Lecture 3 Differential geometry of surfaces 3.1 Definition of surfaces Implicit surfaces F(r,,a)=0 Example: 22+6+2=1 Ellipsoid, see Figure 3.1 Figure 3.1: Ellipsoid · Explicit surfaces If the implicit equation F(, y, a)=0 can be solved for one of the variables as a function of the other two, we obtain an explicit surface, as shown in Figure 3.2. Example: 2= 2(ax+ By2) Parametric surfaces x=ru, v),y=y(u, u), t=z(u, v) Here functions z(u, u), y(u, u), x(u, u) have continuous partial derivatives of the r th order, and the parameters u and v are restricted to some intervals(i.e.,u1≤u≤u2,n≤t≤v2) leading to parametric surface patches. This rectangular domain D of u, v is called parametric space and it is frequently the unit square, see Figure 3.3. If derivatives of the surface are continuous up to the rth order, the surface is said to be of class r, denotedLecture 3 Differential geometry of surfaces 3.1 Definition of surfaces • Implicit surfaces F(x, y, z) = 0 Example: x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 Ellipsoid, see Figure 3.1. x y z Figure 3.1: Ellipsoid. • Explicit surfaces If the implicit equation F(x, y, z) = 0 can be solved for one of the variables as a function of the other two, we obtain an explicit surface, as shown in Figure 3.2. Example: z = 1 2 (αx2 + βy 2 ) • Parametric surfaces x = x(u, v), y = y(u, v), z = z(u, v) Here functions x(u, v), y(u, v), z(u, v) have continuous partial derivatives of the r th order, and the parameters u and v are restricted to some intervals (i.e., u1 ≤ u ≤ u2, v1 ≤ v ≤ v2) leading to parametric surface patches. This rectangular domain D of u, v is called parametric space and it is frequently the unit square, see Figure 3.3. If derivatives of the surface are continuous up to the r th order, the surface is said to be of class r, denoted C r . 2