Upgrade to Pro — share decks privately, control downloads, hide ads and more …

SER431 Lecture 15

SER431 Lecture 15

Advanced Graphics
Hermite and Chaikin Curves
(201810)

B546a9b97d993392e4b22b74b99b91fe?s=128

Javier Gonzalez

October 11, 2018
Tweet

Transcript

  1. jgs SER 431 Advanced Graphics Lecture 15: Hermite and Chaikin

    Curves Javier Gonzalez-Sanchez javiergs@asu.edu PERALTA 230U Office Hours: By appointment
  2. jgs Hermite Curves

  3. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 3 jgs

    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)
  4. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 4 jgs

    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)
  5. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 5 jgs

    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
  6. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 6 jgs

    Hermite Curves P(t)=(2t3-3t2+1)P0 + (2t3+3t2)P1 + (t3-2t2+t)T1 + (t3-t2)T2 Where t is 0 <= t < = 1
  7. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 7 jgs

    Hermite Curves Examples § All tangent vector magnitudes are equal, but the direction of the left tangent vector varies.
  8. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 8 jgs

    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.
  9. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 9 jgs

    Test Yourselves 0 0 0 10 10 0 0 10 0 10 0 0 A B
  10. jgs Source Code

  11. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 11 jgs

    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 };
  12. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 12 jgs

    Step 2 // Points glColor3f(1, 0, 0); glPointSize(3); glBegin(GL_POINTS); glVertex3fv(Geometry[0]); glVertex3fv(Geometry[1]); glEnd();
  13. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 13 jgs

    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();
  14. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 14 jgs

    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();
  15. jgs Chaikin Curves

  16. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 16 jgs

    Definition § Corner cutting algorithm § Generate a curve by successive refinement of a control polygon (a set of control points)
  17. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 17 jgs

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

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

    Examples
  20. jgs Source Code

  21. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 21 jgs

    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<Point> Points;
  22. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 22 jgs

    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));
  23. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 23 jgs

    Step 3 void increasePoints() { if (Points.size() >= 50) return; std::vector<Point> 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; }
  24. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 24 jgs

    Step 4 void decreasePoints() { if (Points.size() <= 5) return; // original frame std::vector<Point> 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
  25. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 25 jgs

    Step 5 // Control Graph void display() { // ... glColor3f(1, 1, 0); glBegin(GL_LINE_STRIP); std::vector<Point>::iterator it = Points.begin(); for ( ; it != Points.end(); ++it) { glVertex3f(it->x, it->y, it->z); } glEnd(); glutSwapBuffers(); }
  26. Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 26 jgs

    Homework § Review the source code on Github
  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.