Envelopes of families of lines

Scientific thinking makes it possible to distinguish between opinion, belief and science. An indispensable tool for every citizen, at the initiative of Henri Broch in the 1980s, it became the subject of scientific methodology teaching for students. He named it zeteticism, thus reviving an old term used by writers and philosophers since Viète. The use of this critical spirit to dismantle paranormal beliefs helped bring zeteticism to public attention. Mathematics, in particular the interpretation of statistics and the famous Simpson's paradox, offer reasoning tools that make it possible to avoid intellectual obscurantism and to demonstrate the perfectly natural character of certain phenomena.

When we consider a family of lines, there often exists a curve that is tangent to each one of them: this is the envelope of that family. Its equation can be computed from the parametrization of the lines. What a delight for the eye to see these geometric figures made of cleverly placed lines along which a beautiful curve winds, seeming to brush against each one of them! Among the curves constructed this way, which can be recognized in folding or in tables of stretched threads, we find some well-known figures such as conics. Light rays reflecting off a surface also draw beautiful curves, called caustics. Passionate about optics and close to Leibniz, the Count of Tschirnhaus understood the importance of differential calculus for studying them. The evolutes of curves, envelopes of normals, also offer beautiful geometry!