Chapter 10: Curves and Surfaces

1. Chapter 10: Curves and Surfaces • Cubic Curves • Hermite Curves • Bézier Curves • B-Spline Curves • Bicubic Surfaces • Subdivision Surfaces Chapter 10: Curves and Surfaces

2. Cubic Curves Straight line segments may be used to approximate curves, but... …the number of line segments required is prohibitively large. Either the approximation has too many sharp angles, or... Chapter 10: Curves and Surfaces

3. Cubic Curves 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. Chapter 10: Curves and Surfaces

4. Cubic Curves Hermite Curves 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: Chapter 10: Curves and Surfaces

5. Cubic Curves Hermite Curve Examples P0´ P1´ P0´ P1´ P0 P1 P1´ P0 P1 P0´ P1 P0 P1´ P0´ P0´ P0 P1 P1´ P0 P1 The big problem with the Hermite approach is the requirement that the tangent vectors at the endpoints must be specified in advance. Chapter 10: Curves and Surfaces

6. Cubic Curves Bézier Curves 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: Chapter 10: Curves and Surfaces

7. Cubic Curves Bézier 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 P5 Discontinuous First Derivative P4 P8 P6 P2 P3 P9 P10 P1 Chapter 10: Curves and Surfaces

8. Cubic Curves Cubic B-Splines 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 Chapter 10: Curves and Surfaces

9. Cubic Curves Cubic B-Spline Second-Order Continuity 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. P7 P5 P4 P8 P6 P2 P3 P9 P11 P10 P1 P0 Chapter 10: Curves and Surfaces

10. Bicubic Surfaces 1000 Triangles 69,451 Triangles 100 Triangles Line segments may be used to approximate surface boundaries, but... • 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! Chapter 10: Curves and Surfaces

11. Bicubic Surfaces Bicubic 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. Chapter 10: Curves and Surfaces

12. Subdivision Surfaces Subdivided Bézier Curves Subdivided Bézier Curves 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)) R4 P4 L4 = R1 = ½(L3+R2) P1 L1 Chapter 10: Curves and Surfaces

13. 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 Chapter 10: Curves and Surfaces