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Conical spiral with an archimedean spiral as floor projection Floor projection: Fermat's spiral Floor projection: logarithmic spiral Floor projection: hyperbolic spiral. In mathematics, a conical spiral, also known as a conical helix, [1] is a space curve on a right circular cone, whose floor projection is a plane spiral.
The slope of a circular helix is commonly defined as the ratio of the circumference of the circular cylinder that it spirals around, and its pitch (the height of one complete helix turn). A conic helix, also known as a conic spiral, may be defined as a spiral on a conic surface, with the distance to the apex an exponential function of the angle ...
Helix. Tendril perversion (a transition between back-to-back helices) Hemihelix, a quasi-helical shape characterized by multiple tendril perversions; Seiffert's spiral [5] Slinky spiral [6] Twisted cubic; Viviani's curve
Two well-known spiral space curves are conical spirals and spherical spirals, defined below. Another instance of space spirals is the toroidal spiral. [8] A spiral wound around a helix, [9] also known as double-twisted helix, [10] represents objects such as coiled coil filaments.
In cylindrical coordinates, the conchospiral is described by the parametric equations: = = =. The projection of a conchospiral on the (,) plane is a logarithmic spiral.The parameter controls the opening angle of the projected spiral, while the parameter controls the slope of the cone on which the curve lies.
Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the right conoid, the helicoid, and the tangent developable of a smooth curve in space. A ruled surface can be described as the set of points swept by a moving straight line.
The angle of the sloped arm remains constant throughout (traces a cone), and setting a different angle varies the pitch of the spiral. This device provides a high degree of precision, depending on the precision with which the device is machined (machining a precise helical screw thread is a related challenge).
Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus , Stokes' theorem and the divergence theorem , are frequently given in a parametric form.