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ENVELOPE OF A FAMILY OF 3D CURVES

If (G) is given by (1): , the envelope exists if and only if the system of 4 equations with 3 unknowns resulting from (1) and (2): has a solution for (_{t}x, y, z) for all values of t. This solution gives the parametrization of the envelope.
If (G M(t,u)),
the value, in function of _{u}t, of the parameter u of the characteristic point is obtained by solving
(the condition of existence of the envelope being the indetermination of this system of 3 equations and two unknowns). |

The *envelope* of a family of curves with one parameter is the locus (G) of the *characteristic points* of the curves G_{t}, limit points when *t*' goes to *t* of the intersection points between (G_{t}) and (G_{t'}); these points only exist if the curve (G* _{t}*) is secant with the infinitely close curves (G

When the curves (G* _{t}*) are straight lines, the envelope exists if and only if the ruled surface generated by the (G

See also the envelopes of surfaces.

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© Robert FERRÉOL 2018