DW 2, 7 Februari 2011

Oefenopgave 1

f1=Resultant[s^2-t^2-x,2 s t -y,s]

f2=Resultant[s^2+t^2 - z,2s t -y,s]

f3=Resultant[f1,f2,t]//Expand

ParametricPlot3D[{s^2-t^2,2s t,s^2+t^2},{s,-100,100},{t,-100,100}]

PolynomialRemainder[f3,x^2+y^2-z^2,x]

Factor[f3]

Oefenopgave 2

f[x_,y_]:=y^2-(x+1)(x-1)(x-2)

<<Graphics`ImplicitPlot`;

K=ImplicitPlot[f[x,y]==0,{x,-5,5},{y,-5,5}]

x0=0; y0=Sqrt[2];

w0=w/.Solve[Coefficient[f[x0+t,y0+t w]//Expand,t]==0,w][[1]]

t0=t/.Solve[PolynomialQuotient[f[x0+t,y0+t w0],t^2,t]==0,t][[1]]

x0=x0+t0//Simplify;y0=y0+t0 w0//Simplify;

{x0,y0}

f[x0,y0]//Simplify

l={{x0,y0}};

For[i=0,i≤4,i++,
w0=w/.Solve[Coefficient[f[x0+t,y0+t w]//Expand,t]==0,w][[1]];
t0=t/.Solve[PolynomialQuotient[f[x0+t,y0+t w0],t^2,t]==0,t][[1]];
x0=x0+t0;y0=y0+t0 w0;
AppendTo[l,{x0,y0}]]

l=l//N

lines=Line[l]

Show[K,Graphics[lines]]