envelope, in mathematics, a curve that is tangential to each one of a family of curves in a plane or, in three dimensions, a surface that is tangent to each one of a family of surfaces. For example, two parallel lines are the envelope of the family of circles of the same radius having centres on a straight line. An example of the envelope of a family of surfaces in space is the circular cone x2 − y2 = z2 as the envelope of the family of paraboloids x2 + y2 = 4a(z − a).