Envelope (mathematics)

In geometry, an envelope of a planar family of curves is a curve that is tangent to each member of the family at some point, and these points of tangency together form the whole envelope.

This idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions.

is the partial derivative of F with respect to t.[1] If t and u, t≠u are two values of the parameter then the intersection of the curves Ct and Cu is given by or, equivalently, Letting u → t gives the definition above.

An important special case is when F(t, x, y) is a polynomial in t. This includes, by clearing denominators, the case where F(t, x, y) is a rational function in t. In this case, the definition amounts to t being a double root of F(t, x, y), so the equation of the envelope can be found by setting the discriminant of F to 0 (because the definition demands F=0 at some t and first derivative =0 i.e. its value 0 and it is min/max at that t).

is the set of points defined at the beginning of this subsection's parent section.

Thus the discriminant is the original curve and its tangent line at γ(0): Next we calculate E1.

Assuming that t ≠ 0 then the intersection is given by Since t ≠ 0 it follows that x = t. The y value is calculated by knowing that this point must lie on a tangent line to the original curve γ: that F(t,(x,y)) = 0.

This point lies on a tangent line if and only if there exists a t such that This is a cubic in t and as such has at least one real solution.

If y = x3 and x = 0, i.e., x = y = 0, then this point has a single tangent line to γ passing through it (this corresponds to the cubic having one real root of multiplicity 3).

It follows that In string art it is common to cross-connect two lines of equally spaced pins.

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: Substituting this value of t into (1) and simplifying gives an equation for the envelope: Or, rearranging into a more elegant form that shows the symmetry between x and y: We can take a rotation of the axes where the b axis is the line y=x oriented northeast and the a axis is the line y=−x oriented southeast.

We obtain, after substitution into (4) and expansion and simplification, which is apparently the equation for a parabola with axis along a=0, or y=x.

To find the discriminant of F we need to compute its partial derivative with respect to t: where κ is the plane curve curvature of γ.

is the envelope of the corresponding family of line segments (that is, the hypotenuses of the triangles), and has Cartesian equation Notice that, in particular, the value

(meaning that all hypotenuses are unit length segments) gives the astroid.

Then the motion gives the following differential dynamical system: which satisfies four initial conditions: Here t denotes motion time, θ is elevation angle, g denotes gravitational acceleration, and v is the constant initial speed (not velocity).

A one-parameter family of surfaces in three-dimensional Euclidean space is given by a set of equations depending on a real parameter a.

The zero level set F(t0,(x,y)) = 0 gives the equation of the tangent line to the parabola at the point (t0,t02).

However, the envelope of this one-parameter family of lines, which is the parabola y = x2, is also a solution to this ODE.

Envelopes can be used to construct more complicated solutions of first order partial differential equations (PDEs) from simpler ones.

A new solution of the differential equation can be constructed by first solving (if possible) for a = φ(x) as a function of x.

The same idea underlies the solution of a first order equation as an integral of the Monge cone.

In Riemannian geometry, if a smooth family of geodesics through a point P in a Riemannian manifold has an envelope, then P has a conjugate point where any geodesic of the family intersects the envelope.

The light rays (shown in blue) are coming from a source at infinity, and so arrive parallel.

When they hit the circular arc the light rays are scattered in different directions according to the law of reflection.

The reflected light rays give a one-parameter family of lines in the plane.

It consists of precisely the points that can be reached from q in time t by travelling at the speed of light.

More generally, the point q0 could be replaced by any curve, surface or closed set in space.