10.5 Point/surface intersection 10.5.1 Point/Implicit(usually algebraic) surface intersection The condition for Ro ntf(r)=0F, where f(R)=0 is an implicit surface If(Ro)/<6,(Ro)l vf(ro) where 6, d are small constants 10.5.2 Point/Rational polynomial surface intersection 1. Implicitization is possible for all such surfaces but computationally expensive and possi bly inaccurate. For a tensor product rational polynomial surface with maximum degrees in u and v equal to m and n, of the form R=R(u,2), the implicit equation is f(x9,y9,29)=0 here q≤2m Therefore,form=m=3→q≤18,m=m=2→q≤8 The above method is useful for special surfaces such as cylindrical and conical ruled surfaces. surfaces of revolution, etc Example (a)Implicitization of a surface of revolution R(t) igure 10.7: Surface of revolution10.5 Point/surface intersection 10.5.1 Point/Implicit (usually algebraic) surface intersection The condition for R0 ∩ {f(R) = 0}, where f(R) = 0 is an implicit surface, is: |f(R0)| < , |f(R0)| | 5 f(R0)| < δ where , δ are small constants. 10.5.2 Point/Rational polynomial surface intersection 1. Implicitization is possible for all such surfaces but computationally expensive and possi￾bly inaccurate. For a tensor product rational polynomial surface with maximum degrees in u and v equal to m and n, of the form R = R(u m, v n ), the implicit equation is f(x q , y q , z q ) = 0 where q ≤ 2mn Therefore, for m = n = 3 −→ q ≤ 18, m = n = 2 −→ q ≤ 8 The above method is useful for special surfaces such as cylindrical and conical ruled surfaces, surfaces of revolution, etc. Examples: (a) Implicitization of a surface of revolution. y x z r r R(t) Figure 10.7: Surface of revolution. 13