jgs SER 431 Advanced Graphics Lecture 16: B-spline Curves Javier Gonzalez-Sanchez [email protected] PERALTA 230U Office Hours: By appointment

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 2 jgs Problem 1 § More control points higher the f(x) degree

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 3 jgs Problem 2 | Continuity Definition of C0, C1, and C2 continuity

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 4 jgs Problem 2 | Continuity § 2 Bezier with C0 continuity § 2 Bezier with C1 continuity § 2 Bezier with C2 continuity

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 5 jgs Problem 3 | Local Control § A problem with Bezier curves is their lack of local control. Simply increasing the number of control points adds little local control to the curve § Bezier combine all the points to create the curve.

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 6 jgs B-splines § B-spline curves are powerful generalization of Bezier curves. § The B in B-spline stands for ”basis”. § B-spline segments are joined at knots. The blending functions uses a knot vector in every computation (instead of all of the control points) § When these knots are spaced evenly, the B-spline is said to be uniform, and non-uniform otherwise.

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 7 jgs B-splines | Advantages § Provide the ability to add control points without increasing the degree of the curve

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 8 jgs B-splines | Advantages § Hermite and Bezier Curves can provide C1 continuity. B-spline Curves can provide C2 continuity § Example, 2 Bezier Curves with 4 control points (thus, Cubic) § C1 because S = W and C2 if we arrange thing to make T = U

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 9 jgs If two Bezier curves are joined at a point S, both their first and second derivatives match at S (i.e., they are C2 continuous) if and only if their control polygons fit an A-Frame.

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 10 jgs B-splines | Advantages § It combine only those points nearest to the current parameter.

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 11 jgs Algorithm 1. Divide each leg of the control polygon in thirds by making 2 “division” points. 2. At each Bi, except the first and last, draw the line segment between the 2 nearest “division” points, and call the midpoint Si. This create an A-Frame with Bi at the top. Include, S0 = B0 and Sn = Bn Y=1/3B0 + 2/3B1 X=2/3B0 + 1/3B1 W=2/3B1 + 1/3B2 Z=1/3B1 + 2/3B2

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 12 jgs Algorithm 3. Si is the average of the ends of its “cross-segment” so that, for instance: S1 = 1/2(Y) + 1/2(W) = 1/6(B0 ) + 2/3(B1 ) + 1/6(B2 ) Y=1/3B0 + 2/3B1 X=2/3B0 + 1/3B1 W=2/3B1 + 1/3B2 Z=1/3B1 + 2/3B2

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 13 jgs Algorithm 4. Finally sketch a cubic Bezier curve from each point Si to the next, using as Bezier control points the two “division” points. Remember, B0 (t)=(1-t)3, B1 (t)=3t(1-t)2, B2 (t)=3t2(1-t), B3 (t)=(t)3

jgs Source Code One Cubic B-spline curve

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 15 jgs bspline.cpp § In red the control points § In yellow and blue the “division” points § In gray the Si midpoints SA SB

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 16 jgs bspline.cpp float Points[4][3] = { { 10, 10, 0 }, { 5, 10, 0 }, { -5, 0, 0 }, { -10, 5, 0 } };

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 17 jgs bspline.cpp glColor3f(0, 1, 0); glBegin(GL_LINE_STRIP); for (int i = 0; i != N; ++i) { float t = (float)i / (N - 1); // the t value inverted float it = 1.0f - t; // calculate blending functions float b0 = it * it*it / 6.0f; float b1 = (3 * t*t*t - 6 * t*t + 4) / 6.0f; float b2 = (-3 * t*t*t + 3 * t*t + 3 * t + 1) / 6.0f; float b3 = t * t*t / 6.0f; // sum the control points mulitplied by their respective blending functions float x = b0*Points[0][0] + b1*Points[1][0] + b2*Points[2][0] + b3*Points[3][0]; float y = b0*Points[0][1] + b1*Points[1][1] + b2*Points[2][1] + b3*Points[3][1]; float z = b0*Points[0][2] + b1*Points[1][2] + b2*Points[2][2] + b3*Points[3][2]; // draw glVertex3f(x, y, z); } glEnd();

jgs Source Code A collection of Cubic B-spline curves

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 19 jgs bspline_connected.cpp § In red the control points § In yellow and blue the “division” points § In gray the Si midpoints SA SB

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 20 jgs bspline.cpp float Points[7][3] = { { 10, 10, 0 }, { 5, 10, 0 }, { 0, 0, 0 }, { -5, -5, 0 }, {-10, 0, 0 }, { -5, 10, 0 }, { 0, 5, 0 } }; #define NUM_POINTS 7 #define NUM_SEGMENTS (NUM_POINTS+1) // curve segments to be draw // 0 0 0 1 // 0 0 1 2 // 0 1 2 3 // 1 2 3 4 // 2 3 4 5 // 3 4 5 6 // 4 5 6 6 // 5 6 6 6 0 1 2 3 4 5 6

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 21 jgs bspline.cpp glBegin(GL_LINE_STRIP); float color=0; for (int segment = -3, j = 0; j != NUM_SEGMENTS; ++j, ++segment) { // for each section of curve, draw N divisions color = !color; glColor3f(color, 1, color); for (int i = 0; i != N; ++i) { float t = (float)i / N; float it = 1.0f - t; // calculate blending functions for cubic bspline. float b0 = it * it*it / 6.0f; float b1 = (3 * t*t*t - 6 * t*t + 4) / 6.0f; float b2 = (-3 * t*t*t + 3 * t*t + 3 * t + 1) / 6.0f; float b3 = t * t*t / 6.0f; // calculate the x,y and z of the curve point float x = b0 * GetPoint(segment + 0)[0] + b1 * GetPoint(segment + 1)[0] + b2 * GetPoint(segment + 2)[0] + b3 * GetPoint(segment + 3)[0]; float y = b0 * GetPoint(segment + 0)[1] + b1 * GetPoint(segment + 1)[1] + b2 * GetPoint(segment + 2)[1] + b3 * GetPoint(segment + 3)[1]; float z = b0 * GetPoint(segment + 0)[2] + b1 * GetPoint(segment + 1)[2] + b2 * GetPoint(segment + 2)[2] + b3 * GetPoint(segment + 3)[2]; // specify the point. glVertex3f(x, y, z); }

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 22 jgs bspline.cpp } // specify the last point on the curve glVertex3fv(Points[NUM_POINTS - 1]); glEnd();

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 23 jgs Test Yourselves § Modify the code in GitHub to draw the following B-splines 23

Javier Gonzalez-Sanchez | SER431 | Fall 2018 | 24 jgs Test Yourselves § Draw the following curve as Cubic B-spline and as Bezier (source code available on GitHub) § Change the position of P 4 as shown § In the picture and compare results 24

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.