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Geometria Computacional

Ole Peter Smith, IME, UFG, [email protected]

24/05/2010www.ime.ufg.br/docentes/olepeter

Life sure is a Mystery to be LivedNot a Problem to be Solved...

These slides were made in LATEX/Beamer

Ole Peter Smith, IME, UFG, [email protected] () Geometria Computacional24/05/2010www.ime.ufg.br/docentes/olepeter 1

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Beautiful Scientific DocumentsClassical Free SoftwareMarkup LanguageThesis, automatic:· Bibliography· Contents· List of Tables, Figures· Index, ReferencesSlideshows: BeamerMikTex, LyX, TexMaker, Kyle ^SL!


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Quero Desenhar µατεµατικα...

Usando SL...Graphical Library· GD· Unices - Ruindows· PHP - C - Perl - ...· http://www.libgd.org/Documentation· http://www.php.net/manual/pt_BR/book.image.phpFormats: GIF - PNG - JPG^ Animated GIF’s_ GIF’s: poucas cores


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http://www.libgd.org/Documentation

http://www.php.net/manual/pt_BR/book.image.php

Apache/CGI

Apache: webserverDocumentos EstáticosDocumentos Dinâmicos

PHPLinguagem para gerar páginas dinâmics

Perl, C, JavaMinha maquina: 127.0.0.1Windows: Windows Apache Mysql Php (WAMP)Localizar ’ServerRoot’


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Estrutura de uma Programa

$image=CreateImageObject;$image− >Initialize;$image− >DrawSomething;$image− >DrawSomethingMore;...$image− >WriteImage;$image− >Close;Pixels... _


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Init Image Object (PHP)

ImageCreate:

$image = imagecreate (400,400)or die ("Cannot Create image");#Black on White - Background first!$bg_color = imagecolorallocate ($image,255, 255, 255);$text_color = imagecolorallocate ($image,0,0,0);

Tell Apache to Expect an Imageheader("Content-type: image/png");

Draw a Lineimageline($image,10,20,100,200,$text_color);

Close and Writeimagepng ($image);

Save in ServerRoot


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Line and Point

void imageline($image,int $x1, int $y1, #first pointint $x2, $int x2, #last pointint $color);

void imagearc($image,int $cx, int $cy, #centerint $a, $int b, #semi-eixosint $ang1, int $ang2, #angle (degrees)int $color);

void imagefilledarc(...);


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Draw Line

Draw Line: y = x + 1, x ∈ [0, 5]$image = imagecreate (400,400)

or die ("Cannot Create image");#Black on White - Background first!$bg_color = imagecolorallocate ($image,255, 255, 255);$text_color = imagecolorallocate ($image,0,0,0);

header("Content-type: image/png");imageline($image,0,1,5,6,$text_color);imagepng ($image);


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Draw Line I - Result


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Draw Line I - Problems

Escalonamento!Flipped: Cima por baixo...

$im=imagecreate (400,400)...imagestring($im,5,15+8,15-5,"1",$txtcol);imagefilledarc($im,15,15,10,10,0,360,$txtcol,IMG_ARC_PIE);imagefilledarc($im,200,200,50,100,0,90,$txtcol,IMG_ARC_PIE);imagestring($im,5,385+8,385-5,"2",$txtcol);imagefilledarc($im,385,385,10,10,0,360,$txtcol,IMG_ARC_PIE);


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Draw Line II - Result


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Scaling

Rx ,Ry ∈ N: Ex. 400,400Minx ,Miny ∈ R: 0.0,0.0Maxx ,Maxy ∈ R: 6.0,6.0Affin Application: (

XY

)=

(axx + bxayy + by

)Conditions:(

0Ry

)=

(axMinx + bxayMiny + by

),

(Rx0

)=

(axMaxx + bxayMaxy + by

)


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Scaling

ax(Maxx −Minx) = Rx − 0 ∧ ay (Maxy −Miny ) = 0− Ry ⇔

ax =Rx

Maxx −Minx∧ ay = − Ry

Maxy −Miny

bx = −axMinx

by = −ayMaxy


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Scaling - Init

function InitCanvas($R,$Min,$Max){

global $a,$b;$a=array();$b=array();

$a[0]=$R[0]/($Max[0]-$Min[0]);$a[1]=-$R[1]/($Max[1]-$Min[1]);

$b[0]=-$a[0]*$Min[0];$b[1]=$a[1]*$Max[1];

}


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Scaling - Convertion

function ScalePoint($p){

global $a,$b;

$pp=array();for ($i=0;$i<2;$i++){

$pp[$i]=((int) ($a[$i]*$p[$i]+$b[$i]));}

return $pp;}


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Now Draw the Line....

$Res=array(400,400);$Min=array(0.0,0.0);$Max=array(6.0,6.0);InitCanvas($R,$Min,$Max);$p1=array(0.0,1.0);$p2=array(5.0,6.0);$pp1=ScalePoint($p1);$pp2=ScalePoint($p2);imageline($pp1[0],$pp1[1],$pp2[0],$pp2[1],$color);


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Draw Line I - Result


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Serial Convertion

Maligma: Código Repetidofunction ScalePoints($ps){

$pps=array();for ($n=0;$n<count($ps);$n++){

$pps[$n]=ScalePoint($ps[$n]);}

return $pps;}


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Funções Trigonométricas

Nada de triângulos...


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Drawing a Circle

Equation:

(x − xc)2 + (y − yc)

2 = r2

Parametrization: (x(t)y(t)

)=

(xc + r cos tyc + r sin t

)t ∈ [−π, π]


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Generating a Circle

function Circle($n,$pc,$r,$ang=0.0){

$dt=2*3.1415927/(1.0*($n-1));$ps=array();for ($i=0,$t=$ang;$i<$n;$i++,$t+=$dt){

$p=array( $pc[0]+$r*cos($t),$pc[1]+$r*sin($t) );array_push($ps,$p);

}return $ps;

}


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Curve as Line Segments

function DrawCurve($im,$ps,$color){

$pps=ScalePoints($ps);for ($n=0;$n<count($pps)-1;$n++){

imageline($im,$pps[$n][0],$pps[$n][1],$pps[$n+1][0],$pps[$n+1][1],$color);

}}


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Drawing Circle

$n=6;$Res=array(400,400);$Min=array(0.0,0.0);$Max=array(6.0,6.0);InitCanvas($Res,$Min,$Max);$center=array(2.0,1.0);$ps=Circle($n,$center,2);DrawCurve($im,$ps,$color);imagestring($im,3,5,5,"n=$n",$color);


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Draw Circle - Result n = 6


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Drawing a Wheel

function DrawWheel($im,$n,$dn,$pc,$r,$R,$color,$ang=0.0){

$p1s=Circle($n,$pc,$r,$ang);DrawCurve($im,$p1s,$color);$p2s=Circle($n,$pc,$R,$ang);DrawCurve($im,$p2s,$color);for ($i=0;$i<$n;$i+=$dn){

$p1=ScalePoint($p1s[$i]);$p2=ScalePoint($p2s[$i]);imageline($im,$p1[0],$p1[1],$p2[0],$p2[1],$color);

}}


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Drawing a Wheel

$n=200;$dn=5;$center=array(2.0,1.0);DrawWheel($im,$n,$dn,$center,$color,3.0,0.2);

Modelo radialMelhorar: Modelo tangencial


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Wheel - Result


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Better Model

Point 1:

r1 =

(r cos θr sin θ

)Tangente ou normal unitário(versor):

n =

(− sin θcos θ

)Ponto 2:

r2 = r1 + ln =

(r cos θ − l sin θr sin θ + l cos θ

)|r2|2 = R2 ⇔ l = ±

√R2 − r2


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Animate Wheel

$nimages=79;$dang=2.0*3.1415927/$nimages;system("/bin/rm Wheel*.gif");for ($ang=0.0,$i=0;$i<$nimages;$i++){

$image = ImageCreate...$center=array(3.0,3.0);DrawWheel($image,200,5,$center,0.2,2.0,$color,$ang);imagegif($image,"Wheel.$i.gif");$ang+=$dang;

}Merda! PHP não tem suporte para GIF animado...


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Animate Wheel

ImageMagick...Continued...system("/usr/bin/convert -delay 2 -loop 0 ".

"Wheel.*.gif Wheel.gif");header("Content-type: text/html");print "<IMG SRC=’Wheel.gif’>\n";

Animate...


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http://127.0.0.1/html/uruacu/6.php

Rotação

Rotação, ângulo θ:(x ′

y ′

)= R(θ)

(xy

)=

(cos θ − sin θsin θ cos θ

)(xy

)=

(x cos θ − y sin θx sin θ + y cos θ

)R(θ1)R(θ2) = R(θ1 + θ2)

⇒ R ortogonal : RTR = I⇔ R−1 = RT

⇒ R(θ)−1 = R(−θ)⇒ R(θ)n = R(nθ), ∀n ∈ Z


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Rotação

^ Pontos fixos: (0.0)^ Retas em retas^ Círcumferências em

Círcumferências^ Elipses em elipses^ Hipérboles em Hipérboles


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Aplicação Linear: Rn 7→ Rn

f (x) = Ax − A = (a1...an)

Composição: f1 ◦ f2 = A1A2Bijeção ⇔ detA 6= 0Núcleo, Ax = 0: dim= n − ρA

Imagen, ger(a1a2): dim Im(f ) = ρA

Inversa seeeeee: A regular, f −1: A−1

Imagem inversa: Se Ax0 = y:

f −1(y) = {x0 + ker f }


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Escalonamento

Dilatação pelo eixo y :

E x =

(λ 00 1

)(xy

)=

(λxy

)λ > 1: Dilataçãoλ = 1: Identidade0 < λ < 1: Contraçãoλ = 0: Projeção no eixo yλ = −1: Reflexão no eixo y


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Escalonamento

Dilatação pelo eixo e = (cos θ, sin θ):1: Rotação ângulo θ: i 7→ e e j 7→ f = e2: Dilatação pelo eixo f3: Rotação ângulo −θEθ

= R−1ER =(cos θ sin θ− sin θ cos θ

)(λ 00 1

)(cos θ − sin θsin θ cos θ

)=

(λ cos2 θ + sin2 θ (1− λ) cos θ sin θ

(1− λ) cos θ sin θ λ sin2 θ + cos2 θ

)LinearEθe E similares


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Escalonamento


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Reflexão


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Projeção


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Reflexão em e = (cos θ, sin θ

R =

(− cos 2θ sin 2θ

sin 2θ cos 2θ

)R2 = I

R−1 = R

Ortogonal: R−1 = R

Ímpar: detR = −1Isometria: x′ · x′ = x · xPreserva comprimentosReverte ângulosNão rígida


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Projeção em e = (cos θ, sin θ)

P =

(sin2 θ sin θ cos θ

sin θ cos θ cos2 θ

)Nilpotente: Pn = P, n ≥ 1Singular: detP = 0ker f = tf, t ∈ R


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Translação, t = (t1, t2)

f (x) = x + t = I x + t_: não linear...Afim: f (x) = A x + b⇒ Geometria Projetiva


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Projective Geometry(

xy

)→

xy1

. E:

xyz

→ (x/zy/z

), z 6= 0

Translation:

1 0 t10 1 t20 0 1

xy1

=

x + t1y + t2

1

^: Translation is Linear!

Rotation:

cos θ − sin θ 0sin θ cos θ 00 0 1

xy1

=

x cos θ − y sin θx sin θ + y cos θ

1

Scaling:

λ1 0 00 λ2 00 0 1

xy1

=

λ1xλ2y1

Ole Peter Smith, IME, UFG, [email protected] () Geometria Computacional

24/05/2010www.ime.ufg.br/docentes/olepeter 44/ 71

Projective Geometry


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CurvatureCurve in R2:

r(t) =

(x(t)y(t)

)− r′(t) =

(x ′(t)y ′(t)

)− r′′(t) =

(x ′′(t)y ′′(t)

)Accompanying Coordenate System:

t(t) =r′(t)|r′(t)|

− n(t) = t(t)

Oscillating Circle, radius ρ - Curvature κ:

1ρ

=r′(t) · r′′(t)|r′(t)|3

=

∣∣∣∣ x ′ x ′′

y ′ y ′′

∣∣∣∣(x ′2 + y ′2)3/2

Curvature Vector κn =

r′(t) · r′′(t)|r′(t)|4

r′(t)


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Calculate Curvature & Center

Vector tan=this->Tangent(t);Vector acc=this->Acceleration(t);

double len=tan.Norm2();double kappa=Determinant2(tan,acc)/(len*len*len);

tan.Normalize();Vector n=tan.Normal2();

n*=kappa;p+=n;

return p;


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EvoluteCenter of Curvature:

rC (t) = r(t) + κn(t)

Center of Curvature:

rC (t) = r(t) +

∣∣∣∣ x ′ x ′′

y ′ y ′′

∣∣∣∣(x ′2 + y ′2)2

(−y ′

x ′

)Evolute: Curve of Centers of Curvature


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Calculate & Draw Evolutedouble dt=(t2-t1)/(1.0*(n-1));//First pointVector rant=this->CalcPoint(t1);Vector revant=this->CenterOfCurvature2(t1);double t=t1+dt;for (int i=1;i<n;i++){

Vector rnext=this->CalcPoint(t);Vector revnext=this->CenterOfCurvature2(t);graph->DrawLine(rant,rnext,color);graph->DrawLine(revant,revnext,colorev);rant=rnext;revant=revnext;t+=dt;

}graph->DrawPoint(rant,color);graph->DrawPoint(revant,colorev);


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Cycloid

Mark Point on Circle, radius a, and Let it Roll

r(t) =

(at − a sin ta − a cos t

)r′(t) =

(a − a cos t

a sin t

)r′′(t) =

(a sin ta cos t

)κ =

a2(cos t − 1)(2a2(1− cos t))3/2

=−1

2√2a(1− cos t)1/2

rc = r + κn =

(at − a sin ta − a cos t

)− 1

4a2(1− cos t)

(−a sin t

a − a cos t

)Cycloids!!!


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http://127.0.0.1/cgi-bin/Graphics/index.pl/Rolling/Cycloid

Define Cycloid

double a;double X(double t) { return a*(t-sin(t)); }double Y(double t) { return a*(1-cos(t)); }double Kappa(double t){

double div=2*sqrt(2)*a*sqrt(1-cos(t));return -1.0/div;

}...Curve *curve=new Curve(2,n);curve->SetR(0,X);curve->SetR(1,Y);curve->SetCurvature(Kappa);curve->DrawWithEvolute(graph,n,0.0,theta);


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Trochoid

Mark Point distance b from Center of Circle, radius a, and Let it Rolla = b: Cycloid

r(t) =

(at − b sin ta − b cos t

)r′(t) =

(a − b cos t

b sin t

)r′′(t) =

(a sin t−b cos t

)κ =

ab cos t − b2

(a2 + b2 − 2ab cos t)3/2rc = r + κn =(

at − b sin ta − b cos t

)+

ab cos t − b2

(a2 + b2 − 2ab cos t)2

(−b sin t

a − b cos t

)Trochoids!!!


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http://127.0.0.1/cgi-bin/Graphics/index.pl/Rolling/Trochoid.1

Define Trochoid

double a,b;double X(double t) { return a*t-b*sin(t); }double Y(double t) { return a*1-b*cos(t); }double Kappa(double t){

double denom=a*a+b*b-2.0*a*b*cos(t);denom=sqrt(denom)*denom;

return (a*b*cos(t)-b*b)/denom;}...Curve *curve=new Curve(2,n);curve->SetR(0,X);curve->SetR(1,Y);curve->SetCurvature(Kappa);curve->DrawWithEvolute(graph,n,0.0,theta);


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Unit Vectors

e(t) =

(cos tsin t

)f(t) =

(− sin tcos t

)Derivatives:

e′(t) = e(t) = f(t)

f ′(t) = f(t) = −e(t)

Dotproducts:

e(t1) · e(t2) = f(t1) · f(t2) = cos (t1 − t2)

e(t1) · f(t2) = −f(t1) · e(t2) = sin (t1 − t2)

Note:e(ηt) = cos (µt)e(t) + sin (µt)f(t)

f(ηt) = − sin (µt)e(t) + cos (µt)f(t)

With: µ = η − 1.Ole Peter Smith, IME, UFG, [email protected] () Geometria Computacional

24/05/2010www.ime.ufg.br/docentes/olepeter 55/ 71

Epicycloid I

Circle, Radius r , Rolls outside Circle, Radius Rr(t) = (

(R + r) cos t − r cos(R+r

r t)

(R + r) sin t − r sin(R+r

r t) ) =

r [η e(t)− e (ηt)]

Dimensionlesses: η =R + r

r> 1 e µ =

Rr

= η − 1 > 0.

r′(t) =rη [f(t)− f (ηt)]

r′′(t) =rη [−e(t) + ηe (ηt)]

r′(t) =rη [−e(t) + e (ηt)]


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Epicycloid II

r′(t) · r′′(t) =

(rη)2 [1 + η − (1 + η) cos (η − 1)t] =

(rη)2(1 + η) [1− cosµt]

r′(t) · r′(t) =(rη)2 [1 + 1− 2 cos (η − 1)t] =

2(rη)2 [1− cosµt] =

κ =(rη)2(1 + η) [1− cosµt]

(2(rη)2 [1− cosµt])3/2=

(1 + η)

2√2rη

[1− cosµt]−1/2


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Epicycloid III

κ = +∞ para µt = 2pπ, p ∈ Zκn = κ

|r(t)| r′(t) =

(rη)2(1 + η) [1− cosµt](2(rη)2 [1− cosµt])2

rη [−e(t) + e (ηt)]

(1 + η)

4rη[1− cosµt]−1 {−e(t) + e (ηt)}

Evolute:

rC (t) = r(t) + κn =

r [η e(t)− e (ηt)] +(1 + η)

4rη[1− cosµt]−1 {−e(t) + e (ηt)}

Epicycloids!!!


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http://127.0.0.1/cgi-bin/Graphics/index.pl/Rolling/Epicycloid.1

Define Epicycloid

double a,b;double X(double t) { return (a+b)*cos(t)-r*cos( (a+b)/r*t ); }double Y(double t) { return (a+b)*sin(t)-r*sin( (a+b)/r*t ); }double Kappa(double t){

double denom=2.0*sqrt(2.0)*b*(b+a)*sqrt(1-cos(a/r*t));double val=(2.0*b+a)/denom;

return val;}...Curve *curve=new Curve(2,n);curve->SetR(0,X);curve->SetR(1,Y);curve->SetCurvature(Kappa);curve->DrawWithEvolute(graph,n,0.0,theta);


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Hipocycloid I

Circle, Radius r , Rolls inside Circle, Radius Rr(t) = (

(R − r) cos t + r cos( r−R

r t)

(R − r) sin t + r sin( r−R

r t) ) =

r [−η e(t) + e (ηt)]

Dimensionlesses: η =r − R

r≤ 0 e µ =

Rr

= 1− η ≥ 1.

r′(t) =rη [−f(t) + f (ηt)]

r′′(t) =rη [e(t)− η e (ηt)]

r′(t) =rη [e(t)− e (ηt)]


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Hipocycloid II

r′(t) · r′′(t) =

(rη)2 [1 + η − (1 + η) cos (η − 1)t] =

(rη)2(1 + η) [1− cosµt]

r′(t) · r′(t) =(rη)2 [1 + 1− 2 cos (η − 1)t] =

2(rη)2 [1− cosµt] =

κ =(rη)2(1 + η) [1− cosµt]

(2(rη)2 [1− cosµt])3/2=

(1 + η)

2√2rη

[1− cosµt]−1/2


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Hipocycloid III

κ = +∞ para µt = π2 + 2pπ, p ∈ Z

κn =(rη)2(1 + η) [1− cosµt](2(rη)2 [1− cosµt])2

rη {−e(t) + e (ηt)} =

(1 + η)

4rη[1− cosµt]−1 {−e(t) + e (ηt)}

Evolute:

rC (t) = r(t) + κn =

r [−η e(t) + e (ηt)] +(1 + η)

4(rη)2[1− cosµt]−1 {−e(t) + e (ηt)}

Hypocycloids!!!


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http://127.0.0.1/cgi-bin/Graphics/index.pl/Rolling/Hypocycloid.1

Define Hypocycloid

double a,b;double X(double t) { return (a-b)*cos(t)+b*cos( (a-b)/r*t ); }double Y(double t) { return (a-b)*sin(t)-b*sin( (a-b)/r*t ); }double Kappa(double t){

double denom=2.0*sqrt(2.0)*b*(b-a)*sqrt(1-cos(a/b*t));double val=(2.0*b-a)/denom;

return val;}...Curve *curve=new Curve(2,n);curve->SetR(0,X);curve->SetR(1,Y);curve->SetCurvature(Kappa);curve->DrawWithEvolute(graph,n,0.0,theta);


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Epicycloid & Hipocycloid

κHypo =

(1 + r−Rr )

2√2r r−R

r

[1− cos (

Rr

t)]−1/2

=

2r − R2√2r(r − R)

[1− cos (

Rr

t)]−1/2

κEpi =

(1 + R+rr )

2√2r R+r

r

[1− cos (

Rr

t)]−1/2

=

2r + R2√2r(r + R)

[1− cos (

Rr

t)]−1/2


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Hypotrochoid & Epitrochoid

Hypotrochoid: r(t) =((R − r) cos t + d cos

( r−Rr t)

(R − r) sin t + d sin( r−R

r t) )

Hypotrochoids...Epitrochoid: r(t) =(

(R + r) cos t − d cos(R+r

r t)

(R + r) sin t − d sin(R+r

r t) )

Epitrochoids...


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http://127.0.0.1/cgi-bin/Graphics/index.pl/Rolling/Cicloid

http://127.0.0.1/cgi-bin/Graphics/index.pl/Rolling/Cicloid

Rolling

Circle, Radius R , Rolls on Curve:

r(t) =

(x(t)y(t)

)

rR(t) =

(x(t)y(t)

)+

R

(x ′(t)2 + y ′(t)2)1/2

(−y ′(t)x(t)

)Oloids...


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http://127.0.0.1/cgi-bin/Graphics/index.pl/Rolling/Rolling

Defining Rolling

Vector Curve::CalcRollingCenter(double t,double a,Vector &rant,int inside){

Vector tan=this->UnitTangent(t);Vector n=tan.Normal2();n*=a;Vector rc;if (inside==1) { rc=rant-n; }else { rc=rant+n; }return rc;

}Vector Curve::CalcRollingPoint(double s,double a,Vector &rc){

Vector *p=Vector2(a*cos(s/a-PI),a*sin(s/a-PI));(*p)+=rc;return *p;

}


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Defining Rolling

double dt=(t2-t1)/(1.0*(n-1));Vector rant=this->CalcPoint(t1);Vector rcant=this->CalcRollingCenter(t1,a,rant,inside);Vector rpant=this->CalcRollingPoint(0.0,a,rcant);double t=t1+dt, s=0.0; //parameter & arc lengthfor (int i=1;i<n;i++){

Vector rnext=this->CalcPoint(t);Vector rcnext=this->CalcRollingCenter(t,a,rnext,inside);Vector rpnext=this->CalcRollingPoint(s,a,rcnext);graph->DrawLine(rpant,rpnext,color);s+=Distance(rant,rnext);rant=rnext;rcant=rcnext;rpant=rpnext;

}


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Defining Rolling

double r;

double RX(double t){

return t;}

double RY(double t){

return r*sin(t);}


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Optimization: Projected Gradients

minx∈GF (x)Forma simplificado:yk+1 = xk − βk∇F (xk)

xk+1 = PG (yk+1)

Iteração externoIteração interno

Visualização


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Thanks!


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