The geometric line
Since Euclid, the line has been the foundation of geometry. Most figures contain them and many theorems have taken up the challenge of proving that certain points are collinear; from this we deduce the existence of remarkable lines, such as those of Euler or Simson (see opposite). Lines seem to "guide" curves and surfaces. A curve is locally assimilated to its tangent, at infinity to its asymptote. It can be defined by a family of lines that all prove to be tangent to it: this is the envelope. Finding lines in a surface leads to better constructing it. Remarkably, some surfaces are even a union of lines while everything seems curved within them!
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Euclid: an introduction to the straight line
Written more than two thousand years ago, Euclid's Elements is a foundational text of classical geometry. The concept of the straight line is central to it. Let's meet this scholar of ancient Greece and take a closer look at his view of the straight line.
Lines that have a name (2)
Some lines have a name, and some mathematicians have their own line.
Miraculous collinearity theorems!
A ruler, a compass, a sharp pencil—and the adventure has already begun! It can come as quite a surprise when three points turn out to lie on the same line. Plane geometry abounds in wondrous collinearity theorems.
Surfaces… made of straight lines!
A plane is generated by straight lines, and one might think it is the only surface that can be constructed in this way. But that is not so! Surfaces generated by straight lines even have a name: ruled surfaces.
Harmonic pencils of lines
How can we express simply that several lines are concurrent? Despite appearances, pure geometry is not the best tool for the job! Introducing a coordinate system and a few equations may prove more useful.
Caustic envelopes
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.
Tangents and asymptotes
From yesterday's "touching line" to today's tangent, from "vanishing quantities" to asymptotes, the story has been a long mathematical epic. Here are a few glorious episodes from this geometric quest to "approximate curves with straight lines."