I generated this animation of Past and Future Light Cones of a stationary point object with vertical line as it's world line using Mathematica. As such the each light cone should sweep a (pencil-like cylendrical) shape as time passes, but I am showing the light cones at any given moment only. Also the cones extend to + ve and -ve infinite time as such.

The vertical axis is time axis with unit t seconds.

The z=0 is the

**NOW**plane showing two (not three) dimensional space. The normalized unit for space axes is ct, where c is the speed of light. Thus distance of 1 in x or y direction is the distance travelled by light in 1 second.
That gives us 45 degree sloped light cones.

The downward moving planes are future moments, becoming NOW and then the past moments.

The top cone is the Future cone.

The bottom cone is past cone.

The vertical line is the world line of a stationary point object.