SER431 Lecture 14

Advanced Graphics
Curves and Splines II
(201810)

Transcript

1. jgs
   SER 431
   Advanced Graphics
   Lecture 14: Curves and Splines II
   Javier Gonzalez-Sanchez
   javiergs@asu.edu
   PERALTA 230U
   Office Hours: By appointment
   

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2. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 2
   jgs
   Bezier Curve
   4 points and the Bezier blending functions
   P(t) = (1-t)3 P0
   + 3t(1-t)2 P1
   + 3t2(1-t) P2
   + (t)3 P3
   Where t is 0 <= t < = 1
   

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3. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 3
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   P(t) = (1-t) P0
   + (t) P1
   Bezier | Linear
   P1
   =(6,6)
   P0
   =(3,3)
   P(0.25) = 0.75(3,3) +
   0.25(6,6)
   P(0.25) =
   (3.75,3.75)
   

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4. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 4
   jgs
   PA
   (t) = (1-t) P0
   + (t) P1
   PB
   (t) = (1-t) P1
   + (t) P2
   PC
   (t) = (1-t) PA
   + (t) PB
   PC
   (t) = (1-t)2 P0
   + 2t(1-t) P1
   + (t)2 P3
   Bezier | Quadratic
   P1
   =(6,6)
   P0
   =(3,3) P2
   =(9,3)
   PA
   PB
   PC
   

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5. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 5
   jgs
   PA
   (t) = (1-t) P0
   + (t) P1
   PB
   (t) = (1-t) P1
   + (t) P2
   PC
   (t) = (1-t) PA
   + (t) PB
   PC
   (t) = (1-t)2 P0
   + 2t(1-t) P1
   + (t)2 P3
   Bezier | Quadratic
   

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6. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 6
   jgs
   PA
   (t) = (1-t) P0
   + (t) P1
   PB
   (t) = (1-t) P1
   + (t) P2
   PC
   (t) = (1-t) P2
   + (t) P3
   PD
   (t) = (1-t) PA
   + (t) PB
   PE
   (t) = (1-t) PB
   + (t) PC
   PF
   (t) = (1-t) PD
   + (t) PE
   PF
   (t) = (1-t)3 P0
   + 3t(1-t)2 P1
   + 3t2(1-t) P2
   + (t)3 P3
   Bezier | Cubic
   P1
   =(6,6)
   P0
   =(3,3)
   P2
   =(9,3)
   PA
   PB
   PD
   P3
   =(7.5,1.5)
   PC
   PE
   PF
   

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7. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 7
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   PA
   (t) = (1-t) P0
   + (t) P1
   PB
   (t) = (1-t) P1
   + (t) P2
   PC
   (t) = (1-t) P2
   + (t) P3
   PD
   (t) = (1-t) PA
   + (t) PB
   PE
   (t) = (1-t) PB
   + (t) PC
   PF
   (t) = (1-t) PD
   + (t) PE
   PF
   (t) = (1-t)3 P0
   + 3t(1-t)2 P1
   + 3t2(1-t) P2
   + (t)3 P3
   Bezier | Cubic
   

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8. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 8
   jgs
   PA
   (t) = (1-t) P0
   + (t) P1
   PB
   (t) = (1-t) P1
   + (t) P2
   PC
   (t) = (1-t) P2
   + (t) P3
   PD
   (t) = (1-t) PA
   + (t) PB
   PE
   (t) = (1-t) PB
   + (t) PC
   PF
   (t) = (1-t) PD
   + (t) PE
   = (1-t) ((1-t) PA
   + (t) PB
   ) + (t) ((1-t) PB
   + (t) PC
   )
   = (1-t) ((1-t) ((1-t) P0
   + (t) P1
   ) + (t) ((1-t) P1
   + (t) P2
   )) +
   (t) ((1-t) ((1-t) P1
   + (t) P2
   ) + (t) ((1-t) P2
   + (t) P3
   )
   = (1-t) (((1-t)(1-t)P0
   + (1-t)(t) P1
   ) + ((t)(1-t) P1
   + (t)(t) P2
   )) +
   (t) (((1-t)(1-t)P1
   + (1-t)(t) P2
   ) + ((t)(1-t) P2
   + (t)(t) P3
   ))
   = (1-t) ((1-t)2P0
   + (1-t)(t)P1
   + (1-t)(t)P1
   +(t)2P2
   ) +
   (t) ((1-t)2P1
   + (1-t)(t)P2
   + (1-t)(t)P2
   +(t)2P3
   )
   = (1-t)(1-t)2P0
   + (1-t)(1-t)(t)P1
   + (1-t)(1-t)(t)P1
   + (1-t)(t)2P2
   +
   (1-t)2(t) P1
   + (1-t)(t)(t) P2
   + (1-t)(t)(t) P2
   +(t)2(t) P3
   = (1-t)3P0
   + (1-t)2(t)P1
   + (1-t)2(t)P1
   + (1-t)(t)2P2
   +
   (1-t)2(t) P1
   + (1-t)(t)2 P2
   + (1-t)(t)2 P2
   +(t)3 P3
   PF
   (t) = (1-t)3 P0
   + 3t(1-t)2 P1
   + 3t2(1-t) P2
   + (t)3 P3
   Bezier | Cubic
   

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9. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 9
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   General Bezier Curves
   § De Casteljau's algorithm – a recursive method
   

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10. jgs
    Source Code
    

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11. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 11
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    Step 1
    // 6 control points
    float Points[6][3] = {
    { 10, 5, 0 },
    { 8, -10, 0 },
    { 6, -10, 0 },
    { 4, 5, 0 },
    { 0, -5, 0 },
    { -10, 5, 0 }
    };
    P0
    P1
    P2
    P3
    P4
    P5
    

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12. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 12
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    Step 2
    // blending functions
    float BezierLinear(float a, float b, float t) { return a * (1.0f - t) + b * t; }
    float BezierQuadratic(float A, float B, float C, float t){
    float AB = BezierLinear(A, B, t); float BC = BezierLinear(B, C, t);
    return BezierLinear(AB, BC, t);
    }
    float BezierCubic(float A, float B, float C, float D, float t) {
    float ABC = BezierQuadratic(A, B, C, t); float BCD = BezierQuadratic(B, C, D, t);
    return BezierLinear(ABC, BCD, t);
    }
    float BezierQuartic(float A, float B, float C, float D, float E, float t) {
    float ABCD = BezierCubic(A, B, C, D, t); float BCDE = BezierCubic(B, C, D, E, t);
    return BezierLinear(ABCD, BCDE, t);
    }
    float BezierQuintic(float A, float B, float C, float D, float E, float F, float t) {
    float ABCDE = BezierQuartic(A, B, C, D, E, t); float BCDEF = BezierQuartic(B, C, D, E, F, t);
    return BezierLinear(ABCDE, BCDEF, t);
    }
    float BezierSextic(float A, float B, float C, float D, float E, float F, float G, float t) {
    float ABCDEF = BezierQuintic(A, B, C, D, E, F, t); float BCDEFG = BezierQuintic(B, C, D, E, F, G, t);
    return BezierLinear(ABCDEF, BCDEFG, t);
    }
    

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13. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 13
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    Step 3
    glBegin(GL_LINE_STRIP);
    glColor3f(0, 1, 0);
    for (int i = 0; i != LOD; ++i) {
    float t = (float)i / (LOD - 1);
    float it = 1.0f - t;
    // calculate the x, y and z of the curve point
    float x = BezierQuintic(Points[0][0], Points[1][0], Points[2][0], Points[3][0], Points[4][0], Points[5][0], t);
    float y = BezierQuintic(Points[0][1], Points[1][1], Points[2][1], Points[3][1], Points[4][1], Points[5][1], t);
    float z = BezierQuintic(Points[0][2], Points[1][2], Points[2][2], Points[3][2], Points[4][2], Points[5][2], t);
    // specify the point
    glVertex3f(x, y, z);
    }
    glEnd();
    

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14. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 14
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    Test Yourselves
    § Is it Bezier Curve (black dots) ?
    § What is its degree?
    § Could you write the source code?
    

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15. 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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