an envelope
Slide a ladder so its foot slips out along the floor while its top slides down the wall, and chalk every position the ladder passes through. The cloud of straight lines you draw does not fill the whole corner — instead it crowds against a graceful curved boundary that each ladder-line just touches. That boundary curve, kissed by every member of the family but belonging to none of them, is the envelope. An envelope is the curve that is tangent to every curve in a whole family, the silhouette the family draws.
Precisely, suppose you have a one-parameter family of curves, each given by an equation F(x, y, c) = 0 where c is a parameter naming which member of the family you mean (different c, different curve). The envelope is a curve that touches each member tangentially. You find it by solving two equations at once: F(x, y, c) = 0 (you are on the member) and the partial derivative of F with respect to c set to zero, that is dF/dc = 0 (neighboring members pile up here). Eliminating c between these two equations gives the equation of the envelope. Intuitively, the envelope is where infinitely close members of the family intersect — the locus of those limiting intersection points — which is why the family seems to 'pinch' against it.
Envelopes are everywhere once you look. The bright cusped curve of light you see in the bottom of a coffee cup (a caustic) is the envelope of reflected light rays. The evolute of a curve is the envelope of that curve's family of normal lines, tying this idea straight back to curvature and centers of curvature. The parabolic safety boundary beyond which a fountain or a cannon (at fixed speed, varying angle) cannot reach is the envelope of all the projectile trajectories. Two honest cautions. First, the envelope is generally NOT a member of the family — it is a separate curve assembled from the family's tangency, so do not look for it among the original curves. Second, the algebraic solution set of F = 0 together with dF/dc = 0 can include extra pieces beyond the true envelope (such as cusp loci or singular points of the members), so one must check that a candidate piece is genuinely tangent to the family rather than an artifact of the equations.
Consider the family of lines x cos c + y sin c = 1, one line for each angle c (each is a tangent to the unit circle at angle c). Solving F = x cos c + y sin c - 1 = 0 together with dF/dc = -x sin c + y cos c = 0 and eliminating c gives x^2 + y^2 = 1: the envelope of this whole family of tangent lines is exactly the unit circle they all touch.
Solve F = 0 and dF/dc = 0 together, eliminate c: the family's tangency curve emerges.
The envelope is usually NOT one of the curves in the family — it is a new curve tangent to all of them. Also, the equations F = 0 and dF/dc = 0 can pick up extra spurious pieces (cusp loci, singular points), so verify a candidate is truly tangent to the family.