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In mathematics, Poinsot's spirals are two spirals represented by the polar equations
where csch is the hyperbolic cosecant, and sech is the hyperbolic secant. [1] They are named after the French mathematician Louis Poinsot.
 The Poinsot spiral r=csch(θ/3). |  The Poinsot spiral r=sech(θ/3). |
In Euclidean geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays are also known as plane angles as they lie in the plane that contains the rays. Angles are also formed by the intersection of two planes; these are called dihedral angles. Two intersecting curves may also define an angle, which is the angle of the rays lying tangent to the respective curves at their point of intersection.
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. Angles in polar notation are generally expressed in either degrees or radians.
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t) respectively, the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t) respectively.
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. Equivalently, in polar coordinates (r, θ) it can be described by the equation
In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. It is a subtype of whorled patterns, a broad group that also includes concentric objects.
Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives.
In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Each is an example of two-dimensional half-space.
In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum, as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation for . The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by Carl Gustav Jakob Jacobi. Carl Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular, but his work was published much later.
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.
The external secant function is a trigonometric function defined in terms of the secant function:
In hyperbolic geometry, angle of parallelism is the angle at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides. The angle depends on the segment length a between the right angle and the vertex of the angle of parallelism.
In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. The hyperbolic secant function is equivalent to the reciprocal hyperbolic cosine, and thus this distribution is also called the inverse-cosh distribution.
In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related:
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle, and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle , the sine and cosine functions are denoted as and .
In physics and in the mathematics of plane curves, a Cotes's spiral is one of a family of spirals classified by Roger Cotes.
The lituus spiral is a spiral in which the angle θ is inversely proportional to the square of the radius r.
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle.
In calculus, the integral of the secant function can be evaluated using a variety of methods and there are multiple ways of expressing the antiderivative, all of which can be shown to be equivalent via trigonometric identities,
In the hyperbolic plane, as in the Euclidean plane, each point can be uniquely identified by two real numbers. Several qualitatively different ways of coordinatizing the plane in hyperbolic geometry are used.
Hypertabastic survival models were introduced in 2007 by Mohammad Tabatabai, Zoran Bursac, David Williams, and Karan Singh. This distribution can be used to analyze time-to-event data in biomedical and public health areas and normally called survival analysis. In engineering, the time-to-event analysis is referred to as reliability theory and in business and economics it is called duration analysis. Other fields may use different names for the same analysis. These survival models are applicable in many fields such as biomedical, behavioral science, social science, statistics, medicine, bioinformatics, medical informatics, data science especially in machine learning, computational biology, business economics, engineering, and commercial entities. They not only look at the time to event, but whether or not the event occurred. These time-to-event models can be applied in a variety of applications for instance, time after diagnosis of cancer until death, comparison of individualized treatment with standard care in cancer research, time until an individual defaults on loans, relapsed time for drug and smoking cessation, time until property sold after being put on the market, time until an individual upgrades to a new phone, time until job relocation, time until bones receive microscopic fractures when undergoing different stress levels, time from marriage until divorce, time until infection due to catheter, and time from bridge completion until first repair.
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