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Lecture

Helical Curves: Theory and Applications

In course

MATH-213: Differential geometry I - curves and surfaces

This course serves as an introduction to classical differential geometry, which studies the geometric properties of curves and surfaces in the Euclidean space. ...

Description

This lecture covers the theory and applications of helical curves, focusing on circular helices with various parameters such as radius, pitch, and twist. The instructor discusses the properties of helical curves, including their curvature and torsion, and demonstrates how these curves can be used in different contexts.

Instructor

Marc Troyanov

Doctorat en mathématiques à l'Université de Genève (1987) - Postdoc à Paris et Salt-Lake City, puis professeur assistant à l'UQAM (Montréal). Venu en 1993 à l'EPFL comme professeur assistant, nommé MER en 1999 puis professeur titulaire en 2005. Domaine de recherche : géométrie différentielle, analyse sur les variétés et sur les espaces métriques.

Official source

Ontological neighbourhood

Mathematics

Geometry: Differential geometry

Related concepts (32)

Curvature

In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point.

Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature Κ of a smooth surface in three-dimensional space at a point is the product of the principal curvatures, κ1 and κ2, at the given point: The Gaussian radius of curvature is the reciprocal of Κ. For example, a sphere of radius r has Gaussian curvature 1/r2 everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus.

Ricci curvature

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space. The Ricci tensor can be characterized by measurement of how a shape is deformed as one moves along geodesics in the space.

Radius of curvature

In differential geometry, the radius of curvature (Rc), R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or combinations thereof. In the case of a space curve, the radius of curvature is the length of the curvature vector. In the case of a plane curve, then R is the absolute value of where s is the arc length from a fixed point on the curve, φ is the tangential angle and κ is the curvature.

Earth radius

Earth radius (denoted as R🜨 or ) is the distance from the center of Earth to a point on or near its surface. Approximating the figure of Earth by an Earth spheroid, the radius ranges from a maximum of nearly (equatorial radius, denoted a) to a minimum of nearly (polar radius, denoted b). A nominal Earth radius is sometimes used as a unit of measurement in astronomy and geophysics, which is recommended by the International Astronomical Union to be the equatorial value. A globally-average value is usually considered to be with a 0.

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