1. Geometric Modeling 91.580.201 Surfaces Mortenson Chapter 6 and Angel Chapter 9

2. Surface Basics Surface: Locus of a point moving with 2 degrees of freedom. Some types of equations to describe curves: –Intrinsic No reliance on external frame of reference. Lack of robustness of surface characteristics under repeated transformations. Discussion deferred until we study more differential geometry. –Explicit Value of dependent variable in terms of independent variable(s) e.g. z = f ( x,y ) Lack of robustness of surface characteristics under repeated transformations. –Implicit e.g. f ( x,y,z ) = 0 –Parametric Express value of each spatial variable in terms of independent variables (the parameters) e.g. for parameters u and w in 3D: x = x (u,w) y = y (u,w) z = z (u,w) source: Mortenson

3. Explicit Form Value of dependent variable in terms of independent variables –e.g. z = f ( x,y ) Axis-dependent Can be hard to represent a transformed and bounded surface. Sample surface- fitting procedure: determine a ij coefficients from data points: source: Mortenson

4. Implicit Form General form: f ( x,y,z ) = 0 –f ( x,y,z ) is polynomial in x, y, z such that: Axis-dependent Examples: –Plane: Equation is linear in all its variables. –Quadric: Second-degree equation. Can represent using vectors, scalars and a type identifier. –Right circular cylinder »One vector gives a point on its axis »One vector defines axis direction »Scalar gives radius Type testing requires robust floating-point computations. source: Mortenson

5. Implicit Form: Quadric Surfaces (continued) Type testing: source: Mortenson Insert equations from p. 181

6. Implicit Form: Quadric Surfaces (continued) Classification: source: Mortenson Insert scanned Table 6.1 from p. 181

7. Implicit Form: Quadric Surfaces (continued) Quadric Surfaces of Revolution: –Rotate conic curve about its axis –Canonical position: Center or vertex at origin Axes of symmetry coincide with coordinate axes. source: Mortenson Insert scanned Table 6.2 from p. 183

8. Parametric Form Express value of each spatial variable in terms of independent variables (the parameters) –e.g. for parameters u, w in 3D: x = x (u,w) y = y (u,w) z = z (u,w) For a rectangular surface patch, typically Patches can be joined to form composite parametric surfaces. source: Mortenson

9. Parametric Form (continued) Sample patch: rectangular segment of x, y plane x = (c - a)u + a y = (d - b)w + b z = 0 Here: –Curves of constant w are horizontal lines. –Curves of constant u are vertical lines. source: Mortenson

10. Parametric Form (continued) Parametric sphere of radius r, centered on ( x 0, y 0, z 0 ): source: Mortenson

11. Parametric Form (continued) Parametric ellipsoid centered on ( x 0, y 0, z 0 ): source: Mortenson

12. Parametric Form (continued) Parametric surface of revolution: partial view source: Mortenson

13. 4 Typical Types of Parametric Curves Interpolating –Curve passes through all control points. Hermite –Defined by its 2 endpoints and tangent vectors at endpoints. –Interpolates all its control points. –Not invariant under affine transformations. –Special case of Bezier and B-Spline. Bezier –Interpolates first and last control points. –Curve is tangent to first and last segments of control polygon. –Easy to subdivide. –Curve segment lies within convex hull of control polygon. –Variation-diminishing. –Special case of B-spline. B-Spline –Not guaranteed to interpolate control points. –Invariant under affine transformations. –Curve segment lies within convex hull of control polygon. –Variation-diminishing. –Greater local control than Bezier. Control points influence curve shape. source: Mortenson, Angel

14. Interpolating Interpolates all control points. Geometric form: Rarely used due to lack of derivative continuity at curve segment join points. source: Angel cubic case with equally spaced parameter values

15. Hermite Geometric form (cubic case): Hermite curves can provide C 1 continuity at curve segment join points. source: Mortenson

16. Bezier Geometric form (cubic case): Bezier curves can provide C 1 continuity at curve segment join points. source: Mortenson Bernstein polynomials. n+1 = number of control points = degree + 1 Adding a control point elevates degree by 1. Convex combination, so Bezier curve points all lie within convex hull of control polygon. Rational form is invariant under perspective transformation: where h i are projective space coordinates (weights)

17. B-Spline Geometric form (non-uniform, non-rational case), where K controls degree ( K -1) of basis functions: Cubic B-splines can provide C 2 continuity at curve segment join points. Convex combination, so B-spline curve points all lie within convex hull of control polygon. Rational form (NURBS) is invariant under perspective transformation, where h i are projective space coordinates (weights). source: Mortenson t i are knot values that relate u to the control points. Uniform case: space knots at equal intervals of u. Repeated knots move curve closer to control points. N N N N N N N N N