HelixA helix ('hiːlɪks; ) is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, and many proteins have helical substructures, known as alpha helices. The word helix comes from the Greek word ἕλιξ, "twisted, curved". A "filled-in" helix – for example, a "spiral" (helical) ramp – is a surface called helicoid.
Differentiable curveDifferential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the synthetic approach. Differential geometry takes another path: curves are represented in a parametrized form, and their geometric properties and various quantities associated with them, such as the curvature and the arc length, are expressed via derivatives and integrals using vector calculus.
CatenaryIn physics and geometry, a catenary (USˈkætənɛri, UKkəˈtiːnəri) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field. The catenary curve has a U-like shape, superficially similar in appearance to a parabola, which it is not. The curve appears in the design of certain types of arches and as a cross section of the catenoid—the shape assumed by a soap film bounded by two parallel circular rings.
Moving frameIn mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space. In lay terms, a frame of reference is a system of measuring rods used by an observer to measure the surrounding space by providing coordinates. A moving frame is then a frame of reference which moves with the observer along a trajectory (a curve).
