We can write a general line as:
x = at + b,
y = ct + d,
z = et + f,
for some constants a, b, c, d, e, f and a parameter t which takes all real values.
If this lies in z = xy, then:
et + f = (at +b)(ct + d)
et + f = act² + (bc + ad)t + bd
for all t.
For the resulting graph to be a straight line, a or c must be zero (eliminating the t² component).
If a is 0, then z = by, so the line can be written as x = b, z = by. Similarly, if c = 0, then the line can be written as y = d, z = dx.
Conversely, it is easy to see that these two families of lines lie in the surface. |
