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cosas.mws

A Package for Drawing Geometric Curves

by Dante Yvn Chavil Montenegro [dante_el_poeta@hotmail.com]

`>`	restart:

`>`	libname := "C:/mylib/CyS", libname:

`>`	with(CyS); with(CyS_n); with(plots):

`>`	Prod_Vect([1,1,2],[1,0,2]);

`>`	P_esc([1,3],[2,1]);

`>`	J([x1,x2]);

`>`	Circu:=a->CIRCUNFERENCIA(a)(t);

`>`	# Traza de la circunferencia de radio 1 centrada en el origen CTraza(Circu(1),[t=0..2*Pi,color=blue,scaling=constrained]);

[Maple Plot]

`>`	eli:=ELIPSE(1.5,1)(t):

`>`	alpha:=KAPPA23(eli)(t):

`>`	*CTraza(alpha, [t=0..2Pi, color=navy, axes=boxed, numpoints=1000,thickness=2,orientation=[33,65],scaling=constrained]);**

[Maple Plot]

`>`	HIPOTROCOIDE(8, 5)(t);

`>`	hip:=HIPOTROCOIDE(8, 5)(t):

`>`	alpha:=KAPPA23(hip)(t):

`>`	*CTraza(alpha, [t=0..2Pi, color=green, axes=boxed, numpoints=1000,thickness=2,orientation=[19,19]]);**

[Maple Plot]

>

`>`	Cn(Circu(a)); # norma simplify(%);

`>`	*Cv(Circu(a))(2/8sqrt(Pi)); # vector velocidad**

`>`	Ca(Circu(a))(0); # vector aceleracion

`>`	Cd(Circu(a),4)(t); # cuarta derivada de la curva

`>`	Elip:=(a,b)->ELIPSE(a,b)(t);

`>`	Cr(Elip(2,1))(Pi/8); # rapidez

(4-3*cos(1/8*Pi)^2)^(1/2)

`>`	CT(Elip(2,1))(Pi/8); # Campo tangente

[-2*sin(1/8*Pi)/(4-3*cos(1/8*Pi)^2)^(1/2), cos(1/8*Pi)/(4-3*cos(1/8*Pi)^2)^(1/2)]

`>`	CN(Elip(2,1))(Pi/8); # Campo normal

[-cos(1/8*Pi), -2*sin(1/8*Pi)]/(4-3*cos(1/8*Pi)^2)^(1/2)

`>`	*CB([sin(t),2cos(t),sin(t)])(Pi/8); # Campo binormal solo para curvas en 3d**

`>`	# Hallar los puntos de irregularidad

`>`	Hallar_puntos_irregularidad(CICLOIDE(1,1)(t),t,true);

Las funciones coordenadas  de la curva son:

Las derivadas de las funciones coordenadas son:

Resolviendo el sistema:

obtenemos:

${t = 2*Pi*_Z1}$

`>`	Hallar_puntos_irregularidad(ASTROIDE_TRI(a)(t),t,true);

Las funciones coordenadas  de la curva son:

Las derivadas de las funciones coordenadas son:

Resolviendo el sistema:

obtenemos:

${t = -1/2*Pi+2*Pi*_Z2}, {t = 1/2*Pi+2*Pi*_Z3}, {t = Pi*_Z4}$

`>`	# Analisis de regularidad en puntos dados

`>`	*Analisis_regularidad(CICLOIDE(1,1)(t),2Pi);**

De la curva:

su vector velocidad es:

Al analizar la regularidad de la curva en:

la curva NO ES REGULAR, ya que su vector velocidad es:

`>`	*Analisis_regularidad(ASTROIDE_TRI(a)(t),Pi/2+2Pi);**

De la curva:

su vector velocidad es:

Al analizar la regularidad de la curva en:

la curva NO ES REGULAR, ya que su vector velocidad es:

`>`	# Hallando reparametrizacin por su longitud de arco

`>`	*Hallar_reparametrizacin([at+b,ct+d],t,0);*

Este programa permite hallar una reparametrizacinde la curva beta por su funcin longitud de arco.La curva

no esta parametrizada por su longitud de arco, ya que:

La funcin longitud de arco, a partir de

es:

La funcin inversa de s es dada por:

t = s/(a^2+c^2)^(1/2)

Por tanto

gamma(s) = [a*s/(a^2+c^2)^(1/2)+b, c*s/(a^2+c^2)^(1/2)+d]

es la curva reparametrizada por su longitud de arco.

`>`	*Hallar_reparametrizacin([at+b,bt+a,at],t,0);**

Este programa permite hallar una reparametrizacinde la curva beta por su funcin longitud de arco.La curva

no esta parametrizada por su longitud de arco, ya que:

La funcin longitud de arco, a partir de

es:

La funcin inversa de s es dada por:

t = s/(2*a^2+b^2)^(1/2)

Por tanto

gamma(s) = [a*s/(2*a^2+b^2)^(1/2)+b, b*s/(2*a^2+b^2)^(1/2)+a, a*s/(2*a^2+b^2)^(1/2)]

es la curva reparametrizada por su longitud de arco.

`>`	# Longitud e interpretacin geometrica de la longitud de arco

`>`	elip:=(a,b)->ELIPSE(a,b)(t):

`>`	*CL(elip(2,1.5))(0,3Pi/2);**

`>`	for i from 2 to 10 do LongGraf(elip(2,1.5),0,3*Pi/2,i,[axes=frame,scaling=constrained],15); end;

[Maple Plot]

`>`	hlice:=(a,b)->HLICE_CIRCULAR(a,b)(t);

`>`	hlice(1,0.2):

`>`	*CL(hlice(1,0.2))(0,2Pi);**

`>`	for i from 2 to 10 do LongGraf(hlice(1,0.2),0,2*Pi,i,[axes=frame,scaling=constrained],15); end;

[Maple Plot]

`>`	# Curvatura y torsin

`>`	Ckappa(OCHO(t))(t);

(-sin(t)*(-cos(t)^2+sin(t)^2)-4*sin(t)*cos(t)^2)/(-3*cos(t)^2+4*cos(t)^4+1)^(3/2)

`>`	# en el espacio

`>`	*alpha:=[sin(t),cos(t),sin(t)cos(t)];**

`>`	Ckappa(alpha)(t);

1/2*(1-6*cos(t)^4+6*cos(t)^2)^(1/2)/((-2*cos(t)^2+2*cos(t)^4+1)^3)^(1/2)

`>`	Ctau(alpha)(t);

-3/2*(-1+2*cos(t)^2)/(-1+6*cos(t)^4-6*cos(t)^2)

`>`	# Evoluta

`>`	Evoluta(Elip(3/2,1))(t); # evoluta de la elipse

`>`	CTraza(expand(%),[t=0..2Pi]): CTraza(Elip(3/2,1),[t=0..2Pi,color=blue]): plots[display](%,%%,scaling=constrained);

[Maple Plot]

`>`	Cisoide:=(a)->CISOIDE(a)(t);

`>`	a1:=expand(Evoluta(Cisoide(1))(t)):

`>`	a2:=expand(Evoluta(a1)(t)):

`>`	a3:=expand(Evoluta(a2)(t)):

`>`	g1:=CTraza(a1,[t=-2..2,scaling=constrained,color=blue]):

`>`	g2:=CTraza(a2,[t=-2..2,scaling=constrained,color=green]):

`>`	g3:=CTraza(a3,[t=-2..2,scaling=constrained,color=maroon]):

`>`	g4:=CTraza(CISOIDE(1),[t=-2..2,scaling=constrained]):

`>`	plots[display](g\|\|(1..4),view=[-4.5..4.5,-4.5..4.5],title="La cisoide y sus tres primeras evolutas");

[Maple Plot]

`>`	# Evolvente

`>`	Evolvente(Circu(1),0)(t);

`>`	CTraza(expand(%),[t=0..2Pi, color=blue]): CTraza(Circu(1),[t=0..2Pi]): plots[display](%,%%,scaling=constrained);

[Maple Plot]

> a1:=expand(Evolvente(Circu(1),0)(t)):
a2:=expand(Evolvente(a1,0)(t)):
a3:=expand(Evolvente(a2,0)(t)):
g1:=CTraza(a1,[t=0..2*Pi,color=maroon,scaling=constrained,axes=none,numpoints=1000,thickness=1,numpoints=1000]):
g2:=CTraza(a2,[t=0..2*Pi,color=maroon,scaling=constrained,axes=none,numpoints=1000,thickness=1,numpoints=1000]):
g3:=CTraza(a3,[t=0..2*Pi,color=maroon,scaling=constrained,axes=none,numpoints=1000,thickness=1,numpoints=1000]):
g4:=CTraza(Circu(1),[t=0..2*Pi,color=black,scaling=constrained,axes=none,numpoints=1000,numpoints=1000]):
plots[display](g||(1..4),title="Una circunferencia y sus tres primeras evolventes");

[Maple Plot]

`>`	# Evolvente numrica # esta solo se usa cuando no se puede obtener resultado simbolico con Evolvente

`>`	Ocho:=OCHO(t):

`>`	*EvolventeNumrica(Ocho,0)(0,2Pi);**

[Maple Plot]

`>`	# Paralela

`>`	Paralela(Elip(a,b),s)(t);

[-b*cos(t)*s/(a^2-a^2*cos(t)^2+b^2*cos(t)^2)^(1/2)+a*cos(t), -a*sin(t)*s/(a^2-a^2*cos(t)^2+b^2*cos(t)^2)^(1/2)+b*sin(t)]

> B:=seq(Paralela(Elip(2,1),0.85*s)(t),s=0..4):
g1:=CTraza(B[1],[t=0..2*Pi,color=green,scaling=constrained]):
g2:=CTraza(B[2],[t=0..2*Pi,color=blue,scaling=constrained]):
g3:=CTraza(B[3],[t=0..2*Pi,color=maroon,scaling=constrained]):
g4:=CTraza(B[4],[t=0..2*Pi,color=violet,scaling=constrained]):
plots[display](g||(1..4),title="Cuatro curvas paralelas a t->Elipse(2,1,t)");

[Maple Plot]

`>`	# Podaria

`>`	a2:=Podaria(Circu(1),[1,1])(t): g1:=CTraza(Circu(1),[t=0..2Pi,scaling=constrained,color=orange]): g2:=CTraza(a2,[t=0..2Pi,scaling=constrained,color=blue]): plots[display](g\|\|(1..2));

[Maple Plot]

`>`	a2:=Podaria(Circu(1),[0.5,0.86])(t): g1:=CTraza(Circu(1),[t=0..2Pi,scaling=constrained,color=orange]): g2:=CTraza(a2,[t=0..2Pi,scaling=constrained,color=blue]): plots[display](g\|\|(1..2));

[Maple Plot]

`>`	Podaria(Circu(1),[0,0])(t): simplify(%);

`>`	# permite obtener una ecuacin implicita de una curva en el plano

`>`	Parametro_Impli(Circu(a),x,y);

$[{t = arctan(y/a,x/a)}, {y^2+x^2-a^2}]$

`>`	Parametro_Impli(Elip(8,3),x,y);

$[{t = arctan(1/3*y,1/8*x)}, {64*y^2+9*x^2-576}]$

`>`	# Curvatura

`>`	Ocho:=OCHO(t);

`>`	Ckappa(Ocho)(t): simplify(%);

-sin(t)*(2*cos(t)^2+1)/(-3*cos(t)^2+4*cos(t)^4+1)^(3/2)

`>`	*plot(%,t=0..2Pi,title="Curvatura de la curva figura de ocho",color=green, thickness=2);**

[Maple Plot]

`>`	# Proporciona informe de algunas propiedades geomtricas de la curva

`>`	Informe(Elip(2,1))(t);

La curva:

posee las siguientes propiedades geomtricas.

   Velocidad:

   Aceleracin:

   Rapidez:

   Campo Tangente:

[-2*sin(t)/(-3*cos(t)^2+4)^(1/2), cos(t)/(-3*cos(t)^2+4)^(1/2)]

   Campo Normal:

[-cos(t), -2*sin(t)]/(-3*cos(t)^2+4)^(1/2)

   Curvatura:

(2*cos(t)^2+2*sin(t)^2)/(-3*cos(t)^2+4)^(3/2)

   Ecuacin de la lnea tangete:

   Ecuacin de la lnea normal:

`>`	Informe([t,t^2,t^3])(t);

La curva:

posee las siguientes propiedades geomtricas.

   Velocidad:

   Aceleracin:

   Rapidez:

   Campo Tangente:

[1/((1+4*t^2+9*t^4)^(1/2)), 2*t/(1+4*t^2+9*t^4)^(1/2), 3*t^2/(1+4*t^2+9*t^4)^(1/2)]

   Campo Normal:

[-t*(2+9*t^2)/(1+9*t^4+9*t^2)^(1/2)/(1+4*t^2+9*t^4)^(1/2), -(9*t^4-1)/(1+9*t^4+9*t^2)^(1/2)/(1+4*t^2+9*t^4)^(1/2), 3*t*(2*t^2+1)/(1+9*t^4+9*t^2)^(1/2)/(1+4*t^2+9*t^4)^(1/2)]

   Campo Binormal:

[3*t^2/(1+9*t^4+9*t^2)^(1/2), -3*t/(1+9*t^4+9*t^2)^(1/2), 1/((1+9*t^4+9*t^2)^(1/2))]

   Curvatura:

2*(1+9*t^4+9*t^2)^(1/2)/(1+4*t^2+9*t^4)^(3/2)

   Torsin:

3/(1+9*t^4+9*t^2)

   Ecuacin de la lnea tangete:

[t+s/(1+4*t^2+9*t^4)^(1/2), t^2+2*s*t/(1+4*t^2+9*t^4)^(1/2), t^3+3*s*t^2/(1+4*t^2+9*t^4)^(1/2)]

   Ecuacin de la lnea normal:

[t-s*t*(2+9*t^2)/(1+9*t^4+9*t^2)^(1/2)/(1+4*t^2+9*t^4)^(1/2), t^2-s*(9*t^4-1)/(1+9*t^4+9*t^2)^(1/2)/(1+4*t^2+9*t^4)^(1/2), t^3+3*s*t*(2*t^2+1)/(1+9*t^4+9*t^2)^(1/2)/(1+4*t^2+9*t^4)^(1/2)]

   Ecuacin de la lnea binormal:

[t+3*s*t^2/(1+9*t^4+9*t^2)^(1/2), t^2-3*s*t/(1+9*t^4+9*t^2)^(1/2), t^3+s/(1+9*t^4+9*t^2)^(1/2)]

`>`	# Animacion de la formacin de la traza de una curva en el plano

`>`	*AnimaC([sin(t),cos(t)],0,2Pi,[color=blue,scaling=constrained]);**

[Maple Plot]

`>`	# Animacion de la formacin de la traza de una curva en el espacio

`>`	*AnimaC([sin(t),cos(t),sin(t)cos(t)],0,3Pi,[color=blue,axes=boxed,scaling=constrained,numpoints=1000]);*

[Maple Plot]

`>`	# Animacin del recorrido del vector tangente

`>`	*AnimaT([sin(t),cos(t)],0,2Pi,0.5);**

[Maple Plot]

`>`	*AnimaT([sin(t),cos(t),sin(t)cos(t)],0,2Pi,0.5);*

[Maple Plot]

`>`	# Animacin del recorrido del vector normal

`>`	*AnimaN([sin(t),cos(t)],0,2Pi,0.5);**

[Maple Plot]

`>`	*AnimaN([sin(t),cos(t),sin(t)cos(t)],0,2Pi,0.5);*

[Maple Plot]

`>`	# Animacin del recorrido del vector binormal

`>`	*AnimaB([sin(t),cos(t),sin(t)cos(t)],0,2Pi,0.5);*

[Maple Plot]

`>`	# obtencin de la representacin grfica en el plano de curva con curvatura prefijada

`>`	kappa:=s+sin(s):

`>`	Curva_Curvatura2d(kappa,-18,18,0,0,0,0,[color=blue,numpoints=1500]);

[Maple Plot]

`>`	*kappa2:=ssin(s):**

`>`	Curva_Curvatura2d(kappa2,-12,12,0,0,0,0,[color=orange,numpoints=2000,scaling=constrained]);

[Maple Plot]

`>`	*kappa3:=ssin(s)2:

`>`	Curva_Curvatura2d(kappa3,-20,20,0,0,0,0,[color=black,numpoints=1500]);

[Maple Plot]

`>`	kappa4:=s:

`>`	g1:=Curva_Curvatura2d(kappa4,-20,20,0,0,0,0,[color=yellow,numpoints=1500]): g2:=Curva_Curvatura2d(kappa4,-20,20,0,0,0,Pi/3,[color=blue,numpoints=1500]): g3:=Curva_Curvatura2d(kappa4,-20,20,0,0,0,2*Pi/3,[numpoints=1500]): display(g\|\|(1..3));

[Maple Plot]

`>`	*kappa5:=ssin(s):**

`>`	Curva_Curvatura2d(kappa5,-40,40,0,0,0,0,[numpoints=1500]);

[Maple Plot]

>

`>`	# obtencin de representacin grfica en el espacio de curva con curvatura y torsin prefijadas

`>`	kappa1:=s*sin(s): tau1:=cos(s):

`>`	setoptions3d(tickmarks=[0,0,0],axes=boxed,thickness=2,orientation=[-13,-99]):

`>`	Curva_Curvatura3d(kappa1,tau1,-10,10,1,0,0,0,1,0,0,0,1,0,[numpoints=1500,color=orange]);

[Maple Plot]

>

`>`	# Campo de vectores tangentes y normales 2d

`>`	elip:=(a,b)->ELIPSE(a,b)(t):

`>`	g1:=seq(CGT(elip(1.5,1),0.5,0,2Pi,2Pii/60,1,[scaling=constrained]),i=1..60): g2:=CTraza(elip(1.5,1),[t=0..2Pi,filled=true,color=gray]): plots[display]({g1,g2},title="Campo de vectores tangente",titlefont=[TIMES,BOLD,12]);

Error, (in plot) invalid arguments

[Maple Plot]

`>`	g1:=seq(CGN(elip(1.5,1),2.5,0,2Pi,2Pii/60,1),i=1..60): g2:=CTraza(elip(1.5,1),[t=0..2Pi,filled=true,color=gray]): plots[display]({g1,g2},title="Campo de vectores normales",titlefont=[TIMES,BOLD,12]);

Error, (in plot) invalid arguments

[Maple Plot]

`>`	# Campo de vectores tangentes normales y binormales para 3d

`>`	hlice:=HLICE_CIRCULAR(1,0.2)(t):

`>`	seq(CGT(hlice,1,0,4Pi,4Pi*i/100,[axes=boxed,orientation=[-62,58],numpoints=1000]),i=1..100): display(%);

[Maple Plot]

`>`	seq(CGN(hlice,1,0,4Pi,4Pi*i/100,[axes=boxed,orientation=[-62,58],numpoints=1000]),i=1..100): display(%);

[Maple Plot]

`>`	seq(CGB(hlice,1,0,4Pi,4Pi*i/100,[axes=boxed,orientation=[-62,58],numpoints=1000]),i=1..100): display(%);

[Maple Plot]

`>`	# tubos alrededor de curvas en el espacio

`>`	alpha:=HLICE_CIRCULAR(2,0.5)(t);

`>`	Tubo_Curva(alpha)(r,theta);

`>`	*Tubo_Grfica(alpha, [t=0..4Pi, radius=0.5, grid=[60,10], scaling=constrained, orientation=[-58,54], lightmodel=light2, style=PATCH,axes=boxed,color=COLOR(RGB,1,0.5,1)]);**

[Maple Plot]

`>`	NUDO_TOROIDAL(a,b,c,p,q)(t);

`>`	*CTraza(NUDO_TOROIDAL(8,3,5,2,5)(t), [t=0..2Pi, grid=[200,100], orientation=[54,16], axes=boxed, scaling=constrained, color=black, thickness=2,numpoints=2000]);**

[Maple Plot]

`>`	*Tubo_Grfica(NUDO_TOROIDAL(8,3,5,2,5)(t), [t=0..2Pi, radius=1.5, grid=[80,10], scaling=constrained, orientation=[53,19], lightmodel=light2, style=PATCH,axes=boxed,color=COLOR(RGB,1,0.5,1)]);**

[Maple Plot]

`>`	Tubo_Grfica(NUDO_TOROIDAL(8,3,5,1,10)(t), [t=0..2*Pi, radius=1.5, grid=[180,12], scaling=constrained, orientation=[53,19], lightmodel=light2, style=PATCHNOGRID,axes=boxed,color=COLOR(RGB,0.5,0.3,0.5)]);

[Maple Plot]

`>`	*OCHO_NUDO(t):=[10(cos(t)+cos(3t))+cos(2t)+cos(4t),6sin(t)+10sin(3t),4sin(t)sin(5t/2)+4sin(4t)-2sin(6t)]:*

`>`	*CTraza(OCHO_NUDO(t), [t=0..2Pi, grid=[200,100],orientation=[54,16], axes=boxed, scaling=constrained,thickness=4,numpoints=2000]);**

[Maple Plot]

`>`	*Tubo_Grfica(OCHO_NUDO(t), [t=0..2Pi, radius=1.5, grid=[180,12], scaling=constrained, orientation=[53,19], axes=boxed]);**

[Maple Plot]

`>`	# curvatura de una curva en el n-simo espacio

`>`	Ckappan:=alpha->proc(t0) local v, a, pe, nume, deno; v:=Cv(alpha)(t0): a:=Ca(alpha)(t0): pe:=diff(Cn(v),t); nume:=Cn(expand(Cn(v)a)-expand(subs(t=t0,pe)v)): deno:=Cn(v)^3: nume/deno; end:

`>`	Ckappan([sin(t),cos(t)])(t); simplify(%);

(sin(t)^2*(sin(t)^2+cos(t)^2)+cos(t)^2*(sin(t)^2+cos(t)^2))^(1/2)/(sin(t)^2+cos(t)^2)^(3/2)

`>`	*Ckappan([sin(t),cos(t),sin(t)cos(t)])(t): simplify(%);**

1/2*(1-6*cos(t)^4+6*cos(t)^2)^(1/2)/((-2*cos(t)^2+2*cos(t)^4+1)^3)^(1/2)

`>`	*Ckappa([sin(t),cos(t),sin(t)cos(t)])(t): simplify(%);**

1/2*(1-6*cos(t)^4+6*cos(t)^2)^(1/2)/((-2*cos(t)^2+2*cos(t)^4+1)^3)^(1/2)

`>`	*Ckappan([sin(t),cos(t),sin(t)cos(t),t^2])(t): simplify(%);**

1/2*(2*t^2+32*cos(t)^2*t^2-32*cos(t)^4*t^2+32*t*sin(t)*cos(t)^3-16*sin(t)*cos(t)*t+5-2*cos(t)^2+2*cos(t)^4)^(1/2)/((-2*cos(t)^2+2*cos(t)^4+2*t^2+1)^3)^(1/2)

`>`	alpha:=[cos(t/2),sin(t/2),t/2]:

`>`	# ecuacin de la linea tangente, normal y binormal (3d)

`>`	CLT(alpha,s)(t);

`>`	CLN(alpha,s)(t);

`>`	CLB(alpha,s)(t);

>

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App Preview:

A Package for Drawing Geometric Curves (documentation in Spanish)

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App Preview:

A Package for Drawing Geometric Curves (documentation in Spanish)

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