ONE FIXED LIGHT CONE OR TWO DIFFERENT LIGHT CONES in the following link. Google the link if does not open or check - Length Contraction and Space-Time animation - on YouTube

From Albert point of view

At 3:59 : vertical line represent the fixed point and x = 0 at different times in his frame of reference

At 4:10: horizontal line represent simultaneous time at different places, t = 0

From Galileo point of view

Although moving but his frame of reference would be static for him but not for Albert , therefore

At 4:51: vertical line represent the fixed point and x′=0 in his frame of reference at different times – horizontal line represent simultaneous time at different places, t = 0

At 4:3: now let the time bubble (event) starts at t1 where vertical (x=0) and oblique (x’=0) lines meet and pause the event at any time t2. Draw the following two light cones (LC) w.r.t Albert and Galileo such that their apexes are at t1. Apothems of their cones represent the light like curves.

Light cone of Albert: Albert sees that

1 - The axis of his cone is vertical (oblique for Galileo)

2 - The surface of his cone is horizontal or perpendicular to the axis of his cone (oblique for Galileo or tilting backward)

Therefore Albert thinks that

1-The axis of the cone of Galileo is oblique (tilting forward)

1-The surface of cone of Galileo is also oblique (tilting forward) or perpendicular to the axis of cone of Galileo

Light cone of Galileo: Galileo sees that

1-The axis of his cone is vertical (oblique for Albert)

2-Surface of his cone is horizontal or perpendicular to the axis of his cone (oblique for Albert or tilting forward)

Therefore Galileo thinks that

1-The axis of cone of Albert is oblique (tilting backward)

2-The surface of cone of Albert is also oblique (tilting backward) or perpendicular to the axis of cone of Albert

Question: There are two different cones. Each has its own surface (base) perpendicular to its axis. Therefore I don not understand why the surfaces of aforesaid both cones always coincides with each other for simultaneity when their axis are not? Similarly, how come Galileo and Albert see each other when the surfaces/ basis of their cones are tilted w.r.t one other (do not coincides each other)?

thanks