Let B be a unit-speed parametrization of the unit circle in the xy plane. onstruct a ruled surface as follows: Move a line L along ß in such a way at L is always orthogonal to the radius of the circle and makes constant gle r14 with B' (Fig. 4.20). (a) Derive this parametrization of the resulting ruled surface M: x(u, v) = B(u) + v(B'lu) + U,). (b) Express x explicitly in terms of v and coordinate functions for B. (c) Deduce that M is given implicitly by the equation x² + y? – z? = 1. (d) Show that if the angle t/4 above is changed to -t/4, the same surface M results. Thus M is doubly ruled. (e) Sketch this surface M showing the two rulings through each of the points (1, 0, 0) and (2, 1, 2).
Let B be a unit-speed parametrization of the unit circle in the xy plane. onstruct a ruled surface as follows: Move a line L along ß in such a way at L is always orthogonal to the radius of the circle and makes constant gle r14 with B' (Fig. 4.20). (a) Derive this parametrization of the resulting ruled surface M: x(u, v) = B(u) + v(B'lu) + U,). (b) Express x explicitly in terms of v and coordinate functions for B. (c) Deduce that M is given implicitly by the equation x² + y? – z? = 1. (d) Show that if the angle t/4 above is changed to -t/4, the same surface M results. Thus M is doubly ruled. (e) Sketch this surface M showing the two rulings through each of the points (1, 0, 0) and (2, 1, 2).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Let B be a unit-speed parametrization of the unit circle in the xy plane.
onstruct a ruled surface as follows: Move a line L along ß in such a way
at L is always orthogonal to the radius of the circle and makes constant
gle r14 with B' (Fig. 4.20).
(a) Derive this parametrization of the resulting ruled surface M:
x(u, v) = B(u) + v(B'lu) + U,).
(b) Express x explicitly in terms of v and coordinate functions for B.
(c) Deduce that M is given implicitly by the equation
x² + y? – z? = 1.
(d) Show that if the angle t/4 above is changed to -t/4, the same surface
M results. Thus M is doubly ruled.
(e) Sketch this surface M showing the two rulings through each of the
points (1, 0, 0) and (2, 1, 2).
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