which can be rewritten in matrix form as follows b o Co-r Co-y bo 0000 0 Co-y a」Lt3 The maximum degree of t in the above vector is determined by the degree m of the r polynomial and the degree n of the y polynomial, and is given by m+n-1. In this case m+n-1=3 A necessary and sufficient condition for the above system to be solvable is b 0 f(a, y) Co-y bo The equation f(, y)=0 is the implicit equation of the curve. Consequently in an exact arithmetic context, we need to check if f(o, yo)=0, to verify if(o, yo)is on In general, if x=x(t"),y=y(t")→F(x",y")=0 where n is the total degree Inversion: If f(a, y)=0 then we could use the first 3 equations 10.4 o bo ba0」Lt3 v(x0,30) Where o and y are polynomials in to and yo, and To, yo satisfy f(xo,3o)=0 The method is efficient and (usually) accurate for n 3(but no real guaran- tees on accuracy and robustness exist if the method is implemented in Hoating point) Subdivision methods are preferable for higher n, and as we will see later when coupled with rounded interval arithmetic are robust, accurate and efficient Intersection of points(o, yo, zo) and 3D polynomial curves R=R(t) via implicit ivolves a process of projection on y plane and finding to by inversion and verification of z0=i(to) 11which can be rewritten in matrix form as follows: ⇒      c0 − x b0 a0 0 0 c0 − x b0 a0 c 0 0 − y b 0 0 a 0 0 0 0 c 0 0 − y b 0 0 a 0 0           1 t t 2 t 3      =      0 0 0 0      (10.4) The maximum degree of t in the above vector is determined by the degree m of the x polynomial and the degree n of the y polynomial, and is given by m + n − 1. In this case m + n − 1 = 3. A necessary and sufficient condition for the above system to be solvable is          c0 − x b0 a0 0 0 c0 − x b0 a0 c 0 0 − y b 0 0 a 0 0 0 0 c 0 0 − y b 0 0 a 0 0          = f(x, y) = 0 The equation f(x, y) = 0 is the implicit equation of the curve. Consequently in an exact arithmetic context, we need to check if f(x0, y0) = 0, to verify if (x0, y0) is on the initial curve. In general, if x = x(t n ), y = y(t n ) ⇒ F(x n , y n ) = 0 where n is the total degree. • Inversion: If f(x, y) = 0 then we could use the first 3 equations 10.4:    b0 a0 0 c0 − x0 b0 a0 b 0 0 a 0 0 0       t t 2 t 3    = −    c0 − x0 0 c 0 0 − y0    ⇒ t = φ(x0, y0) ψ(x0, y0) Where φ and ψ are polynomials in x0 and y0, and x0, y0 satisfy f(x0, y0) = 0 – The method is efficient and (usually) accurate for n ≤ 3 (but no real guaran￾tees on accuracy and robustness exist if the method is implemented in floating point). – Subdivision methods are preferable for higher n, and as we will see later when coupled with rounded interval arithmetic are robust, accurate and efficient. Intersection of points (x0, y0, z0) and 3D polynomial curves R = R(t) via implicit￾ization of such curves involves a process of projection on x, y plane and finding t0 by inversion and verification of z0 = z(t0). 11