a line cut itself

how to find the intersection point of a line cut itself

[Mathieu NOE]

see

find the point of intersection of one curve with itselfs - MATLAB Answers - MATLAB Central (mathworks.com)

Answer 1

[John D'Errico]

Please stop asking the same question. This is just a continuation of your last question.

syms T
Fx=3*T.^2-12*T+16;
Fy=2-9*T+6*T.^2-T.^3;
fplot(Fx,Fy,[0,5])

You are asking how to find the self-intersection point on this curve. Said another way, you want to find the point where the curve crosses itself. Mathematically, what does that mean? Mathematically, it asks for a pair of distinct values of T, such that Fx(T1) == Fx(T2) and Fy(T1) == Fy(T2).

syms T1 T2

Fxeq = subs(Fx,T,T1) - subs(Fx,T,T2)

Fxeq = 

Fyeq = subs(Fy,T,T1) - subs(Fy,T,T2)

Fyeq = 

We now have two equations in the two unknowns T1 and T2.

[T1sol,T2sol] = solve(Fxeq,Fyeq,[T1,T2])

T1sol = 

T2sol = 

As expected, we can ignore the case where T1=T2 =0. And there is complete symmetry here, so solutions 2 and 3 are the same solution, just swapped around.

xsol = expand(subs(Fx,T,T1sol(2)))

xsol = 

13

ysol = expand(subs(Fy,T,T1sol(2)))

ysol = 

0

fplot(Fx,Fy,[0,5])
hold on
plot(double(xsol),double(ysol),'rs')
title("[ " + double(xsol) + " , " + double(ysol)+ " ]")
xline(13,'g')
yline(0,'g')

The intersection lies at (x,y) = (13,0).