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 , and thus , are "constant" in t — ie, where "adjacent" family members intersect, which is another feature of the envelope.
where variation of the parameter p gives the different curves of the family.
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:
Now differentiate F(x,y,t) with respect to t and set the result equal to zero, to get
These two equations jointly define the equation of the envelope. From (2) we have t = (−y + x + k)/2. Substituting this value of t into (1) and simplifying gives an equation for the envelope in terms of x and y only:
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).
Another example: is a tangent of a parametrised curve . If we take then and gives when . 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.)
depending on a real parameter a (see ). For example the tangent planes to a surface along a curve in the surface form such a family.
Two surfaces corresponding to different values a and a' intersect in a common curve defined by
In the limit as a' approaches a, this curve tends to a curve contained in the surface at a
This curve is called the characteristic of the family at a. As a varies the locus of these characteristic curves defines a surface called the envelope of the family of surfaces.
Asymptotic behavior for numerical solutions of a semilinear parabolic equation with a nonlinear boundary condition.(Report)
Jan 01, 2008; AMS subject classifications. 35B40, 35B50, 35K60, 65M06 1. Introduction. Consider the following initial-boundary value problem,...