Differential Geometry.

Differential Geometry.

Exercise 2.1.24. Suppose M: x(u, v) = 5(a) + v6(u) is a ruled surface with lfl’l = 1 and I6]
1. Also suppose 5’ yé 0, so that M is non-cylindrical. Show that M may be reparametrized by
y(u, w) = y(u) + w6(u), where y’ – 5’ = 0. Note that y may not be unit speed. A curve such as
y is called a line of striction for M. Show that any point on M where x“ x xv = 0 must lie on
the line ofstriction. Hint: write y(u) = flu) + r(u)6(u), use y’ – 8’ = 0 and solve for r(u). Let
w = v – r(u).

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