Gc.d day
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The document discusses using the LaTeX typesetting system and the GD graphics library in PHP to generate geometric diagrams and images computationally. It covers initializing image objects in PHP-GD, drawing lines and curves, scaling drawings, and generating circles by dividing them into line segments. Examples are provided of drawing a line, email as an icon, and a circle to demonstrate these techniques.
![A
L TEX
Geometria Computacional Init Image Object
Primeiros GDs Email as Icon
Escalonamento Draw Line I
Curvas Problem
Inventing the Wheel Draw Line II
Show Off: Os Oids...
Never Publish Emails on www
$id=$_GET["ID"];
$person=Mysql_SelectUniqueHash("People","ID=’".$id."’");
$email=$person[ "Email" ];
header("Content-type: image/png");
$hgt=imagefontheight(3);
$wdt=imagefontwidth(3*strlen($email);
$image = ImageCreate ($wdt+10, $hgt+10)
or die ("Cannot Create image");
$bg_color = imagecolorallocate ($image,255, 255, 255);
$text_color = imagecolorallocate ($image,34,139,34);
imagecolortransparent($image,$bg_color);
imagestring ($image,$fontsize, 5, 5, $email, $text_color);
imagepng ($image);
Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional](https://image.slidesharecdn.com/gc-dday-130201023511-phpapp02/85/Gc-d-day-6-320.jpg)
![A
L TEX
Geometria Computacional Init Image Object
Primeiros GDs Email as Icon
Escalonamento Draw Line I
Curvas Problem
Inventing the Wheel Draw Line II
Show Off: Os Oids...
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 ($image);
Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional](https://image.slidesharecdn.com/gc-dday-130201023511-phpapp02/85/Gc-d-day-7-320.jpg)
![A
L TEX
Scaling
Geometria Computacional
Programmed
Primeiros GDs
Initialization
Escalonamento
Convertion
Curvas
Now Draw
Inventing the Wheel
Convertions
Show Off: Os Oids...
Scaling - Init
function InitCanvas($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]*$Max[1];
}
Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional](https://image.slidesharecdn.com/gc-dday-130201023511-phpapp02/85/Gc-d-day-14-320.jpg)
![A
L TEX
Scaling
Geometria Computacional
Programmed
Primeiros GDs
Initialization
Escalonamento
Convertion
Curvas
Now Draw
Inventing the Wheel
Convertions
Show Off: Os Oids...
Scaling - Convertion
function ScalePoint($p)
{
global $a,$b;
$pp=array();
for ($i=0;$i<2;$i++)
{
$pp[$i]=((int) ($a[$i]*$p[$i]+$b[$i]));
}
return $pp;
}
Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional](https://image.slidesharecdn.com/gc-dday-130201023511-phpapp02/85/Gc-d-day-15-320.jpg)
![A
L TEX
Scaling
Geometria Computacional
Programmed
Primeiros GDs
Initialization
Escalonamento
Convertion
Curvas
Now Draw
Inventing the Wheel
Convertions
Show Off: Os Oids...
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],$color);
Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional](https://image.slidesharecdn.com/gc-dday-130201023511-phpapp02/85/Gc-d-day-16-320.jpg)
![A
L TEX
Scaling
Geometria Computacional
Programmed
Primeiros GDs
Initialization
Escalonamento
Convertion
Curvas
Now Draw
Inventing the Wheel
Convertions
Show Off: Os Oids...
Serial Convertion
Maligma: C´digo Repetido
o
function ScalePoints($ps)
{
$pps=array();
for ($n=0;$n<count($ps);$n++)
{
$pps[$n]=ScalePoint($ps[$n]);
}
return $pps;
}
Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional](https://image.slidesharecdn.com/gc-dday-130201023511-phpapp02/85/Gc-d-day-18-320.jpg)
![A
L TEX
Geometria Computacional Drawing a Circle
Primeiros GDs Generate
Escalonamento Line Segments
Curvas Code
Inventing the Wheel Results
Show Off: Os Oids...
Drawing a Circle
Equation:
(x − xc )2 + (y − yc )2 = r 2
Parametrization:
x(t) xc + r cos t
=
y (t) yc + r sin t
t ∈ [−π, π]
Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional](https://image.slidesharecdn.com/gc-dday-130201023511-phpapp02/85/Gc-d-day-19-320.jpg)
![A
L TEX
Geometria Computacional Drawing a Circle
Primeiros GDs Generate
Escalonamento Line Segments
Curvas Code
Inventing the Wheel Results
Show Off: Os Oids...
Generating a Circle
function Circle($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 $ps;
}
Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional](https://image.slidesharecdn.com/gc-dday-130201023511-phpapp02/85/Gc-d-day-20-320.jpg)
![A
L TEX
Geometria Computacional Drawing a Circle
Primeiros GDs Generate
Escalonamento Line Segments
Curvas Code
Inventing the Wheel Results
Show Off: Os Oids...
Curve as Line Segments
function DrawCurve($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],
$color
);
}
Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional](https://image.slidesharecdn.com/gc-dday-130201023511-phpapp02/85/Gc-d-day-21-320.jpg)
![A
L TEX
Geometria Computacional
Primeiros GDs Drawing a Wheel
Escalonamento Animate...
Curvas Solution
Inventing the Wheel
Show Off: Os Oids...
Drawing a Wheel
function DrawWheel($im,$n,$dn,$pc,$r1,$r2,$color,$ang=0.0)
{
$p1s=Circle($n,$pc,$r1,$ang);
DrawCurve($im,$p1s,$color);
$p2s=Circle($n,$pc,$r2,$ang);
DrawCurve($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);
}
}
Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional](https://image.slidesharecdn.com/gc-dday-130201023511-phpapp02/85/Gc-d-day-26-320.jpg)
Gc.d day
- 1. A L TEX Geometria Computacional Primeiros GDs Escalonamento Curvas Inventing the Wheel Show Off: Os Oids... Geometria Computacional Ole Peter Smith, IME, UFG, ole@mat.ufg.br Dia Debian UEG, UnU Laranjeiras 17/08/2012 Life sure is a Mystery to be Lived Not a Problem to be Solved... Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 2. A L TEX Geometria Computacional Primeiros GDs Escalonamento Curvas Inventing the Wheel Show Off: Os Oids... Beautiful Scientific Documents Classical Free Software Markup Language Thesis, automatic: · Bibliography · Contents · List of Tables, Figures · Index, References SL! Slideshows: Beamer MikTex, LyX, TexMaker, Kyle Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 3. A L TEX Geometria Computacional Primeiros GDs Geometria Computacional Escalonamento GD Curvas Inventing the Wheel Show Off: Os Oids... Quero Desenhar µατ µατ ικα... Usando SL... Graphical Library · GD · Unices - Ruindows · PHP - C - Perl - ... · http://www.libgd.org/Documentation · http://www.php.net/manual/pt BR/book.image.php Formats: GIF - PNG - JPG Animated GIF’s GIF’s: poucas cores Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 4. A L TEX Geometria Computacional Primeiros GDs Geometria Computacional Escalonamento GD Curvas Inventing the Wheel Show Off: Os Oids... Estrutura: $image=CreateImageObject; $image− >DrawSomething; ... $image− >WriteImage; $image− >Close; Pixels... Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 5. A L TEX Geometria Computacional Init Image Object Primeiros GDs Email as Icon Escalonamento Draw Line I Curvas Problem Inventing the Wheel Draw Line II Show Off: Os Oids... 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 Image header("Content-type: image/png"); Draw a Line imageline($image,10,20,100,200,$text_color); Close and Write imagepng ($image); Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 6. A L TEX Geometria Computacional Init Image Object Primeiros GDs Email as Icon Escalonamento Draw Line I Curvas Problem Inventing the Wheel Draw Line II Show Off: Os Oids... Never Publish Emails on www $id=$_GET["ID"]; $person=Mysql_SelectUniqueHash("People","ID=’".$id."’"); $email=$person[ "Email" ]; header("Content-type: image/png"); $hgt=imagefontheight(3); $wdt=imagefontwidth(3*strlen($email); $image = ImageCreate ($wdt+10, $hgt+10) or die ("Cannot Create image"); $bg_color = imagecolorallocate ($image,255, 255, 255); $text_color = imagecolorallocate ($image,34,139,34); imagecolortransparent($image,$bg_color); imagestring ($image,$fontsize, 5, 5, $email, $text_color); imagepng ($image); Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 7. A L TEX Geometria Computacional Init Image Object Primeiros GDs Email as Icon Escalonamento Draw Line I Curvas Problem Inventing the Wheel Draw Line II Show Off: Os Oids... 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 ($image); Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 8. A L TEX Geometria Computacional Init Image Object Primeiros GDs Email as Icon Escalonamento Draw Line I Curvas Problem Inventing the Wheel Draw Line II Show Off: Os Oids... Draw Line I - Result Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 9. A L TEX Geometria Computacional Init Image Object Primeiros GDs Email as Icon Escalonamento Draw Line I Curvas Problem Inventing the Wheel Draw Line II Show Off: Os Oids... Draw Line I - Problem Escalonamento! Flipped: Cima por baixo... Draw Point: filledarc void imagearc($im, int $cx, int $cy, #center int $a, $int b, #semi-eixos int $ang1, int $ang2, #angle (degrees) int $color); void imagefilledarc($im, int $cx, int $cy, int $a, int $b, int $ang1, int $ang2, int $color,int $style); Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 10. A L TEX Geometria Computacional Init Image Object Primeiros GDs Email as Icon Escalonamento Draw Line I Curvas Problem Inventing the Wheel Draw Line II Show Off: Os Oids... Draw Line II $im=imagecreate (400,400); $bgcol = imagecolorallocate ($im,255, 255, 255); $txtcol = imagecolorallocate ($im,0,0,0); header("Content-type: image/png"); 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) imagepng($im); Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 11. A L TEX Geometria Computacional Init Image Object Primeiros GDs Email as Icon Escalonamento Draw Line I Curvas Problem Inventing the Wheel Draw Line II Show Off: Os Oids... Draw Line II - Result Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 12. A L TEX Scaling Geometria Computacional Programmed Primeiros GDs Initialization Escalonamento Convertion Curvas Now Draw Inventing the Wheel Convertions Show Off: Os Oids... Scaling Rx , Ry ∈ N: Ex. 400,400 Minx , Miny ∈ R: 0.0,0.0 Maxx , Maxy ∈ R: 6.0,6.0 Affin Application: X ax x + bx = Y ay y + by Conditions: 0 ax Minx + bx Rx ax Maxx + bx = , = Ry ay Miny + by 0 ay Maxy + by Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 13. A L TEX Scaling Geometria Computacional Programmed Primeiros GDs Initialization Escalonamento Convertion Curvas Now Draw Inventing the Wheel Convertions Show Off: Os Oids... Scaling ax (Maxx − Minx ) = Rx − 0 ∧ ay (Maxy − Miny ) = 0 − Ry ⇔ Rx Ry ax = ∧ ay = − Maxx − Minx Maxy − Miny bx = −ax Minx by = −ay Maxy Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 14. A L TEX Scaling Geometria Computacional Programmed Primeiros GDs Initialization Escalonamento Convertion Curvas Now Draw Inventing the Wheel Convertions Show Off: Os Oids... Scaling - Init function InitCanvas($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]*$Max[1]; } Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 15. A L TEX Scaling Geometria Computacional Programmed Primeiros GDs Initialization Escalonamento Convertion Curvas Now Draw Inventing the Wheel Convertions Show Off: Os Oids... Scaling - Convertion function ScalePoint($p) { global $a,$b; $pp=array(); for ($i=0;$i<2;$i++) { $pp[$i]=((int) ($a[$i]*$p[$i]+$b[$i])); } return $pp; } Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 16. A L TEX Scaling Geometria Computacional Programmed Primeiros GDs Initialization Escalonamento Convertion Curvas Now Draw Inventing the Wheel Convertions Show Off: Os Oids... 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],$color); Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 17. A L TEX Scaling Geometria Computacional Programmed Primeiros GDs Initialization Escalonamento Convertion Curvas Now Draw Inventing the Wheel Convertions Show Off: Os Oids... Draw Line I - Result Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 18. A L TEX Scaling Geometria Computacional Programmed Primeiros GDs Initialization Escalonamento Convertion Curvas Now Draw Inventing the Wheel Convertions Show Off: Os Oids... Serial Convertion Maligma: C´digo Repetido o function ScalePoints($ps) { $pps=array(); for ($n=0;$n<count($ps);$n++) { $pps[$n]=ScalePoint($ps[$n]); } return $pps; } Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 19. A L TEX Geometria Computacional Drawing a Circle Primeiros GDs Generate Escalonamento Line Segments Curvas Code Inventing the Wheel Results Show Off: Os Oids... Drawing a Circle Equation: (x − xc )2 + (y − yc )2 = r 2 Parametrization: x(t) xc + r cos t = y (t) yc + r sin t t ∈ [−π, π] Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 20. A L TEX Geometria Computacional Drawing a Circle Primeiros GDs Generate Escalonamento Line Segments Curvas Code Inventing the Wheel Results Show Off: Os Oids... Generating a Circle function Circle($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 $ps; } Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 21. A L TEX Geometria Computacional Drawing a Circle Primeiros GDs Generate Escalonamento Line Segments Curvas Code Inventing the Wheel Results Show Off: Os Oids... Curve as Line Segments function DrawCurve($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 ); } Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 22. A L TEX Geometria Computacional Drawing a Circle Primeiros GDs Generate Escalonamento Line Segments Curvas Code Inventing the Wheel Results Show Off: Os Oids... Drawing Circle $n=6; $center=array(2.0,1.0); $ps=Circle($n,$center,2); DrawCurve($im,$ps,$color); imagestring($im,3,5,5,"n=$n",$color); Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 23. A L TEX Geometria Computacional Drawing a Circle Primeiros GDs Generate Escalonamento Line Segments Curvas Code Inventing the Wheel Results Show Off: Os Oids... Draw Circle - Result n = 6 Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 24. A L TEX Geometria Computacional Drawing a Circle Primeiros GDs Generate Escalonamento Line Segments Curvas Code Inventing the Wheel Results Show Off: Os Oids... Draw Circle - Result n = 20 Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 25. A L TEX Geometria Computacional Drawing a Circle Primeiros GDs Generate Escalonamento Line Segments Curvas Code Inventing the Wheel Results Show Off: Os Oids... Draw Circle - Result n = 500 Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 26. A L TEX Geometria Computacional Primeiros GDs Drawing a Wheel Escalonamento Animate... Curvas Solution Inventing the Wheel Show Off: Os Oids... Drawing a Wheel function DrawWheel($im,$n,$dn,$pc,$r1,$r2,$color,$ang=0.0) { $p1s=Circle($n,$pc,$r1,$ang); DrawCurve($im,$p1s,$color); $p2s=Circle($n,$pc,$r2,$ang); DrawCurve($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); } } Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 27. A L TEX Geometria Computacional Primeiros GDs Drawing a Wheel Escalonamento Animate... Curvas Solution Inventing the Wheel Show Off: Os Oids... Drawing a Wheel $n=200; $dn=5; $center=array(2.0,1.0); DrawWheel($im,$n,$dn,$center,$color,3.0,0.2); Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 28. A L TEX Geometria Computacional Primeiros GDs Drawing a Wheel Escalonamento Animate... Curvas Solution Inventing the Wheel Show Off: Os Oids... Wheel - Result Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 29. A L TEX Geometria Computacional Primeiros GDs Drawing a Wheel Escalonamento Animate... Curvas Solution Inventing the Wheel Show Off: Os Oids... Animate Wheel $nimages=79; $dt=2.0*3.1415927/$nimages; system("/bin/rm Wheel*.gif"); for ($t=0.0,$i=0;$i<$nimages;$i++) { $image = ImageCreate... $center=array(3.0,3.0); DrawWheel($image,200,5,$center,0.2,2.0,$color,$t); imagegif($image,"Wheel.$i.gif"); $t+=$dt; } //Merda! PHP nao tem suporte para GIF animado... Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 30. A L TEX Geometria Computacional Primeiros GDs Drawing a Wheel Escalonamento Animate... Curvas Solution Inventing the Wheel Show Off: Os Oids... 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"; uruacu/Wheel.gif Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 31. A L TEX Geometria Computacional Primeiros GDs Escalonamento Curvas Inventing the Wheel Show Off: Os Oids... Cycloid Icons/Cycloid.gif Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 32. A L TEX Geometria Computacional Primeiros GDs Escalonamento Curvas Inventing the Wheel Show Off: Os Oids... Trochoid Icons/Trochoid_1.gif Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 33. A L TEX Geometria Computacional Primeiros GDs Escalonamento Curvas Inventing the Wheel Show Off: Os Oids... Trochoid Icons/Trochoid_2.gif Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 34. A L TEX Geometria Computacional Primeiros GDs Escalonamento Curvas Inventing the Wheel Show Off: Os Oids... Epicycloid Icons/Epicycloid_1.gif Icons/Epicycloid_2.gif Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 35. A L TEX Geometria Computacional Primeiros GDs Escalonamento Curvas Inventing the Wheel Show Off: Os Oids... Hypocycloid Icons/Hypocycloid_1.gif Icons/Hypocycloid_2.gif Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional
- 36. A L TEX Geometria Computacional Primeiros GDs Escalonamento Curvas Inventing the Wheel Show Off: Os Oids... Oloid Icons/Oloid.gif Please Always Enjoy... Animations - Made in C++ Ole Peter Smith, IME, UFG, ole@mat.ufg.br Geometria Computacional