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ENVELOPE OF A FAMILY OF 3D CURVES
If (Gt) is given by (1): If (Gt) is parametrically defined by (M(t,u))u,
the value, in function of t, of the parameter u of the characteristic point is obtained by solving |
The envelope of a family of curves with one parameter is the locus (G) of the characteristic points of the curves Gt, limit points when t' goes to t of the intersection points between (Gt) and (Gt'); these points only exist if the curve (Gt) is secant with the infinitely close curves (Gt+dt); (G) is then tangent at each of its points to a curve (Gt) and, in general, any curve Gt is tangent at at least one point to (G).
When the curves (Gt) are straight lines, the envelope exists if and only if the ruled surface generated by the (Gt) is developable and the envelope is the cuspidal edge of this developable surface.
See also the envelopes of surfaces.
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© Robert FERRÉOL 2018