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 ENVELOPE OF A FAMILY OF PLANE CURVES


Notion studied by Leibnitz in 1694 and Taylor in 1715.

 
If (Gt) is defined by the Cartesian equation (1): f(x, y, t) = 0, the equation of the envelope is obtained by eliminating t in (1) and the equation (2): (equivalent to the visible outline along Oz of the surface f(x,y,z) = 0).
If (Gt) is defined parametrically by (M(u,t))u, the value of the parameter u as a function of t is obtained by solving .

The envelope of a family of curves with one parameter (Gt) is the family (G) of the characteristic points of the curves Gt, limit points when t' goes to t of the intersection points between (Gt) and (Gt'); the curve (G) is tangent, at any of its points, to a curve (Gt) and, "in general", any curve Gt is tangent at at least a point to (G); the restrictive cases are the following:
    - on an interval, the curves (Gt) pass by one or several fixed points, in which case this point belongs to the envelope.
    - the curves do not intersect (for example, concentric circles, or curves for which the intersection points are imaginary).

With the notations above in the parametric case, since the condition  is symmetrical with respect to u and t, the envelope of the curves (G'u) loci of the points (M(u,t))t is the same as that of the curves (Gt); the envelope is in fact the locus of the points where a curve of the first family is tangent to a curve of the second one. The two configurations are said dual of one another.
 
Example: the envelope of a circle (Gt) with constant radius the centre of which describes a parabola is a curve parallel to the parabola. 

We took M(t, u) = (t + cos u, t² + sin u)


The envelope of the circles (Gt) ...

... is also that of the parabolas (G'u)
See also a nice example of duality at involute of a circle.

The envelope can also be seen as the singular solution of the differential equation of which the curves (Gt) are solutions.

Special case: the envelope of a family of lines is a curve for which this family is the family of the tangents.

 Envelopes of lines can be physically produced thanks to tables of wires.

 Examples:
 - Every curve is the envelope of its tangents, or of its osculating circles.
 - The evolute of a curve is the envelope of the normals of the curve.
 - The caustics are the envelopes of rays reflected by an optical system.
 - The curves parallel to a curve are the envelopes of circles with constant radii centred on this curve.
 - The pedal of a curve (G) with respect to a point O is the envelope of the circles with diameter [OM] when M describes (G), and the orthotomic curve is the envelope of the circles with centre M passing by O.
 - The pedal of a curve (G) with respect to a circle (C) is the envelope of the polar lines with respect to (C) of the points on (G).
 - An anallagmatic curve is the envelope of the circles orthogonal to the inversion circle and centred on the initial curve.
 - The anticaustic of a curve is the envelope of a circle centred on the curve and the radius of which keeps a constant ratio with the distance between the centre of the circle and a fixed point.
 - The envelope of a segment line [AB] the ends of which move along the axes Ox and Oy is
        - an astroid when the length AB is a constant equal to a (see, more generally, the Cremona generation of epi- and hypocycloids)
        - the reunion of four arcs of parabola  when OA + OB is a constant equal to a (see at Lamé curve)
        - the reunion of two rectangular hyperbolas when the area of the triangle OAB is a constant equal to .

- In the movement of a plane over a fixed plane, it is often interesting to consider the envelope in the fixed plane of a line in the moving plane.

 - The shooting parabola is the envelope of all the trajectories of shots taken at a given point with a constant initial speed.

See also the envelopes of surfaces and the envelopes of 3D curves.
 
Set of the osculating circles of an Archimedean spiral.
There is no need to trace the envelope...

 
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© Robert FERRÉOL, Jacques MANDONNET 2017