INTERSECCIÓN RECTA-PLANO.

Recta AB

Punto A --> (x(A_1), -y(A_1), y(A_2))
Punto B --> (x(B_1), -y(B_1), y(B_2))

(x, y, z) = (x(A_1), -y(a_1), y(A_2)) + t ((x(B_1)-x(A_1)), (-y(B_1)+y(A_1)), (y(B_2)-y(A_2)))

x = x(A_1) + t(x(B_1)-x(A_1))
y = -y(A_1) + t(-y(B_1)+y(A_1))
z = y(A_2) + t(y(B_2)-y(A_2))

---------------

PLANO AVB

A = (x(A), -y(A), 0)
V = (x(V), 0, 0)
B = (x(B), 0, y(B))

VA = ((x(A)-x(V)),-y(A), 0)
VB = ((x(B)-x(V)), 0, y(B))

x = x(V)+λ(x(A)-x(V))+η(x(B)-x(V))
y = 0-λy(A)+η0
z = 0+λ0+η(y(B))

x = x(V)+λ(x(A)-x(V))+η(x(B)-x(V))
y = 0-λy(A)+0
z = 0+0+η(y(B))

! x-x(V)   x(A)-x(V)   x(B)-x(V) !
!   y        -y(A)         0     ! = 0
!   z          0          y(B)   !

(x-x(V))(-y(A)y(B)) - (x(A)-x(V))y·y(B) + (x(B)-x(V))z·y(A) = 0

Ecuación del plano
x·(-y(A)y(B)) + x(V)y(A)y(B) - y·y(B)(x(A)-x(V)) + z·y(A)(x(B)-x(V)) = 0

---------------

Sustituyendo por los valores x, y, z de la recta:

(x(A_1) + t(x(B_1)-x(A_1)))(-y(A)y(B)) + x(V)y(A)y(B) - (-y(A_1)+t(-y(B_1)+y(A_1)))y(B)(x(A)-x(V)) + (y(A_2)+t(y(B_2)-y(A_2))y(A)(x(B)-x(V)) = 0

-x(A_1)y(A)y(B) -t((x(B_1)-x(A_1))y(A)y(B)) + x(V)y(A)y(B) + y(A_1)y(B)(x(A)-x(V)) - t(y(A_1)-y(B_1))y(B)(x(A)-x(V)) + y(A_2)y(A)(x(B)-x(V)) + t(y(B_2)-y(A_2))y(A)(x(B)-x(V))) = 0

-x(A_1)y(A)y(B)+x(V)y(A)y(B)+y(A_1)y(B)(x(A)-x(V))+y(A_2)y(A)(x(B)-x(V)) = t ((x(B_1)-x(A_1))y(A)y(B)+(y(A_1)-y(B_1))y(B)(x(A)-x(V))-(y(B_2)-y(A_2))y(A)(x(B)-x(V)))

t = (-x(A_1)y(A)y(B)+x(V)y(A)y(B)+y(A_1)y(B)(x(A)-x(V))+y(A_2)y(A)(x(B)-x(V))) / ((x(B_1)-x(A_1))y(A)y(B)+(y(A_1)-y(B_1))y(B)(x(A)-x(V))-(y(B_2)-y(A_2))y(A)(x(B)-x(V)))

SOLUCIÓN ------> P=(P_1, P_2)

x(P_1) = x(A_1) + t(x(B_1)-x(A_1))
y(P_1) = y(A_1) + t(y(B_1)-y(A_1))
x(P_2) = x(P_1)
y(P_2) = y(A_2) + t(y(B_2)-y(A_2))

x(P_1) = x(A_1) + (x(B_1)-x(A_1))(-x(A_1)y(A)y(B)+x(V)y(A)y(B)+y(A_1)y(B)(x(A)-x(V))+y(A_2)y(A)(x(B)-x(V))) / ((x(B_1)-x(A_1))y(A)y(B)+(y(A_1)-y(B_1))y(B)(x(A)-x(V))-(y(B_2)-y(A_2))y(A)(x(B)-x(V)))
y(P_1) = y(A_1) + (y(B_1)-y(A_1))(-x(A_1)y(A)y(B)+x(V)y(A)y(B)+y(A_1)y(B)(x(A)-x(V))+y(A_2)y(A)(x(B)-x(V))) / ((x(B_1)-x(A_1))y(A)y(B)+(y(A_1)-y(B_1))y(B)(x(A)-x(V))-(y(B_2)-y(A_2))y(A)(x(B)-x(V)))

x(P_2) = x(P_1)
y(P_2) = y(A_2) + (y(B_2)-y(A_2))(-x(A_1)y(A)y(B)+x(V)y(A)y(B)+y(A_1)y(B)(x(A)-x(V))+y(A_2)y(A)(x(B)-x(V))) / ((x(B_1)-x(A_1))y(A)y(B)+(y(A_1)-y(B_1)y(B)(x(A)-x(V))-(y(B_2)-y(A_2))y(A)(x(B)-x(V)))

----------------

Textos utilizados

P_1 = (x(A_1) + (x(B_1)-x(A_1))(-x(A_1)y(A)y(B)+x(V)y(A)y(B)+y(A_1)y(B)(x(A)-x(V))+y(A_2)y(A)(x(B)-x(V))) / ((x(B_1)-x(A_1))y(A)y(B)+(y(A_1)-y(B_1))y(B)(x(A)-x(V))-(y(B_2)-y(A_2))y(A)(x(B)-x(V))),y(A_1) + (y(B_1)-y(A_1))(-x(A_1)y(A)y(B)+x(V)y(A)y(B)+y(A_1)y(B)(x(A)-x(V))+y(A_2)y(A)(x(B)-x(V))) / ((x(B_1)-x(A_1))y(A)y(B)+(y(A_1)-y(B_1))y(B)(x(A)-x(V))-(y(B_2)-y(A_2))y(A)(x(B)-x(V))))

P_2 = (x(A_1) + (x(B_1)-x(A_1))(-x(A_1)y(A)y(B)+x(V)y(A)y(B)+y(A_1)y(B)(x(A)-x(V))+y(A_2)y(A)(x(B)-x(V))) / ((x(B_1)-x(A_1))y(A)y(B)+(y(A_1)-y(B_1))y(B)(x(A)-x(V))-(y(B_2)-y(A_2))y(A)(x(B)-x(V))),y(A_2) + (y(B_2)-y(A_2))(-x(A_1)y(A)y(B)+x(V)y(A)y(B)+y(A_1)y(B)(x(A)-x(V))+y(A_2)y(A)(x(B)-x(V))) / ((x(B_1)-x(A_1))y(A)y(B)+(y(A_1)-y(B_1))y(B)(x(A)-x(V))-(y(B_2)-y(A_2))y(A)(x(B)-x(V))))