10.4 Point/curve intersection 10.4.1 Point/Implicit curve intersection Ro∩{z=0,f(x,y)=0} where f(a, y) is usually a polynomial(and f(a, y)=0 represents an algebraic curve). In an exact arithmetic context, we can substitute Ro in a, f(a, y)=0 and verify if the results are zero. Similarly, we could handle Ro∩{f(R)=9(R)=0} where f(r)=g(r=0 represents an implicit 3D space curve What does verify mean in"Hoating point"arithmetic? · Example A et zo=0 and o, yo satisfy f(xo,y0)<∈<1 where e is a small constant and If(a, y) s l in the domain of interest including(ao, yo) then a"distance"check can be performed by f(xo,3∥d 1 10.2 provided|f(xo,o川≠0. Equation10.1 is called the“ algebraic distance” and Equa tion 10.2 is called the non-algebraic distance". The true geometric distance is given d= min R-Rol; where R=(, y), f(r)=0 (10.3) The true geometric distance is difficult and expensive to compute(particularly for implicit f(r =0 and involves computing the global minimum of R-Rol. Equation 10.2 results from the first order approximation of Equation 10.3 as derived by Taylor expansion and is exact when f(R)is represents a plane10.4 Point/curve intersection 10.4.1 Point/Implicit curve intersection R0 ∩ {z = 0, f(x, y) = 0} where f(x, y) is usually a polynomial (and f(x, y) = 0 represents an algebraic curve). In an exact arithmetic context, we can substitute R0 in {z, f(x, y) = 0} and verify if the results are zero. Similarly, we could handle: R0 ∩ {f(R) = g(R) = 0} where f(R) = g(R) = 0 represents an implicit 3D space curve. What does verify mean in “floating point” arithmetic? • Example A: Let z0 = 0 and x0, y0 satisfy |f(x0, y0)| <   1 (10.1) where  is a small constant and |f(x, y)| ≤ 1 in the domain of interest including (x0, y0), then a “distance” check can be performed by: |f(x0, y0)| | 5 f(x0, y0)| < δ  1 (10.2) provided | 5 f(x0, y0)| 6= 0. Equation 10.1 is called the “algebraic distance” and Equa￾tion 10.2 is called the “non-algebraic distance”. The true geometric distance is given by: d = min|R − R0|; where R = (x, y), f(R) = 0 (10.3) The true geometric distance is difficult and expensive to compute (particularly for implicit f(R) = 0 and involves computing the global minimum of |R−R0|. Equation 10.2 results from the first order approximation of Equation 10.3 as derived by Taylor expansion and is exact when f(R) is represents a plane. 8