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ENVELOPES, CHARACTERISTICS, TANGENT SURFACE OF A SPACE CURVE, RULED SURFACES, DEVELOPABLE SURFACES

Envelope. The envelope of a one-parameter family of curves is a curve that is tangent to
(has a common tangent with) every curve of the family. Its equation is obtained by eliminating the
parameter between the equation of the curve and the partial derivative of this equation with respect
to this parameter. The envelope of the circles (x - a)^{2} + y^{2} - 1 = 0 is y =
1. The envelope of a
one-parameter family of surfaces is the surface that is tangent to (has a common tangent plane
with) each of the surfaces of the family along their characteristics; the locus of the characteristic
curves of the family.

James/James. Mathematics Dictionary.

Characteristic of a one-parameter family of surfaces. The limiting curve of intersection of two neighboring members of the family as they approach coincidence — i.e. as the two values of the parameter determining the two members of the family of surfaces approach a common value. The equations of a given characteristic curve are the equation of the family taken with the partial derivative of this equation with respect to the parameter, each equation being evaluated for a particular value of the parameter. The locus of the characteristic curves, as the parameter varies, is the envelope of the family of surfaces. E.g., if the family of surfaces consists of all spheres of a given, fixed radius with centers on a given line, the characteristic curves are circles having their centers on the line, and the envelope is the cylinder generated by these circles.

James/James. Mathematics Dictionary.

Let the equation of a one-parameter family of surfaces be

1) f(x_{1}, x_{2}, x_{3}, t) = 0

where t is the parameter and f(x_{1}, x_{2}, x_{3}, t) is a real, single-valued, analytic function. Consider an
arbitrary surface S of the family and assume that it is intersected by a neighboring surface S' of the
family in a curve C'. If S and S' correspond to the values t and t + Δt of the parameter, the curve C'
is represented by the simultaneous equations

1) f(x_{1}, x_{2}, x_{3}, t) = 0

f(x_{1}, x_{2}, x_{3}, t + Δt) = 0 .

It is also represented by the equations

since the surface f(x_{1}, x_{2}, x_{3}, t + Δt) - f(x_{1}, x_{2}, x_{3}, t) = 0 goes through the curve common to the
two surfaces f(x_{1}, x_{2}, x_{3}, t) = 0 and f(x_{1}, x_{2}, x_{3}, t + Δt) = 0 and meets neither one in any other
point.

When S' approaches S as a limit, that is, when Δt 0, the curve C' will, in general, approach a curve C. Since

is f_{t}(x_{1}, x_{2}, x_{3}, t), this curve C has the equations

3) f(x_{1}, x_{2}, x_{3}, t) = 0

f_{t}(x_{1}, x_{2}, x_{3}, t) = 0 .

Because curve C' always lies of S, the curve C lies on S. It is known as the characteristic curve on S.

When the parameter t varies, the characteristic curve C will, in general, vary with S and generate a surface E. This surface E is defined to be the envelope of the given family 1). It can, in fact, be proved that E is tangent to each surface S of the family at every point of the characteristic curve C on S.

Graustein. Differential Geometry.

Ruled surface. A surface that can be generated by a moving straight line. The cone, cylinder, hyperbolic paraboloid, and hyperboloid of one sheet are ruled surfaces.

James/James. Mathematics Dictionary.

Developable surface. The envelope of a one-parameter family of planes; a surface that can be developed, or rolled out, on a plane without stretching or shrinking; a surface for which the total curvature vanishes identically.

James/James. Mathematics Dictionary.

Tangent surface of a space curve. The tangent surface of a space curve C (which is not a straight line) is that surface S that is swept out by the tangent line to curve C at a point P as P moves along C. It is also the envelope of the family of osculating planes of the space curve.

Equation of the tangent surface of a space curve. Let a curve C be given by the parametric equations

x_{1} = x_{1}(s)

x_{2} = x_{2}(s)

x_{3} = x_{3}(s)

or, in vector notation,

where s is arc length as measured from some specified point on the curve. Let

be the equation of the unit tangent vector to C i.e.

Then the equation of the tangent surface of C is given by the equation

where r is a parameter along with s. For a fixed s, r varies, tracing out the tangent line at that point. For each value of s, r traces out a tangent line; thus with increasing s a surface S is swept out. The tangent line at each point P of C thus corresponds to a ruling in the surface S. The normal N to surface S at point (r, s) — which corresponds to point P on C — is parallel to the binormal B of curve C at P. Proof. Thus at point (r, s) surface S is tangent to the osculating plane of C at P. As a consequence, surface S is the envelope of the family of osculating planes of C.

Theorem. A developable surface is the tangent surface of a twisted curve, or a cone, or a cylinder.

Rectifying developable of a space curve. The envelope S of the rectifying planes of the space curve C. This developable surface S is called the rectifying developable of C because the process of developing S on a plane results in rolling C out along a straight line.

James/James. Mathematics Dictionary.

References.

1. Graustein. Differential Geometry.

2. James/James. Mathematics Dictionary.

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