Envelope (mathematics)

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This article is about envelopes in mathematics. For other uses, see Envelope (disambiguation).

Construction of the envelope of a family of curves.

In mathematics, an envelope of a family of manifolds (especially a family of curves) is a manifold that is tangent to each member of the family at some point.

The simplest formal expression for an envelope of curves in the $(x, y)$ -plane is the pair of equations

$F(x,y,t)=0\qquad\qquad(1)\,$

$\frac{\partial F(x,y,t)}{\partial t}=0\qquad\qquad(2)\,$

where the family is implicitly defined by (1). Obviously the family has to be "nicely" — differentiably — indexed by t.

The logic of this form may not be obvious, but in the vulgar: solutions of (2) are places where $F (x, y, t)$ , and thus $(x, y)$ , are "constant" in t — ie, where "adjacent" family members intersect, which is another feature of the envelope.

For a family of plane curves given by parametric equations $(x(t, p), y(t, p))\,$ , the envelope can be found using the equation

${\partial x\over\partial t}{\partial y\over\partial p} = {\partial y\over\partial t}{\partial x\over\partial p}$

where variation of the parameter p gives the different curves of the family.

1 Examples
- 1.1 Example 1
- 1.2 Example 2
2 See also
3 External links

[edit] Examples

[edit] Example 1

In string art it is common to cross-connect two lines of equally spaced pins. What curve is formed?

For simplicity, set the pins on the x- and y-axes; a non-orthogonal layout is a rotation and scaling away. A general straight-line thread connects the two points (0, k−t) and (t, 0), where k is an arbitrary scaling constant, and the family of lines is generated by varying the parameter t. From simple geometry, the equation of this straight line is y = −(k − t)x/t + k − t. Rearranging and casting in the form F(x,y,t) = 0 gives:

$F(x,y,t)=t^2 + t(y-x-k) + kx = 0\,$

(1)

Now differentiate F(x,y,t) with respect to t and set the result equal to zero, to get

$\frac{\partial F(x,y,t)}{\partial t}=2t+ y-x-k = 0\,$

(2)

These two equations jointly define the equation of the envelope. From (2) we have t = (−y + x + k)/2. Subsituting this value of t into (1) and simplifying gives an equation for the envelope in terms of x and y only:

$x^2 - 2xy + y^2 -2kx - 2ky + k^2 = 0\,$

This is the familiar implicit conic section form, in this case a parabola. Parabolae remain parabolae under rotation and scaling; thus the string art forms a parabolic arc ("arc" since only a portion of the full parabola is produced). In this case an anticlockwise rotation through 45° gives the orthogonal parabolic equation y = x²/(k√2) + k/(2√2). Note that the final step of eliminating t may not always be possible to do analytically, depending on the form of F(x,y,t).

[edit] Example 2

Another example: $(x - u) v' = (y - v) u'$ is a tangent of a parametrised curve $(u (t), v (t))$ . If we take $F (x, y, t) = (x - u) v' - (y - v) u'$ then $F t (x, y, t) = x v'' - y u'' - u v'' + v u''$ and $F = F t = 0$ gives $(x, y) = (u, v)$ when $v''u'\ne u''v'$ . So a curve is the envelope of its own tangents except where its curvature is zero. (This could also be read as a validation of this analytical form.)