The Envelope of a Family of Curves in the Plane

The concept of the envelope of a family of curves plays a significant role in differential geometry, mathematical analysis, and applied mathematics. An envelope represents a curve that is tangent to each member of a given family of curves at some point. This paper presents a systematic study of envelopes of one-parameter families of curves in the plane. Theoretical foundations, analytical methods for determining envelopes, and geometric interpretations are discussed in detail.