CAP 4703 Computer Graphic Methods Prof. Roy Levow Chapter 10

Curves and Surfaces  Explicit representation  Curves – y = f(x) in two dimensions – y = f(x), z = g(x) in three dimensions x = f(t), y = g(t), z = h(t) parametric  Surfaces –z = f(x,y)

Curves and Surfaces.2  Implicit Representation –f(x, y) = 0 –g(x, y, z) = 0 –Surface is “algebraic” if function is polynomial in x, y, z  Most common are quadratic

Parametric Form  Curve p(u) = [x(u), y(u), z(u)] T –can easily compute derivatives wrt to u  Surfaces require two parameters – p(u,v) = [x(u,v), y(u,v), z(u,v)] T can easily compute pratial derivatives wrt to u,v –n = dp/du x dp/dv

Parametric Polynomials  Curves – p(u) is a polynomial in u  can be described by coefficient vector  degree n has n+1 coefficients  Surfaces –p(u,v) is polynomial in u, v –can be described by 3(n+1)(m+1) coefficients where n is degree in u and m is degree in v. If m=n, 3(n+1) 2

Surface Patch  Parametric surface where 0 <= u, v <= 1 0 <= u, v <= 1 –Can be viewed as collection of lines by fixing either u or v constant in range

Design Criteria  Local control of shape  Smoothness and continuity  Ability to evaluate derivatives  Stability  Ease of rendering

Smoothness  Generally curve and surface elements will be smooth  Problems can arise at boundary of curve or surface elements, join points  Usual definition of smoothness is in terms of change in the derivative  Small changes in input should produce small changes in souput

Control Points  Used to define a curve or surface –Pass through some –Interpolate, come “close” to others  Curve is often “over specified” –more points that can match for given degree of polynomial