SER431 Lecture 15

Advanced Graphics
Hermite and Chaikin Curves
(201810)

Transcript

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   SER 431
   Advanced Graphics
   Lecture 15: Hermite and Chaikin Curves
   Javier Gonzalez-Sanchez
   javiergs@asu.edu
   PERALTA 230U
   Office Hours: By appointment
   

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   Hermite Curves
   

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3. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 3
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   Hermite Curves
   We want curves that fit together smoothly.
   To accomplish this, we would like to specify a curve by providing:
   § The 2 end points, and
   § The 2 tangents vectors (first derivatives at these endpoints)
   

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4. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 4
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   Hermite Curves Examples
   0 0 0
   10 10 0
   0 100 0
   0 100 0
   0 0 0
   10 10 0
   0 10 0
   10 0 0
   P1=(0, 0, 0)
   P2=(10,10, 0)
   T1=(0,10, 0) T2=(10,0,0)
   P1=(0, 0, 0)
   P2=(10,10, 0)
   T1=(0,100, 0)
   T2=(100,0,0)
   

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5. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 5
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   Hermite blending functions
   § A curve spline is specified as
   P(t) = p0
   *B0
   (t) + p1
   *B1
   (t) + p2
   *B2
   (t) + p3
   *B3
   (t)
   § Hermite Curve
   B0
   (t)= 2*t3 – 3*t2 + 1,
   B1
   (t)= -2*t3 + 3*t2,
   B2
   (t)= t3 - 2*t2 + t,
   B3
   (t)= t3 – t2
   

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6. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 6
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   Hermite Curves
   P(t)=(2t3-3t2+1)P0
   + (2t3+3t2)P1
   + (t3-2t2+t)T1
   + (t3-t2)T2
   Where t is 0 <= t < = 1
   

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7. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 7
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   Hermite Curves Examples
   § All tangent vector magnitudes are equal, but the direction of the left
   tangent vector varies.
   

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8. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 8
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   Hermite Curves Examples
   § The set of Hermite curves that have the same values for the endpoints P0
   and P1, tangent vectors R0 and R1 of the same direction, but with
   different magnitudes for R0. The magnitude of R1 remains fixed.
   

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9. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 9
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   Test Yourselves
   0 0 0
   10 10 0
   0 10 0
   10 0 0
   A
   B
   

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    Source Code
    

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11. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 11
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    Step 1
    float Geometry[4][3] = {
    { -10, -10, 0 }, // Point 1
    { 10, 10, 0 }, // Point 2
    { -10, 20, 0 }, // Tangent 1
    { 10, 0, 0 } // Tangent 2
    };
    

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12. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 12
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    Step 2
    // Points
    glColor3f(1, 0, 0);
    glPointSize(3);
    glBegin(GL_POINTS);
    glVertex3fv(Geometry[0]);
    glVertex3fv(Geometry[1]);
    glEnd();
    

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13. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 13
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    Step 3
    glColor3f(0, 1, 0);
    glBegin(GL_LINE_STRIP);
    for (int i = 0; i != N; ++i) {
    float t = (float)i / (N - 1);
    // blending functions
    float b0 = 2 * t*t*t - 3 * t*t + 1;
    float b1 = -2 * t*t*t + 3 * t*t;
    float b2 = t*t*t - 2 * t*t + t;
    float b3 = t*t*t - t * t;
    // x, y and z of the curve point
    float x = b0*Geom[0][0] + b1*Geom[1][0] + b2*Geom[2][0] + b3*Geom[3][0];
    float y = b0*Geom[0][1] + b1*Geom[1][1] + b2*Geom[2][1] + b3*Geom[3][1];
    float z = b0*Geom[0][2] + b1*Geom[1][2] + b2*Geom[2][2] + b3*Geom[3][2];
    // the point
    glVertex3f(x, y, z);
    }
    glEnd();
    

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14. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 14
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    Step 4 (extra)
    // Control Graph
    glColor3f(0, 0, 1);
    glPushMatrix();
    glTranslatef(Geometry[0][0], Geometry[0][1], Geometry[0][2]);
    glBegin(GL_LINES);
    glVertex3f(0, 0, 0);
    glVertex3fv(Geometry[2]);
    glEnd();
    glPopMatrix();
    glPushMatrix();
    glTranslatef(Geometry[1][0], Geometry[1][1], Geometry[1][2]);
    glBegin(GL_LINES);
    glVertex3f(0, 0, 0);
    glVertex3fv(Geometry[3]);
    glEnd();
    glPopMatrix();
    

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    Chaikin Curves
    

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16. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 16
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    Definition
    § Corner cutting algorithm
    § Generate a curve by successive refinement of a control polygon (a set of
    control points)
    

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17. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 17
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    Method
    § Given a control polygon {P0, P1, P2, …, Pn}
    § Refine it by generating a new sequence of control point as
    {Q0, R0, Q1, R1, …, Qn-1, Rn-1}
    Where each pair Qi, Ri are to be at a ratio of ¼ and ¾ between the end
    points of the line segment Pi Pi+1
    Qi=3/4Pi + 1/4Pi+1
    Ri=1/4Pi + 3/4Pi+1
    

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18. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 18
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    Examples
    

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19. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 19
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    Examples
    

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    Source Code
    

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21. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 21
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    Step 1
    class Point {
    public:
    float x, y, z;
    Point() { x = 0; y = 0; z = 0; }
    Point(const float a, const float b, const float c) { x = a; y = b; z = c; }
    Point(const Point& p) { x = p.x; y = p.y; z = p.z; }
    };
    std::vector Points;
    

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22. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 22
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    Step 2
    // Initial Points
    Points.push_back(Point( 0, 10, 0));
    Points.push_back(Point( -10, 10, 0));
    Points.push_back(Point(- 10, -10, 0));
    Points.push_back(Point( 10, -10, 0));
    Points.push_back(Point( 10, 10, 0));
    

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23. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 23
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    Step 3
    void increasePoints() {
    if (Points.size() >= 50) return;
    std::vector NewPoints;
    NewPoints.push_back(Points[0]); // keep the first point
    for (unsigned int i = 0; i<(Points.size() - 1); ++i) {
    const Point& p0 = Points[i];
    const Point& p1 = Points[i + 1];
    Point Q, R;
    Q.x = 0.75f*p0.x + 0.25f*p1.x; Q.y = 0.75f*p0.y + 0.25f*p1.y;
    Q.z = 0.75f*p0.z + 0.25f*p1.z;
    R.x = 0.25f*p0.x + 0.75f*p1.x; R.y = 0.25f*p0.y + 0.75f*p1.y;
    R.z = 0.25f*p0.z + 0.75f*p1.z;
    NewPoints.push_back(Q);
    NewPoints.push_back(R);
    }
    NewPoints.push_back(Points[Points.size() - 1]);
    Points = NewPoints;
    }
    

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24. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 24
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    Step 4
    void decreasePoints() {
    if (Points.size() <= 5) return; // original frame
    std::vector NewPoints;
    NewPoints.push_back(Points[0]);
    for (unsigned int i = 1; i<(Points.size() - 1); i += 2) {
    const Point& pLast = NewPoints[NewPoints.size()-1]; // get last
    const Point& p0 = Points[i]; // get 2 original point
    Point Q;
    Q.x = p0.x - 0.75f * pLast.x; Q.y = p0.y - 0.75f * pLast.y;
    Q.z = p0.z - 0.75f * pLast.z;
    Q.x = Q.x * 4.0f; Q.y = Q.y * 4.0f;
    Q.z = Q.z * 4.0f;
    NewPoints.push_back(Q);
    }
    Points = NewPoints;
    } Qi=3/4Pi + 1/4Pi+1
    Ri=1/4Pi + 3/4Pi+1
    

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25. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 25
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    Step 5
    // Control Graph
    void display() {
    // ...
    glColor3f(1, 1, 0);
    glBegin(GL_LINE_STRIP);
    std::vector::iterator it = Points.begin();
    for ( ; it != Points.end(); ++it) {
    glVertex3f(it->x, it->y, it->z);
    }
    glEnd();
    glutSwapBuffers();
    }
    

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26. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 26
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    Homework
    § Review the source code on Github
    

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27. 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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