jgs SER 431 Advanced Graphics Lecture 15: Hermite and Chaikin Curves Javier Gonzalez-Sanchez [email protected] PERALTA 230U Office Hours: By appointment

jgs Hermite Curves

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)

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)

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

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

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.

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.

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

jgs Source Code

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 };

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();

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();

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();

jgs Chaikin Curves

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)

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

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

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

jgs Source Code

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 Points;

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));

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 23 jgs 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; }

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 24 jgs 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

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 25 jgs 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(); }

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

jgs SER431 Advanced Graphics Javier Gonzalez-Sanchez [email protected] 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.