Geometria Computacional e Diferencial: As Curvas da Bosta

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MADE INLA_{TEX. O}LEPETERSMITH, IME/UFG -OLE@UFG.BR

Geometria Computacional e Diferencial:

As Curvas da Bosta

Ole Peter Smith,

ole@ufg.br, http://olepeter.mat.ufg.br

Instituto de Matemática e Estatística, Universidade Federal de Goiás

FLISOL, Goiânia-GO

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Introduction

•

Playing with curves....

•

PHP (should become Python...)

•

Graphical Library: GD - Open Source!

•

Math... the truely divine!

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Image

•

Resolução: (R

x

,

R

y

).

•

Drawing Area: (x

0,

y

0), (x1,

y

1). •

Coordinates ↔ Pixels

xp

y

p

=

ax

0

0 a

y

x

y

+

bx

b

y

⇔

ax

ay

=

−

Rx x1−x0 Ry y1−y0

!

∧

bx

by

=

Rx

− ax

· x0

−ay

· y

0

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Image, Init Graphics

O Conhecimento se aprende com os Livros e os Mestres A Sabedoria se aprende com os humildes Cora Coralina-GO

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Image, Init Canvas

function Image_Init_Canvas() { $this->Image_InitGraphics $this->A=array ( -1.0*$this->R[0]/($this->r1[0]-$this->r0[0]), 1.0*this->R[0]/($this->r1[1]-$this->r0[1]) ); $this->b=array ( 1.0*$this->R[0]-$this->A[0]*$this->r0[0] -$this->A[1]*$this->r0[1] ); }

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Image, Point2Pixels

function Point2Pixels($p) { return array ( $this->A[0]*p[0]+$this->b[0], $this->A[1]*p[1]+$this->b[1], ); } function Points2Pixels($ps) { foreach ($ps as $id => $p) { $ps[ $id ]=$this->Point2Pixel($p); } }

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I’m a Curve because I... Curve

Equation:

F (x , y ) = 0,

(x , y ) ∈ Ω

Parametrization:

r(t) =

x(t)

y (t)

,

t ∈ I

•

Derivatives:

• 1st order Derivative • Tangent, Unit Tangent • Unit Normal

• Fórmula de Taylor • 2nd order Derivative • Curvature

•

Integrals:

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Parabola

•

Equation:

y = ax

2

+

bx + c

•

Parametrization:

x(t)

y (t)

=

t

at

2

+

bt + c

,

t ∈ R

•

Implementation:

function Parabola($t) { return array ( $t,

$this->a*$t*$t + $this->b*t + $this->c, );

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Ellipse

•

Equation:

x

2

a

2

+

y

2

b

2

=

1

•

Parametrization:

x(t)

y (t)

=

a cos t

b sin t

,

t ∈ R([0, 2π[)

•

Implementation:

function Ellipse($t) { return array ( $this->$a*cos($t), $this->$b*sin($t), ); }

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Hyperbola

•

Equation:

x

2

a

2

−

y

2

b

2

=

1

•

Parametrization:

x(t)

y (t)

=

a cosh t

b sinh t

,

t ∈ R

•

Implementation:

function Hyperbola($t) { return array ( $this->$a*cosh($t), $this->$b*sinh($t), ); }

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Divide Interval

function Get_ts($N,$t1,$t2) {

$dt=($t2-$t1)/(1.0*($N-1)); $ts=array();

for ($i=0,$t=$t1 ; $i<=$N ; $t+=$dt,$i++) {

array_push($rs,$t); }

return $ts; }

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Gather Curve Points

•

$R: ’Lambda’ function.

•

Good old eval...

•

As unsave as your Programming Language and

Environment...

function CurvePoints($R,$N,$t1,$t2) { foreach ($this->Get_ts($N,$t1,$t2) as $t) { array_push($ps,$this->$R($t)); } return $ps; }

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Curve Animation

function CurveAnimate($R,$N,$t1,$t2) { $n=0; foreach ($this->Get_ts($N,$t1,$t2) as $t) { $this->Image_Init(); $ps=$this->CurvePoints($R,$n,$t1,$t); $this->DrawPoints($ps); $this->Image_Destroy($R,$n++); } system (

"usr/bin/convert -loop 0 -delay 5 ". $R."*.png ".$R.".gif"

); }

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Cycloids

•

Circle Rolling on a Line

r(t) =

x(t)

y (t)

=

r (t − sin t

r (1 − cos t

,

t > 0

• function Cicloid($t) { return array ( $this->r*($t-sin($t)), $this->r*(1.0-cos($t)) ); }

O bem que a violência faz é passageiro. O mal que ela faz é permanente. Mahatma

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Cycloid

Figure:Animations/Cycloid.gif

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Cycloid

Figure: Animations/Cycloid_Poop.gif

Dia que bosta vale ouro Os pobres todos nascerão sem... Grafiti

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Physics....

•

Velocity:

r

0

(t) =

v(t) =

x

0

_(t)

y

0

(t)

•

Acceleration:

r

00

(t) =

a(t) =

x

00

_(t)

y

00

(t)

•

Taylor’s Formula:

r(t) = r(t

0

) +

r

0

(t

0

)(t − t

0

) +

1

2 r

00

_(t

0

)(t − t

0

)

2

+

o

(t − t

0

)

2

•

Newton II:

m

r

00

(t) =

F

_ext

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Derivatives, Cycloid

Figure:Animations/Cycloid_Derivatives.gif

Deus não preocupa-se com as nossas dificuldades matemáticas... Ele integra empiricalmente. Einstein

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Derivatives

function Df($func,$t,$eps) { return ($this->$func($t+$eps)-$this->$func($t-$eps)) / (2.0*$eps); } function D2f($func,$t,$eps) { return ( $this->Df($func,$t+$eps)-$this->Df($func,$t-$eps) ) / (2.0*$eps); }

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Accompanying Coordinate System

•

Unit Tangent:

t(t) =

r

0

_(t)

|r

0

_(t)|

•

Transverse:

a =

a

x

a

y

b

a =

−a

y

a

x

•

Unit Normal:

n(t) = bt(t)

•

Sistema ortonormal: (

r(t), t(t), n(t)).

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ACS, Cycloid

Figure:Animations/Cycloid_ACS.gif

Vive como se fosse morrer amanhã. Estude como se fosse viver para sempre. Einstein

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Curvature

•

Lines doesn’t curve...

•

Circles curves constantly: κ = 1/R.

•

Regular Curve:

r

0

(t) 6=

0.

•

Curvature:

κ(t) =

x

0

_y

00

_{− x}

00

_y

0

x

02

₊

_y

02

3/2

• Ratio: ρ(t) = 1/κ(t) • Curvature Vector: c(t) = ρ(t)c(t) • Center of Curvature: r_c(t) =r(t) + c(t) • Oscillating Circle: (x − xx)2+ (y − yc)2= ρ2 • Evolute: [ t∈I r_c(t)

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Oscillating Circle

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Oscillating Circle, Curvature Vector

Figure: Animations/Circle_Oscillating.gif

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Cycloid & Evolute

Figure:Animations/Cycloid_Evolute.gif

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Trochoids

•

Rolling circle ratio, a > 0.

•

Poop ratio, b > 0.

•

r(t) =

x(t)

y (t)

=

at − (a + b) sin t

a − (a + b) cos t

• function Trochoid($t) { return array ( $this->a*$t-($this->a+$this->b)*sin($t)), $this->a -($this->a+$this->b)*cos($t)) ); }

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Trochoids

Figure: Animations/Trochoid.gif

A seriedade dos acontecimentos da minha epoca, me enche de esperanças... Marx

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Trochoids

Figure:Animations/Trochoids.gif

Divide um maçã, que cada um fica com metade. Divide uma idéia... Compartilhe seus conhecimentos!

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Epicycloid

•

Fixed Circle, radius R;

•

Circle, radius r , rolling outside;

•

ρ =

R + r e ω =

R+r_r

=

ρ_r •

r(t) =

x(t)

y (t)

=

ρ cos t − r cos ωt

ρ

sin t − r sin ωt

• function Epicicloid($t) { return array ( $this->rho*cos($t)-$this->r*cos($this->omega*$t), $this->rho*sin($t)-$this->r*sin($this->omega*$t) ); }

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Cardioid:

R = r .

Figure: Animations/Cardioid.gif

É melhor que o povo não saiba, como se fazem leis e salsichas Otto von Bismarck

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Epitrochoid

Poop size b > 0.

•

ρ =

R + r ;

•

σ =

R + r + b;

r(t) =

x(t)

y (t)

=

ρ cos t − σ cos ωt

ρ

sin t − σ sin ωt

= ρ

e(t) − σe(ωt)

•

Angular Velocity: ω =

ρ_r

;

• function Epicicloid($t) { return array ( $this->rho*cos($t)-$this->sigma*cos($this->omega*$t), $this->rho*sin($t)-$this->sigma*sin($this->omega*$t) ); }

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Figure: Animations/Epitrochoid.gif, R = 5r , r = 5b.

Quem leva braincadeira somente por brincadeira. E sério somente serio. De fato, desentendeu ambos. Piet Hein

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Epitrochoids

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Hypocycloid

•

Circle, radius r , rolling inside a circle, radius R;

•

ω =

R−r_r •

r(t) =

x(t)

y (t)

=

(R + r ) cos t + r cos ωt

(R + r ) sin t + r sin ωt

• function Hypocicloid($t) { return array ( $this->rho*cos($t)+ $this->r*cos($this->omega*$t), $this->rho*sin($t)+ $this->r*sin($this->omega*$t) ); }

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Hypotrochoid

•

Poop size b > 0

•

r(t) =

x(t)

y (t)

=

(R + r ) cos t + (b + r ) cos ωt

(R + r ) sin t + (b + r ) sin ωt

• function Hypotrochoid($t) { return array ( $this->rho*cos($t)+ ($this->r+$this->b)*cos($this->omega*$t), $this->rho*sin($t)+ ($this->r+$this->b)*sin($this->omega*$t) ); }

Geometria Computacional e Diferencial: As Curvas da Bosta