Since antiquity, mathematicians have worked to approximate an arbitrary curve by another considered "simpler" to handle. As reference curves, they have often chosen the most elementary and best-known curves, namely lines (see La Droite, Bibliothèque Tangente 59, 2017) or circles (see Le Cercle, Bibliothèque Tangente 36, 2009). They established that a plane curve can generally be seen as nearly coinciding, near one of its points, with the tangent line or with the osculating circle at that point (see our feature "La saga des courbes" [The Saga of Curves], Tangente 146, 2012).
These two geometric objects, the line and the circle, obviously play an important role in the study of curves and are widely used in many contexts (in geometry, analysis, physics, economics, finance…). Let's take the time to rediscover them by observing a curve, in the immediate neighborhood of one of its points, using magnifying glasses that produce enlarged images of the regions studied.
The case of the parabola
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To show that the method used is both natural and effective, let's put it into action by working with a first, very simple curve, the parabola (P) with equation y = *x 2.
First, let's observe (P) near its vertex, the origin O with coordinates (0, 0) in the plane, using different magnifying glasses. To clearly perceive the magnifying effect of these tools, let's gather on a single figure the parabola (P) drawn in blue, together with the images produced by magnifying glasses of power ten (in red), a hundred (in green), and a thousand (in gray).
