Curves and lines
The infinite variety of trajectories in the plane seemed to escape any classification. Yet, smooth curves share a common property: when one "zooms in" on a generic point, one "sees" a line! The notions of tangent and derivative made it possible, quite belatedly, to formalize this observation. The tangent remains today a powerful universal tool for studying curves and trying to uncover their properties. However, be careful not to confuse the tangent to a curve with its asymptote at infinity. A priori, no confusion is possible. But beware of hasty conclusions: projective geometry has surprises in store for you!
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Observing curves with magnifying glasses
Used appropriately, extremely powerful magnifying glasses make most curves appear to coincide locally with a tangent line or an osculating circle.
Deviation of a curve from a tangent
Newton and Leibniz, the founders of differential and integral calculus, studied how a curve deviates from a tangent. To this end, they drew on the notion of curvature. The fundamental ideas of these two scholars gave rise to contemporary mathematical analysis.
Tangent or asymptote?
Confusing "tangent" and "asymptote" is a very common mistake: what student has never made it? Yet the distinction between the two notions is perfectly clear. An exploration of certain constructions in projective geometry will nonetheless force us to call our certainties into question.
Tangents and derivatives
For many people, tangents and derivatives go hand in hand.
A remarkable line: the asymptote
A look back at some properties of asymptotes...
Envelope of a parametrized family of lines | Tangente
Using tangency, a family of lines can also define a curve.