In the differential geometry of surfaces, an asymptotic curve is a curve always tangent to an asymptotic direction of the surface (where they exist). It is sometimes called an asymptotic line, although it need not be a line. An asymptotic direction is one in which the normal curvature is zero. Which is to say: for a point on an asymptotic curve, take the plane which bears both the curve's tangent and the surface's normal at that point.

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  • In the differential geometry of surfaces, an asymptotic curve is a curve always tangent to an asymptotic direction of the surface (where they exist). It is sometimes called an asymptotic line, although it need not be a line. An asymptotic direction is one in which the normal curvature is zero. Which is to say: for a point on an asymptotic curve, take the plane which bears both the curve's tangent and the surface's normal at that point. The curve of intersection of the plane and the surface will have zero curvature at that point. Asymptotic directions can only occur when the Gaussian curvature is negative (or zero). There will be two asymptotic directions through every point with negative Gaussian curvature, these directions are symmetric about the principal directions. The direction of the asymptotic direction are the same as the asymptotes of the hyperbola of the Dupin indicatrix. A related notion is a curvature line, which is a curve always tangent to a principal direction.
  • Asymptotická křivka je taková křivka, k níž se bod probíhající danou křivku neomezeně přibližuje.
  • Асимптотическая кривая — кривая <math>\gamma</math> на регулярной поверхности <math>F</math>, нормальная кривизна которой вдоль <math>\gamma</math> равна нулю. Асимптотическая кривая определяется дифференциальным уравнением: <math>\mathrm{I\!I}_{\gamma(t)}(\dot\gamma,\dot\gamma)=0</math> где <math>\mathrm{I\!I}</math> — вторая фундаментальная форма поверхности.
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  • Asymptotic Curve
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  • AsymptoticCurve
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  • In the differential geometry of surfaces, an asymptotic curve is a curve always tangent to an asymptotic direction of the surface (where they exist). It is sometimes called an asymptotic line, although it need not be a line. An asymptotic direction is one in which the normal curvature is zero. Which is to say: for a point on an asymptotic curve, take the plane which bears both the curve's tangent and the surface's normal at that point.
  • Asymptotická křivka je taková křivka, k níž se bod probíhající danou křivku neomezeně přibližuje.
  • Асимптотическая кривая — кривая <math>\gamma</math> на регулярной поверхности <math>F</math>, нормальная кривизна которой вдоль <math>\gamma</math> равна нулю.
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  • Asymptotic curve
  • Asymptotická křivka
  • Асимптотическая кривая
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