Введение
Математическая функция, связывающая круговые и гиперболические функции
Mathematical function relating circular and hyperbolic functions
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s by Johann Heinrich Lambert, and later named for Christoph Gudermann who also described the relationship between circular and hyperbolic functions in 1830. The gudermannian is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant
The real inverse Gudermannian function can be defined for as the integral of the (circular) secant
The hyperbolic angle measure is called the anti gudermannian of or sometimes the lambertian of , denoted In the context of geodesy and navigation for latitude , (scaled by arbitrary constant ) was historically called the meridional part of (French: latitude croissante). It is the vertical coordinate of the Mercator projection. The two angle measures and are related by a common stereographic projection
and this identity can serve as an alternative definition for and valid throughout the complex plane:
TOC
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s by Johann Heinrich Lambert, and later named for Christoph Gudermann who also described the relationship between circular and hyperbolic functions in 1830. The gudermannian is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant
The real inverse Gudermannian function can be defined for as the integral of the (circular) secant
The hyperbolic angle measure is called the anti gudermannian of or sometimes the lambertian of , denoted In the context of geodesy and navigation for latitude , (scaled by arbitrary constant ) was historically called the meridional part of (French: latitude croissante). It is the vertical coordinate of the Mercator projection. The two angle measures and are related by a common stereographic projection
and this identity can serve as an alternative definition for and valid throughout the complex plane:
TOC
В математике гудерманновская функция связывает меру гиперболического угла с круговой мерой угла, называемой гудерманновским и обозначаемой . Гудерманновская функция демонстрирует тесную связь между круговыми и гиперболическими функциями. Она была введена в 1760-х годах Иоганном Генрихом Ламбертом, а позднее названа в честь Кристофа Гудерманна, который также описал связь между круговыми и гиперболическими функциями в 1830 году. Гудерманновский иногда называют гиперболической амплитудой как предельный случай эллиптической амплитуды Якоби, когда параметр равен .
Mathematical function relating circular and hyperbolic functions
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s by Johann Heinrich Lambert, and later named for Christoph Gudermann who also described the relationship between circular and hyperbolic functions in 1830. The gudermannian is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant
The real inverse Gudermannian function can be defined for as the integral of the (circular) secant
The hyperbolic angle measure is called the anti gudermannian of or sometimes the lambertian of , denoted In the context of geodesy and navigation for latitude , (scaled by arbitrary constant ) was historically called the meridional part of (French: latitude croissante). It is the vertical coordinate of the Mercator projection. The two angle measures and are related by a common stereographic projection
and this identity can serve as an alternative definition for and valid throughout the complex plane:
TOC
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s by Johann Heinrich Lambert, and later named for Christoph Gudermann who also described the relationship between circular and hyperbolic functions in 1830. The gudermannian is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant
The real inverse Gudermannian function can be defined for as the integral of the (circular) secant
The hyperbolic angle measure is called the anti gudermannian of or sometimes the lambertian of , denoted In the context of geodesy and navigation for latitude , (scaled by arbitrary constant ) was historically called the meridional part of (French: latitude croissante). It is the vertical coordinate of the Mercator projection. The two angle measures and are related by a common stereographic projection
and this identity can serve as an alternative definition for and valid throughout the complex plane:
TOC
Реальная гудерманновская функция обычно определяется как интеграл гиперболического секанса .
Mathematical function relating circular and hyperbolic functions
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s by Johann Heinrich Lambert, and later named for Christoph Gudermann who also described the relationship between circular and hyperbolic functions in 1830. The gudermannian is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant
The real inverse Gudermannian function can be defined for as the integral of the (circular) secant
The hyperbolic angle measure is called the anti gudermannian of or sometimes the lambertian of , denoted In the context of geodesy and navigation for latitude , (scaled by arbitrary constant ) was historically called the meridional part of (French: latitude croissante). It is the vertical coordinate of the Mercator projection. The two angle measures and are related by a common stereographic projection
and this identity can serve as an alternative definition for and valid throughout the complex plane:
TOC
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s by Johann Heinrich Lambert, and later named for Christoph Gudermann who also described the relationship between circular and hyperbolic functions in 1830. The gudermannian is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant
The real inverse Gudermannian function can be defined for as the integral of the (circular) secant
The hyperbolic angle measure is called the anti gudermannian of or sometimes the lambertian of , denoted In the context of geodesy and navigation for latitude , (scaled by arbitrary constant ) was historically called the meridional part of (French: latitude croissante). It is the vertical coordinate of the Mercator projection. The two angle measures and are related by a common stereographic projection
and this identity can serve as an alternative definition for and valid throughout the complex plane:
TOC
Реальная обратная гудерманновская функция может быть определена как интеграл (круглого) секанса .
Mathematical function relating circular and hyperbolic functions
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s by Johann Heinrich Lambert, and later named for Christoph Gudermann who also described the relationship between circular and hyperbolic functions in 1830. The gudermannian is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant
The real inverse Gudermannian function can be defined for as the integral of the (circular) secant
The hyperbolic angle measure is called the anti gudermannian of or sometimes the lambertian of , denoted In the context of geodesy and navigation for latitude , (scaled by arbitrary constant ) was historically called the meridional part of (French: latitude croissante). It is the vertical coordinate of the Mercator projection. The two angle measures and are related by a common stereographic projection
and this identity can serve as an alternative definition for and valid throughout the complex plane:
TOC
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s by Johann Heinrich Lambert, and later named for Christoph Gudermann who also described the relationship between circular and hyperbolic functions in 1830. The gudermannian is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant
The real inverse Gudermannian function can be defined for as the integral of the (circular) secant
The hyperbolic angle measure is called the anti gudermannian of or sometimes the lambertian of , denoted In the context of geodesy and navigation for latitude , (scaled by arbitrary constant ) was historically called the meridional part of (French: latitude croissante). It is the vertical coordinate of the Mercator projection. The two angle measures and are related by a common stereographic projection
and this identity can serve as an alternative definition for and valid throughout the complex plane:
TOC
Меру гиперболического угла называют антигудерманновским или иногда ламбертианским, обозначаемым . В контексте геодезии и навигации для широты , (умноженная на произвольную постоянную) исторически называлась меридиональной частью (фр. latitude croissante). Это вертикальная координата проекции Меркатора. Два угла и связаны общей стереографической проекцией, и это тождество может служить альтернативным определением для и , справедливым на всей комплексной плоскости:
Mathematical function relating circular and hyperbolic functions
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s by Johann Heinrich Lambert, and later named for Christoph Gudermann who also described the relationship between circular and hyperbolic functions in 1830. The gudermannian is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant
The real inverse Gudermannian function can be defined for as the integral of the (circular) secant
The hyperbolic angle measure is called the anti gudermannian of or sometimes the lambertian of , denoted In the context of geodesy and navigation for latitude , (scaled by arbitrary constant ) was historically called the meridional part of (French: latitude croissante). It is the vertical coordinate of the Mercator projection. The two angle measures and are related by a common stereographic projection
and this identity can serve as an alternative definition for and valid throughout the complex plane:
TOC
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s by Johann Heinrich Lambert, and later named for Christoph Gudermann who also described the relationship between circular and hyperbolic functions in 1830. The gudermannian is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant
The real inverse Gudermannian function can be defined for as the integral of the (circular) secant
The hyperbolic angle measure is called the anti gudermannian of or sometimes the lambertian of , denoted In the context of geodesy and navigation for latitude , (scaled by arbitrary constant ) was historically called the meridional part of (French: latitude croissante). It is the vertical coordinate of the Mercator projection. The two angle measures and are related by a common stereographic projection
and this identity can serve as an alternative definition for and valid throughout the complex plane:
TOC
Обобщение
Функцию Гудерманна можно рассматривать как отображение точек на одной ветке гиперболы в точки на полукруге. Точки на одном листе n-мерного гиперболоида двухполостного могут аналогичным образом отображаться на n-мерное полушарие посредством стереографической проекции. Полушарийная модель гиперболического пространства использует такое отображение для представления гиперболического пространства.
The Gudermannian function can be thought of mapping points on one branch of a hyperbola to points on a semicircle. Points on one sheet of an n dimensional hyperboloid of two sheets can be likewise mapped onto a n dimensional hemisphere via stereographic projection. The hemisphere model of hyperbolic space uses such a map to represent hyperbolic space.