Curves and Surfaces

Limitations of Polygonal Meshes planar facets

Some Non-Polygonal Modeling Tools Extrusion Surface of Revolution Spline Surfaces/Patches Quadrics and other implicit polynomials

Continuity definitions: C0 continuous curve/surface has no breaks/gaps/holes "watertight" C1 continuous curve/surface derivative is continuous "looks smooth, no facets" C2 continuous curve/surface 2nd derivative is continuous Actually important for shading

Splines... Classic problem: How to draw smooth curves? Spline curve: smooth curve that is defined by a sequence of points Requirements: … H&B 8-8:420-425

Splines 1 Spline curve: smooth curve that is defined by a sequence of points Approximating spline Interpolating spline H&B 8-8:420-425

Splines 2 Convex hull: Smallest polygon that encloses all points Approximating spline Interpolating spline H&B 8-8:420-425

Splines 3 Control graph: polyline through sequence of points Approximating spline Interpolating spline H&B 8-8:420-425

Splines 4 Splines in computer graphics: Piecewise cubic splines Segments H&B 8-8:420-425

Splines 5 Segments have to match ‘nicely’. Given two segments P(u) en Q(v). We consider the transition of P(1) to Q(0). Zero-order parametric continuity C0: P(1) = Q(0). Endpoint of P(u) coincides with startpoint Q(v). P(u) Q(v) H&B 8-8:420-425

Splines 6 Segments have to match ‘nicely’. Given two segments P(u) en Q(v). We consider the transition of P(1) to Q(0). First order parametric continuity C1: dP(1)/du = dQ(0)/dv. Direction of P(1) coincides with direction of Q(0). P(u) Q(v) H&B 8-8:420-425

Splines 7 First order parametric continuity gives a smooth curve. Sometimes good enough, sometimes not. line segment circle arc Suppose that you are bicycling over the curve. What to do with the steering rod at the transition? Turn around! Discontinuity in the curvature! H&B 8-8:420-425

Splines 8 Given two segments P(u) and Q(v). We consider the transition of P(1) to Q(0). Second order parametric continuity C2: d2P(1)/du2 = d2Q(0)/dv2. Curvatures in P(1) and Q(0) are equal. P(u) Q(v) H&B 8-8:420-425

Splines 9 So far: considered parametric continuity. Here the vectors are exactly equal. It suffices to require that the directions are the same: geometric continuity. Q(v) P(u) Q(v) P(u) H&B 8-8:420-425

Splines 10 Given two segments P(u) en Q(v). We consider the transition of P(1) to Q(0). First order geometric continuity: G1: dP(1)/du =  dQ(0)/dv with  >0. Direction of P(1) coincides with direction Q(0). P(u) Q(v) H&B 8-8:420-425

Representation cubic spline 1 H&B 8-8:420-425 U: Powers of u C: Coefficient matrix

Representation cubic spline 2 Control points or Control vectors H&B 8-8:420-425 Matrix Mspline :’translates’ geometric info to coefficients

Representation cubic spline 3 H&B 8-8:420-425

Representation cubic spline 4 H&B 8-8:420-425

Representatie cubic spline 5 Puzzle: Describe a line segment between the points P0 en P1 with those three variants. P1 u=1 P0 u u=0 H&B 8-8:420-425

Representation cubic spline 6 H&B 8-8:420-425

Representation cubic spline 7 H&B 8-8:420-425

Representation cubic spline 8 H&B 8-8:420-425

Spline surface 1 P33 P03 P30 P20 P10 H&B 8-8:420-425 P00

Spline surface 2 H&B 8-8:420-425

Spline surface 2 P33 P03 P30 P20 P10 H&B 8-8:420-425 P00

Spline surface 3 v u H&B 8-8:420-425

Spline surface 4 v du := 1/nu; // nu: #facets u-direction u dv := 1/nv; // nv: #facets v-direction for i := 0 to nu1 do u := i*du; for j := 0 to nv  1 do v := j*dv; DrawQuad(P(u,v), P(u+du, v), P(u+du, v+dv), P(u, v+dv)) u H&B 8-8:420-425

Spline surface 5 // Alternative: calculate points first for i := 0 to nu do for j := 0 to nv do Q[i, j] := P(i/nu, j/nv); for i := 0 to nu  1 do for j := 0 to nv  1 do DrawQuad(Q[i, j], Q[i+1, j], Q[i+1, j+1], Q[i, j+1]) v u H&B 8-8:420-425

Spline surface 6 // Alternative: calculate points first, // triangle version for i := 0 to nu do for j := 0 to nv do Q[i, j] := P(i/nu, j/nv); for i := 0 to nu  1 do for j := 0 to nv  1 do DrawTriangle(Q[i, j], Q[i+1, j], Q[i+1, j+1]); DrawTriangle(Q[i, j], Q[i+1, j+1], Q[i, j+1]); v u H&B 8-8:420-425

Spline surface 7 // Alternative: calculate points first, // triangle variant, triangle strip for i := 0 to nu do for j := 0 to nv do Q[i, j] := P(i/nu, j/nv); for i := 0 to nu  1 do glBegin(GL_TRIANGLE_STRIP); for j := 0 to nv  1 do glVertex(Q[i, j]); glVertex(Q[i, j+1]); glEnd; v u H&B 8-8:420-425

Bézier spline curves 1 H&B 8-10:432-441

Bézier spline curves 2 P1 P0 H&B 8-10:432-441

Bézier spline curves 3 P1 P2 P0 H&B 8-10:432-441

Bézier spline curves 4 P2 P3 P1 P0 H&B 8-10:432-441

Bézier spline curves 5 P2 P3 P1 P0 H&B 8-10:432-441

Bézier spline curves 6 P2 P3 P1 P0 H&B 8-10:432-441

Bézier spline curves 7 P2 P3 P1 P0 H&B 8-10:432-441

Bézier spline curves 8 P2 P3 Q1 P1 Q0 Q2 Q3 P0 H&B 8-10:432-441

Bézier spline curves 8 P2 P3 Q1 P1 Q0 Q2 Q3 P0 H&B 8-10:432-441

Bézier spline curves 9 P2 P3 Q1 P1 Q0 Q2 Q3 P0 H&B 8-10:432-441

Bézier spline curves 9 P2 P3 P1 Q1 Q0 Q2 Q3 P0 H&B 8-10:432-441

Bézier spline curves 10 H&B 8-10:432-441

Bézier surface 1 P33 P03 P30 P20 P10 P00 H&B 8-10:432-441

Bézier surface 2 P33 P30 Q33 P03 P00 Q30 Q03 H&B 8-10:432-441 Q00

Bézier surface 2 P33 P30 Q33 P03 P00 Q03 Q30 H&B 8-10:432-441 Q00

Bézier surface 3 P33 P30 P03 P00 Q30 H&B 8-10:432-441 Q00

Bézier surface 3 P33 P30 P03 P00 Q30 H&B 8-10:432-441 Q00

Bézier surface 3 P33 P30 P03 P00 Q30 H&B 8-10:432-441 Q00