COMPUTER GRAPHICS CHAPTERS 22-23 CS 482 – Fall 2017 SPLINES CUBIC CURVES HERMITE CURVES BéZIER CURVES B-SPLINES BICUBIC SURFACES SUBDIVISION SURFACES

CUBIC CURVES LINEAR LIMITATIONS Straight line segments may be used to approximate curves, but... Either the approximation has too many sharp angles, or... …the number of line segments required is prohibitively large. CS 482 – Fall 2017 CHAPTERS 22-23: SPLINES PAGE 181

CUBIC CURVES PARAMETERIZED CUBICS Cubics are the lowest order polynomials capable of illustrating maxima, minima, concavity, and inflection points. Notice that each equation has four unknowns, so four pieces of information would define each cubic equation! By defining the cubic curve parametrically, restricting the parameter t to the [0,1] interval, we obtain a concise structure to contain a good approximation to a desired curve. CS 482 – Fall 2017 CHAPTERS 22-23: SPLINES PAGE 182

HERMITE CURVES ENDPOINTS & TANGENT VECTORS The Hermite form of the cubic polynomial curve is constrained by the endpoints (P0 and P1) and the tangent vectors at the endpoints (P0´ and P1´). P0´ P0 P1 P1´ Similar solutions for the y- and z-coordinates yield: CS 482 – Fall 2017 CHAPTERS 22-23: SPLINES PAGE 183

HERMITE CURVES EXAMPLES The big problem with the Hermite approach is the requirement that the tangent vectors at the endpoints must be specified in advance. CS 482 – Fall 2017 CHAPTERS 22-23: SPLINES PAGE 184

BéZIER CURVES ENDPOINTS & CONTROL POINTS P2 The Bézier form of the cubic polynomial curve indirectly specifies the tangent vectors at endpoints P1 and P4 by specifying two intermediate points (P2 and P3) that are not on the curve. P2 P4 P1 P3 Calculations similar to those derived for the Hermite form yield: CS 482 – Fall 2017 CHAPTERS 22-23: SPLINES PAGE 185

BéZIER CURVES FIRST-ORDER DISCONTINUITY The Bézier form has zero-order continuity, but it lacks first-order continuity (unless the triple of vertices around the “knot” happen to be collinear). Discontinuous First Derivative P7 Discontinuous First Derivative P5 P4 P8 P6 P2 P3 P9 P10 P1 CS 482 – Fall 2017 CHAPTERS 22-23: SPLINES PAGE 186

B-SPLINES CONTROL POINTS ONLY To ensure first-order (and even second-order) continuity at the “knots” adjoining consecutive cubic curve segments, a B-spline approach is taken, with the drawback that the curve passes through none of the control points. P2 P4 P1 P3 CS 482 – Fall 2017 CHAPTERS 22-23: SPLINES PAGE 187

SECOND-ORDER CONTINUITY B-SPLINES SECOND-ORDER CONTINUITY P7 P5 P4 P8 P6 P2 P3 P9 P11 P10 P1 The B-spline knots have first-order continuity (i.e., smooth tangents) and second-order continuity (i.e., smooth concavity), but require three times as many parameterizations. P0 CS 482 – Fall 2017 CHAPTERS 22-23: SPLINES PAGE 188

BICUBIC SURFACES APPROXIMATING SMOOTH SURFACES Line segments may be used to approximate surface boundaries, but... 100 Triangles 1000 Triangles 69,451 Triangles Determining the segment endpoint coordinates is a big job! It takes a lot of triangles to yield a decent image, even with shading! The resulting storage and processing costs could be prohibitive! CS 482 – Fall 2017 CHAPTERS 22-23: SPLINES PAGE 189

BICUBIC SURFACES PATCHES Patches made up of orthogonal cubic curves can be used to approximate surfaces. Hermite patches require partial derivatives at every vertex. Bézier patches require collinearity between adjacent patches. B-spline patches require nine times as many parameterizations. CS 482 – Fall 2017 CHAPTERS 22-23: SPLINES PAGE 190

Subdivided Bézier Curves SUBDIVISION SURFACES USING BEZIER PATCHES Patches can be refined by creating a larger grid of control points from the original grid of control points. Original Bézier Curve P3 L1 = P1 R4 = P4 P2 R2 L4=R1 L3 L2 = ½(P1+P2) R3 = ½(P3+P4) R3 L2 L3 = ½(L2+½(P2+P3)) R2 = ½(R3+½(P2+P3)) Subdivided Bézier Curves P4 R4 L4 = R1 = ½(L3+R2) L1 P1 CS 482 – Fall 2017 CHAPTERS 22-23: SPLINES PAGE 191

SUBDIVISION SURFACES APPLICATIONS Reverse engineering on sparse data (e.g., limited medical scans) Controlling surface quality according to needs and processing abilities Transmission and compression of 3D mesh data CS 482 – Fall 2017 CHAPTERS 22-23: SPLINES PAGE 192