why this program is not running ? Thank you.

## Barycentric coordinate conversion test i...

"with(geometry);

with(LinearAlgebra);

with(plots);"

"_EnvHorizontalName := 'x'"

"_EnvVerticalName := 'y'"

xA := 1: yA := 0: xB := 0: yB := 0: xC := 0: yC := 1:
Mat := Matrix(3,3,[xA,xB,xC,yA,yB,yC,1,1,1]);
"v := <x, y, 1>"

"Mcb := MatrixVectorMultiply(1/Mat, v)"

"Mbc := MatrixVectorMultiply(Mat, Mcb)"

"xA := 4;

yA := 10;

xB := 0;

yB := -2;

xC := 13;

yC := 0;

xP := 5;

yP := 4;

Mat := Matrix(3, 3, [xA, xB, xC, yA, yB, yC, 1, 1, 1]);"

"Mcb := MatrixVectorMultiply(1/Mat, v)"

"Mbc := MatrixVectorMultiply(Mat, Mcb)"

"point(A, xA, yA);

point(B, xB, yB);

point(C, xC, yC);

point(P, xP, yP);

triangle(Tr, [A, B, C]);"

"line(AP, [A, P]);

line(BP, [B, P]);

line(CP, [C, P]);"

"alpha := FindAngle(BP, CP)"

"beta := FindAngle(CP, AP)"

"eta := FindAngle(BP, AP)"

"line(AB, [A, B]);

line(BC, [B, C]);

line(CA, [C, A]);"

"NULL"

"Aa := FindAngle(AB, CA)"

"Bb := FindAngle(BC, AB)"

"Cc := FindAngle(BC, CA)"

"Mcb := <1/(cot(Aa) - cot(alpha)), 1/(cot(Bb) - cot(beta)),

1/(cot(Cc) - cot(eta))>"

"Mbc := MatrixVectorMultiply(Mat, Mcb)"

"t3 := thickness = 3;

L3 := linestyle = 1;

cbl := color = blue;"

"dr := draw([Tr(t3), AP(cbl, L3), BP(cbl, L3), CP(cbl, L3)]),

textplot([[coordinates(A)[], "A"], [coordinates(B)[], "B"],

[coordinates(C)[], "C"], [coordinates(P)[], "P"]], align =

{above, right})"

"display({dr}, title = "Conversion", scaling = constrained,

axes = none, view = [-1 .. 14, -1 .. 11])"

Do you want to tell me why the program is not running ? Thank you very much.

## it’s not quite what I want...

How do you get these results? Thank you.

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