Using tangency, a family of lines can also define a curve.
Articles recommended for you.
Some geometric figures are a delight to behold: a host of skilfully arranged lines, intertwined with a graceful curve that seems barely to touch each one. What mathematical theory lies behind them? The theory of envelopes of lines.
Every regular curve is the envelope of all its tangent lines. For conics, these tangent lines are particularly easy to construct geometrically, making it possible to produce them by folding.
The envelope of a family of lines can be understood in two ways. The second lends itself better to computation, but both give rise to superb geometric questions.
Lines whose direction varies continuously may reveal the curve to which they are all tangent. Such curves, enveloped by straight lines, appear in optics as caustics. Their properties give rise to geometric construction methods.
Discussion
Sign in to post a comment and talk with other readers.
No comments yet. Be the first to respond.
