Representing Curves and Surfaces 1 Introduction We need

Representing Curves and Surfaces 1

Introduction • We need smooth curves and surfaces in many applications: – model real world objects – computer-aided design (CAD) – high quality fonts – data plots – artists sketches 2

Introduction • Most common representation for surfaces: – polygon mesh – parametric surfaces – quadric surfaces • Solid modeling – don’t miss the next episode. . . 3

Introduction • Polygon mesh: – set of connected planar surfaces bounded by polygons – good for boxes, cabinets, building exteriors – bad for curved surfaces – errors can be made arbitrarily small at the cost of space and execution time – enlarged images show geometric aliasing 4

Introduction • Parametric polynomial curves: – point on 3 D curve = (x(t), y(t), z(t)) – x(t), y(t), and z(t) are polynomials – usually cubic: cubic curves 5

Introduction • Parametric bivariate (two-variable) polynomial surface patches: – point on 3 D surface = (x(u, v), y(u, v), z(u, v)) – boundaries of the patches are parametric polynomial curves – many fewer parametric patches than polynomial patches are needed to approximate a curved surface to a given accuracy – more complex algorithms though 6

Parametric cubic curves • Polylines and polygons: – large amounts of data to achieve good accuracy – interactive manipulation of the data is tedious • Higher-order curves: – more compact (use less storage) – easier to manipulate interactively • Possible representations of curves: – explicit, implicit, and parametric 7

Parametric cubic curves • Explicit functions: – y = f(x), z = g(x) – impossible to get multiple values for a single x • break curves like circles and ellipses into segments – not invariant with rotation • rotation might require further segment breaking – problem with curves with vertical tangents • infinite slope is difficult to represent 8

Parametric cubic curves • Implicit equations: – f(x, y, z) = 0 – equation may have more solutions than we want • circle: x² + y² = 1, half circle: ? – problem to join curve segments together • difficult to determine if their tangent directions agree at their joint point 9

Parametric cubic curves • Parametric representation: – x = x(t), y = y(t), z = z(t) – overcomes problems with explicit and implicit forms – no geometric slopes (which may be infinite) – parametric tangent vectors instead (never infinite) – a curve is approximated by a piecewise polynomial curve 10

Parametric cubic curves • Why cubic? – lower-degree polynomials give too little flexibility in controlling the shape of the curve – higher-degree polynomials can introduce unwanted wiggles and require more computation – lowest degree that allows specification of endpoints and their derivatives – lowest degree that is not planar in 3 D 11

Parametric cubic curves • Kinds of continuity: – G 0: two curve segments join together – G 1: directions of tangents are equal at the joint – C 1: directions and magnitudes of tangents are equal at the joint – Cn: directions and magnitudes of n-th derivative are equal at the joint 12

Parametric cubic curves • Major types of curves: – Hermit • defined by two endpoints and two tangent vectors – Bezier • defined by two endpoints and two other points that control the endpoint tangent vectors – Splines • several kinds, each defined by four points • uniform B-splines, non-uniform B-splines, ß-splines 13

Parametric cubic curves • General form: 14

Parametric cubic curves • It is not necessary to choose a single representation, since it is possible to convert between them. • Interactive editors provide several choices, but internally they usually use NURBS, which is the most general. 15

Parametric bicubic surfaces • Generalization of parametric cubic curves. • For each value of s there is a family of curves in t. • Major kinds of surfaces: – Hermit, Bezier, B-spline 16

Parametric bicubic surfaces • Displaying bicubic surfaces: – brute-force iterative evaluation is very expensive (the surface is evaluated 20, 000 times if step in parameters is 0. 01) – forward-difference methods are better, but still expensive – fastest is adaptive subdivision, but it might create cracks 17

Quadric surfaces • Implicit form: • Particularly useful for molecular modeling. • Alternative to rational surfaces if only quadric surfaces are being represented. 18

Quadric surfaces • Reasons to use them: – easy to compute normal – easy to test point inclusion – easy to compute z given x and y – easy to compute intersections of one surface with another 19

Summary • Polygon meshes – well suited for representing flat-faced objects – seldom satisfactory for curved-faced objects – space inefficient – simpler algorithms – hardware support 20

Summary • Piecewise cubic curves and bicubic surfaces – permit multiple values for a single x or y – represent infinite slopes – easier to manipulate interactively – can either interpolate or approximate – space efficient – more complex algorithms – little hardware support 21