4902.sagews

Construction of conics

PROBLEM: Determine the equation of a conic if two tangents and three its points are given.

x,y,a,b,c,d,e,f=var('x,y,a,b,c,d,e,f')

A=(0,0)
B=(1,0)
C=(4,3)
t1=x-3*y+1==0
t2=-2*x+y+4==0

plot([point(A,size=50),point(B,size=50),point(C,size=50)],figsize=5,aspect_ratio=1)+implicit_plot(t1,(x,-1,5),(y,-1,5),color='red')+implicit_plot(t2,(x,-1,5),(y,-1,5),color='red')

k=a*x^2+b*x*y+c*y^2+d*x+e*y+f==0

h1=k.substitute(x=A[0],y=A[1]); h2=k.substitute(x=B[0],y=B[1]); h3=k.substitute(x=C[0],y=C[1])
h1; h2; h3

f == 0 a + d + f == 0 16*a + 12*b + 9*c + 4*d + 3*e + f == 0

t1x=solve(t1,x, solution_dict=True); t2x=solve(t2,x, solution_dict=True)
t1x; t2x

[{x: 3*y - 1}] [{x: 1/2*y + 2}]

kt1=k.substitute(t1x[0]).lhs(); expand(kt1)

9*a*y^2 + 3*b*y^2 + c*y^2 - 6*a*y - b*y + 3*d*y + e*y + a - d + f

A1=expand(kt1).coeff(y,2); B1=expand(kt1).coeff(y,1); C1=expand(kt1).coeff(y,0)
Disc1=B1^2-4*A1*C1; h4=Disc1 
h4

-4*(a - d + f)*(9*a + 3*b + c) + (6*a + b - 3*d - e)^2

kt2=k.substitute(t2x[0]).lhs(); expand(kt2)

1/4*a*y^2 + 1/2*b*y^2 + c*y^2 + 2*a*y + 2*b*y + 1/2*d*y + e*y + 4*a + 2*d + f

A2=expand(kt2).coeff(y,2); B2=expand(kt2).coeff(y,1); C2=expand(kt2).coeff(y,0)
Disc2=B2^2-4*A2*C2; h5=Disc2 
h5

-(a + 2*b + 4*c)*(4*a + 2*d + f) + 1/4*(4*a + 4*b + d + 2*e)^2

Coeff=solve([h1,h2,h3,h4,h5],[a,b,c,d,f],solution_dict=True); Coeff

[{f: 0, b: -13/9*e, a: 2/3*e, d: -2/3*e, c: 19/27*e}, {f: 0, b: -21/17*e, a: 6/17*e, d: -6/17*e, c: 43/51*e}, {f: 0, b: -149/97*e, a: 54/97*e, d: -54/97*e, c: 283/291*e}, {f: 0, b: -7/11*e, a: 2/11*e, d: -2/11*e, c: 3/11*e}]

Q1=expand(k.substitute(Coeff[0])/e); Q2=expand(k.substitute(Coeff[1])/e); Q3=expand(k.substitute(Coeff[2])/e); Q4=expand(k.substitute(Coeff[3])/e)
Q1; Q2; Q3; Q4

2/3*x^2 - 13/9*x*y + 19/27*y^2 - 2/3*x + y == 0 6/17*x^2 - 21/17*x*y + 43/51*y^2 - 6/17*x + y == 0 54/97*x^2 - 149/97*x*y + 283/291*y^2 - 54/97*x + y == 0 2/11*x^2 - 7/11*x*y + 3/11*y^2 - 2/11*x + y == 0

plot([point(A,size=50),point(B,size=50),point(C,size=50)],figsize=10)+implicit_plot(t1,(x,-3,6),(y,-3,6),color='red')+implicit_plot(t2,(x,-3,6),(y,-3,6),color='red')+implicit_plot(Q1,(x,-3,6),(y,-3,6),color='black')+implicit_plot(Q2,(x,-3,6),(y,-3,6),color='green')+implicit_plot(Q3,(x,-3,6),(y,-3,6),color='violet')+implicit_plot(Q4,(x,-3,6),(y,-3,6),color='brown')