Curve and Surface Design

1. Curve and Surface Design CVGLab

2. Goals of the Chapter • To develop tools for representing and designing curves. • To determine key properties of curves, such as their “smoothness”. • To develop the mathematical properties of Bezier and B-spline curves. • To develop tools to design Bezier, B-spline, and NURBS surface patches.

3. Introduction • Parametric Curves as Trajectories Specifying the path of a camera in an animation.

4. Introduction • Smoothness of Motion -. t: the passage of time. -. P(t): moves along the curve as t increases. -. v(t): a vector (the speed and direction of P(t) as it traverses the curve.) The line passes through P(t0) “at” u = 0 and “moves” in the direction v(t0).

5. Introduction The velocity vectors at t = a- (“just before” t = a) and at t = a+ (“just after” t = a) are, respectively,

6. Introduction • Parametric continuity (Ck-continuity or “k-smoothness”) -. We say that a curve P(t) has kth-order parametriccontinuity everywhere in the t-interval [a, b] if all derivatives of the curve, up through the kth, exist and are continuous at all points inside [a, b]. To express this succinctly, we say that P( ) is k-smooth in [a, b] • Geometric Continuity (Gk-continuity) -. G0 continuity is the same as 0-smoothness; it simply means that p(t) is continuous with respect to t throughout the interval [a, b] of interest. -. G1 continuity in [a, b] means that P`(c-) = kP`(c+) for same constant k and for every c in the interval [a, b]. -. G2 continuity in [a, b] means that both the first and second derivatives of a curve have continuous directions: P`(c-) = kP`(c+) and P"(c-) = mP"(c+) for constants k and m and for each c in the interval [a, b].

7. Describing Curves by Means of Polynomials • Polynomial Curves of Degree 1 • Polynomial Curves of Degree 2 • Implicit Forms of Degree Two F(x, y) = Ax2 + 2Bxy + Cy2 + Dx + Ey + F. if AC – B2 > 0, the curve is an ellipse. if AC – B2 = 0, the curve is a parabola. if AC – B2 < 0, the curve is a hyperbola. • Rational Parametric Forms P0, P1, P2 – control points.

8. Describing Curves by Means of Polynomials • The curve is one of the conic sections, and the type depends on the value of w: if w < 1, the curve is an ellipse. if w = 1, the curve is a parabola. if w > 1, the curve is a hyperbola.