The picture below shows the twisted cubic union a line as the intersection of
two quadrics, one a cone with green exterior and reddish interior
and the other smooth with blue exterior.
The two families of lines on the smooth quadric are shown.
Note that lines slanting up to the right hit the twisted cubic 1 time,
while those slanting up to the left hit it 2 times, so this is a curve
of type (1,2) on the Segre surface (see Harris, page 28). The line contained
in the intersection of the two quadrics is tangent to the cubic.