SER431 Lecture 22

Advanced Graphics
Fractals
(201811)

Transcript

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   SER 431
   Advanced Graphics
   Lecture 22: Fractals
   Javier Gonzalez-Sanchez
   javiergs@asu.edu
   PERALTA 230U
   Office Hours: By appointment
   

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2. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 1
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   Definition
   Fractal:
   “a fragmented geometric shape that can be split into parts, each of which is
   (at least approximately) a reduced-size copy of the whole.”
   Fundamental component of fractal geometry is recursion, i.e., fractals all have
   a recursive definition
   

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3. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 2
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   Definition
   Fractal:
   “a fragmented geometric shape that can be split into parts, each of which is
   (at least approximately) a reduced-size copy of the whole.”
   Fundamental component of fractal geometry is recursion, i.e., fractals all have
   a recursive definition
   https://goo.gl/images/XC8RWU
   

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4. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 3
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   https://goo.gl/images/aPyG6i
   

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5. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 4
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   Definition
   § Symmetrical fractal
   The parts are, in fact, exact replicas of the whole.
   § Stochastic fractal
   Built out of probabilities and randomness.
   

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   Case 1: Symmetrical
   Stack boxes
   

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7. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 6
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   Code
   // create a pile of boxes as fractal
   void stackBoxes(float x, float y, float z) {
   if (y >= 0) {
   glPushMatrix();
   glTranslatef(x, y*100, z*100/2);
   glCallList(display2);
   glPopMatrix();
   stackBoxes(x-50, y-1, z-1);
   stackBoxes(x-50, y-1, z+1);
   stackBoxes(x+50, y-1, z-1);
   stackBoxes(x+50, y-1, z+1);
   }
   }
   

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8. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 7
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   Case 1
   

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   Case 2: Stochastic ● ArrayList
   Lightning bolt
   

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10. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 9
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    Case 1
    Level = 0 Level = 1 Level = 2
    Level = 3 Level = 4
    

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11. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 10
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    Code
    vector thunderbolt;
    void createBolt(Vec3f p1, Vec3f p2, int level) {
    thunderbolt.push_back(Line(p1.x,p1.y,p1.z, p2.x,p2.y, p2.z));
    for (int t = 0; t < level; t++) {
    int tam = thunderbolt.size();
    Vec3f middle; int i;
    for (i = 0; i < tam; i++) {
    p1.x = thunderbolt[0].x1; p1.y = thunderbolt[0].y1; p1.z = thunderbolt[0].z1;
    p2.x = thunderbolt[0].x2; p2.y = thunderbolt[0].y2; p2.z = thunderbolt[0].z2;
    thunderbolt.erase(thunderbolt.begin());
    middle = calculateMiddle(p1, p2, t);
    thunderbolt.push_back(Line(p1.x, p1.y, p1.z, middle.x,middle.y, middle.z));
    thunderbolt.push_back(Line(middle.x, middle.y, middle.z, p2.x,p2.y, p2.z));
    }
    // extension line
    Vec3f direction = middle - p1;
    Vec3f pin = middle + direction * 0.7;
    thunderbolt.push_back(Line(middle.x, middle.y, middle.z, pin.x, pin.y, pin.z));
    }
    }
    

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12. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 11
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    // calculate middle points with a displacement for the thunderbolt
    Vec3f calculateMiddle(Vec3f p1, Vec3f p2, int level) {
    Vec3f O = (p2+p1)/2;
    float m1 = (p2.y - p1.y) / (p2.x - p1.x);
    int signo = (random_number()>0.5) ? 1 : -1;
    // higher level, small displacement
    float x = O.x + signo*((cos(PI/2 + atan(m1)))/ pow(2, level));
    float y = O.y + signo*((sin(PI/2 + atan(m1)))/ pow(2, level));
    return Vec3f(x, y, (signo*random_number() / pow(2, level)) );
    }
    

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13. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 12
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    void display(void) {
    // ...
    glPushMatrix();
    glTranslatef(-300,0,300);
    glScalef(100, 100, 100);
    for (Line line : thunderbolt) {
    glBegin(GL_QUADS);
    glVertex3f(line.x1, line.y1, line.z1);
    glVertex3f(line.x1+0.1, line.y1, line.z1 + 0.1);
    glVertex3f(line.x2, line.y2, line.z2);
    glVertex3f(line.x2 +0.1, line.y2, line.z2 + 0.1);
    glEnd();
    }
    glPopMatrix();
    // ...
    }
    

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    Case 3 Stochastic ● DisplayList
    Tree
    

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15. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 14
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    Case 1
    Level = 0 Level = 1 Level = 2
    Level = 3 Level = 10
    

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16. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 15
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    Code
    // Create display lists for a leaf, a set of leaves, and a stem
    void CreateTreeLists(void) {
    // steam
    glNewList(STEM, GL_COMPILE);
    glPushMatrix();
    glRotatef(-90, 1, 0, 0);
    gluCylinder(cylquad, 0.1, 0.08, 1, 10, 2);
    glPopMatrix();
    glEndList();
    // leaf
    glNewList(LEAF, GL_COMPILE);
    glBegin(GL_TRIANGLES);
    // vertex here
    glEnd();
    glEndList();
    

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17. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 16
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    // steam & leaf
    glNewList(STEMANDLEAVES, GL_COMPILE);
    glPushMatrix();
    glCallList(STEM);
    for (i = 0; i < 3; i++) {
    glTranslatef(0, 0.333, 0); glRotatef(90, 0, 1, 0);
    glPushMatrix();
    glRotatef(0, 0, 1, 0); glRotatef(50, 1, 0, 0); glCallList(LEAF);
    glPopMatrix();
    glPushMatrix();
    glRotatef(180, 0, 1, 0); glRotatef(60, 1, 0, 0); glCallList(LEAF);
    glPopMatrix();
    }
    glPopMatrix();
    glEndList();
    // full tree
    glNewList(FULLTREE, GL_COMPILE);
    glPushMatrix();
    glTranslatef(0, -1, 0);
    fractalTree(0);
    glPopMatrix();
    glEndList();
    }
    

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18. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 17
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    void fractalTree(int level) {
    long savedseed;
    if (level == MAX_LEVEL) {
    glPushMatrix();
    glRotatef(random_number() * 180, 0, 1, 0); glCallList(STEMANDLEAVES);
    glPopMatrix();
    } else {
    glCallList(STEM);
    glPushMatrix();
    glRotatef(random_number() * 180, 0, 1, 0); glTranslatef(0, 1, 0); glScalef(0.7, 0.7, 0.7);
    glPushMatrix();
    glRotatef(110 + random_number() * 40, 0, 1, 0);
    glRotatef(30 + random_number() * 20, 0, 0, 1);
    fractalTree(level + 1);
    glPopMatrix();glPushMatrix();
    glRotatef(-130 + random_number() * 40, 0, 1, 0);
    glRotatef(30 + random_number() * 20, 0, 0, 1);
    fractalTree(level + 1);
    glPopMatrix();glPushMatrix();
    glRotatef(-20 + random_number() * 40, 0, 1, 0);
    glRotatef(30 + random_number() * 20, 0, 0, 1);
    fractalTree(level + 1);
    glPopMatrix();
    glPopMatrix();
    }
    }
    

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    Test Yourselves
    

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20. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 19
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    Test Yourselves
    § Create a forest
    It could be possible to have 2 or 3 slightly different trees
    § Create a lightning bolt
    Is is possible to add glow?
    What about animation?
    How to create a Koch snowflake?
    

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21. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 20
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    Homework
    https://github.com/javiergs/SER431/blob/master/Lecture22/fractals.cpp
    § Review the source code
    

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22. jgs
    SER431 Advanced Graphics
    Javier Gonzalez-Sanchez
    javiergs@asu.edu
    Fall 2018
    Disclaimer. These slides can only be used as study material for the class SER431 at ASU. They cannot be
    distributed or used for another purpose.
    

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