Geometria Computacional e Diferencial: As Curvas da Bosta

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1 Geometria Computacional e Diferencial: As Curvas da Bosta Ole Peter Smith, ole@ufg.br, Instituto de Matemática e Estatística, Universidade Federal de Goiás FLISOL, Goiânia-GO 16/04/2016

2 Please allow me to pay a tribute to Geogebra...

3 Introduction Playing with curves... PHP (should become Python...) Graphical Library: GD - Open Source! Math... the truely divine! Figure: Ellipse Evolute, Animations/Intro.gif

4 Image Resolução: (R x, R y ). Drawing Area: (x 0, y 0 ), (x 1, y 1 ). Coordinates Pixels ( ) ( ) ( ) xp ax 0 x = + y p 0 a y y ( ax a y ) = ( ) R x x 1 x 0 R y y 1 y 0 ( bx b y ( bx b y ) ) ( ) Rx a = x x 0 a y y 0

5 Image, Init Graphics function Image_Init_Graphics() { $this->image=imagecreate($this->r[0],$this->r[1]); } function Image_Save($fname,$n,$destroy=TRUE) { imagepng($this->image,$fname.".".$n.".png"); if ($destroy) { imagedestroy($this->img); } } O Conhecimento se aprende com os Livros e os Mestres A Sabedoria se aprende com os humildes Cora Coralina-GO

6 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] ); } TTT: Things Takes Time Piet Hein

7 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); } }

8 Equation: Parametrization: I m a Curve because I... Curve F(x, y) = 0, r(t) = (x, y) Ω ( ) x(t), t I y(t) Derivatives: 1st order Derivative Tangent, Unit Tangent Unit Normal Fórmula de Taylor 2nd order Derivative Curvature Integrals: Curve (Arc) Length

9 Equation: Parametrization: ( ) ( x(t) = y(t) Implementation: Parabola y = ax 2 + bx + c t at 2 + bt + c ), t R function Parabola($t) { return array ( $t, $this->a*$t*$t + $this->b*t + $this->c, ); }

10 Equation: Parametrization: ( ) x(t) = y(t) Ellipse x 2 a 2 + y 2 b 2 = 1 ( ) a cos t, t R([0, 2π[) b sin t Implementation: function Ellipse($t) { return array ( $this->$a*cos($t), $this->$b*sin($t), ); }

11 Equation: Parametrization: ( ) x(t) = y(t) Hyperbola x 2 a 2 y 2 b 2 = 1 ( ) a cosh t, t R b sinh t Implementation: function Hyperbola($t) { return array ( $this->$a*cosh($t), $this->$b*sinh($t), ); }

12 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; }

13 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; }

14 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" ); } $this->curveanimate("ellipse",200,0.0, );

15 Cycloids Circle Rolling on a Line ( ) ( ) x(t) r(t sin t r(t) = =, t > 0 y(t) r(1 cos t 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 Sorry, not a Matter of Opinion!

16 Cycloid Figure: Animations/Cycloid.gif Se Adão e Eva foram chineses, teriam comido a cobra... Facebook

17 Cycloid Figure: Animations/Cycloid_Poop.gif Dia que bosta vale ouro Os pobres todos nascerão sem... Grafiti

18 Physics... Velocity: Acceleration: Taylor s Formula: r (t) = v(t) = r (t) = a(t) = ( x ) (t) y (t) ( x ) (t) y (t) r(t) = r(t 0 ) + r (t 0 )(t t 0 ) r (t 0 )(t t 0 ) 2 + o ((t t 0 ) 2) Newton II: m r (t) = F ext

19 Derivatives, Cycloid Figure: Animations/Cycloid_Derivatives.gif Deus não preocupa-se com as nossas dificuldades matemáticas... Ele integra empiricalmente. Einstein

20 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); }

21 Accompanying Coordinate System Unit Tangent: Transverse: Unit Normal: a = ( ax a y t(t) = r (t) r (t) ) n(t) = t(t) Sistema ortonormal: (r(t), t(t), n(t)). ( ) ay â = a x

22 ACS, Cycloid Figure: Animations/Cycloid_ACS.gif Vive como se fosse morrer amanhã. Estude como se fosse viver para sempre. Einstein

23 Curvature Lines doesn t curve... Circles curves constantly: κ = 1/R. Regular Curve: r (t) 0. Curvature: κ(t) = x y x y ( x 2 + y 2) 3/2 Ratio: ρ(t) = 1/κ(t) Curvature Vector: c(t) = ρ(t)c(t) Center of Curvature: Oscillating Circle: Evolute: r c (t) = r(t) + c(t) (x x x ) 2 + (y y c ) 2 = ρ 2 r c (t) t I

24 Oscillating Circle Figure: Credit WikiPedia

25 Oscillating Circle, Curvature Vector Figure: Animations/Circle_Oscillating.gif Como libertar um povo que preza sues correntes? Voltaire

26 Cycloid & Evolute Figure: Animations/Cycloid_Evolute.gif»The Evolute of a Cycloid is a Cycloid«

27 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)) ); }

28 Trochoids Figure: Animations/Trochoid.gif A seriedade dos acontecimentos da minha epoca, me enche de esperanças... Marx

29 Trochoids Figure: Animations/Trochoids.gif Divide um maçã, que cada um fica com metade. Divide uma idéia... Compartilhe seus conhecimentos!

30 Fixed Circle, radius R; Epicycloid Circle, radius r, rolling outside; ρ = R + r e ω = R+r r = ρ r ( ) ( ) x(t) ρ cos t r cos ωt r(t) = = y(t) 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) ); }

31 Figure: Animations/Epicycloid.gif, R = 5r

32 Cardioid: R = r. Figure: Animations/Cardioid.gif É melhor que o povo não saiba, como se fazem leis e salsichas Otto von Bismarck

33 Poop size b > 0. ρ = R + r; σ = R + r + b; ( ) x(t) r(t) = y(t) = Epitrochoid ( ) ρ cos t σ cos ωt = ρe(t) σe(ωt) ρ sin t σ sin 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) ); }

34 Figure: Animations/Epitrochoid.gif, R = 5r, r = 5b. Quem leva braincadeira somente por brincadeira. E sério somente serio. De fato, desentendeu ambos. Piet Hein

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

36 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) ); }

37 Figure: Animations/Hypocycloid.gif, R = 9r

38 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) ); }

39 Figure: Animations/Hypotrochoid.gif, R = 7r, r = 5b.

40 Circle, radius R, Rolling on a Curve, r(t) Arc Length, s(t): s(t) s(t 0 ) = Rolling Circle Center: t t 0 r (t) dt R c (t) = r(t) + Rn(t) Rolling point angle: Unit Vector: Rolling Point: θ = s(t) s(t 0) R e(θ) = ( ) cos θ sin θ R(t) = r(t) + Rn(t) + Re(θ)

41 Circle Rolling on a Parabola Figure: Animations/Rolling_Parabola.gif

42 Rolling on y = sin(x) Figure: Animations/Rolling_Sinus.gif

43 Rolling on y = x(x 1)(x 2) Figure: Animations/Rolling_Polinomia.gif

44 Rolling outside a Hiperbola Figure: Animations/Rolling_Hiperbola.gif

45 Rolling inside a Hiperbola Figure: Animations/Rolling_Hiperbola2.gif

46 Lisajous Figure: Animations/Lisajous.gif

47 Life sure is a Mystery to be Lived Not a Problem to be Solved. Kierkegård