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I quaternioni forniscono una notazione mat … I quaternioni forniscono una notazione matematica conveniente per la rappresentazione di orientamenti e rotazioni di oggetti in tre dimensioni. In confronto agli angoli di Eulero presentano funzioni più semplici da comporre ed evitano il problema del blocco cardanico. Confrontati con le matrici di rotazione essi sono più stabili numericamente e forse più efficienti. Quando vengono usati per rappresentare le rotazioni, i quaternioni vengono anche chiamati versori o quaternioni di rotazione, quando invece vengono usati per rappresentare le direzioni sono chiamati quaternioni di orientamento.sono chiamati quaternioni di orientamento.
, Les quaternions unitaires fournissent une … Les quaternions unitaires fournissent une notation mathématique commode pour représenter l'orientation et la rotation d'objets en trois dimensions. Comparés aux angles d'Euler, ils sont plus simples à composer et évitent le problème du blocage de cardan. Comparés aux matrices de rotations, ils sont plus stables numériquement et peuvent se révéler plus efficaces. Les quaternions ont été adoptés dans des applications en infographie, robotique, navigation, dynamique moléculaire et la mécanique spatiale des satellites.e et la mécanique spatiale des satellites.
, Кватерніон можна представити у вигляді пари скаляра та 3-вимірного вектора: , множення кватерніонів буде виражатись через скалярний та векторний добутки 3-вимірних векторів: Виразимо векторний добуток через добуток кватерніонів:
, 사원수(쿼터니언)는 3차원 공간에서 물체의 회전을 표현하는 편리한 수학적 표 … 사원수(쿼터니언)는 3차원 공간에서 물체의 회전을 표현하는 편리한 수학적 표기법으로 사용된다. 쿼터니언은 임의의 축을 중심으로 회전한 상태를 4개의 숫자로 표현한다. 회전을 표현하는 자세쿼터니언은 컴퓨터 그래픽, 컴퓨터 비젼, 항법, 분자동역학, 비행동역학, 인공위성의 궤도역학,과 자세역학, 결정(crystallographic) 질감 해석에 사용된다. 회전을 나타내는 단위 쿼터니언은 회전쿼터니언이라고도 불린다. 어떤 물체의 자세를 기준좌표계에 대하여 표현할때, 이를 표현한 쿼터니언은 자세쿼터니언이라고 불린다. 단위벡터 에 대하여 만큼 회전한 자세는 쿼터니언으로 로 표현된다. 회전행렬에 9개의 숫자와 비교하여 쿼터니언은 4개만의 훨씬 적은 갯수의 숫자로 회전을 표현하여 더 효율적이면 수치적으로도 더 안정하다. 오일러각과 비교해서는 김벌락이라는 현상이 생기지 않는다. 다만, 오일러각이 직관적으로 자세를 이해할 수 있는 반면, 쿼터니언은 직관적인 값을 제공하지는 않는다. 그리고 삼각함수의 주기성으로 인해 같은 자세를 나타내는 쿼터니언은 유일하지 않다. 삼각함수의 주기성으로 인해 같은 자세를 나타내는 쿼터니언은 유일하지 않다.
, Unit quaternions, known as versors, provid … Unit quaternions, known as versors, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation about an arbitrary axis. Rotation and orientation quaternions have applications in computer graphics, computer vision, robotics, navigation, molecular dynamics, flight dynamics, orbital mechanics of satellites, and crystallographic texture analysis. When used to represent rotation, unit quaternions are also called rotation quaternions as they represent the 3D rotation group. When used to represent an orientation (rotation relative to a reference coordinate system), they are called orientation quaternions or attitude quaternions. A spatial rotation around a fixed point of radians about a unit axis that denotes the Euler axis is given by the quaternion , where and . Compared to rotation matrices, quaternions are more compact, efficient, and numerically stable. Compared to Euler angles, they are simpler to compose. However, they are not as intuitive and easy to understand and, due to the periodic nature of sine and cosine, rotation angles differing precisely by the natural period will be encoded into identical quaternions and recovered angles in radians will be limited to .red angles in radians will be limited to .
, Кватернионы предоставляют удобное математи … Кватернионы предоставляют удобное математическое обозначение ориентации пространства и вращения объектов в этом пространстве. В сравнении с углами Эйлера кватернионы позволяют проще комбинировать вращения, а также избежать проблемы, связанной с невозможностью поворота вокруг оси независимо от совершённого вращения по другим осям (на иллюстрации). В сравнении с матрицами поворота они обладают большей вычислительной устойчивостью и могут быть более эффективными. Кватернионы нашли своё применение в компьютерной графике, робототехнике, навигации, молекулярной динамике.технике, навигации, молекулярной динамике.
, Los cuaterniones unitarios proporcionan un … Los cuaterniones unitarios proporcionan una notación matemática para representar las orientaciones y las rotaciones de objetos en tres dimensiones. Comparados con los ángulos de Euler, son más simples de componer y evitan el problema del bloqueo del cardán. Comparados con las matrices de rotación, son más eficientes y más estables numéricamente. Los cuarteniones son útiles en aplicaciones de gráficos por computadora, robótica, navegación y mecánica orbital de satélites.avegación y mecánica orbital de satélites.
, 单位四元数（Unit quaternion）可以用于表示三维空间裡的旋转。它与常用的另外两种表示方式（三维正交矩阵和欧拉角）是等价的，但是避免了欧拉角表示法中的万向锁问题。比起三维正交矩阵表示，四元数表示能够更方便地给出旋转的转轴与旋转角。
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Los cuaterniones unitarios proporcionan un … Los cuaterniones unitarios proporcionan una notación matemática para representar las orientaciones y las rotaciones de objetos en tres dimensiones. Comparados con los ángulos de Euler, son más simples de componer y evitan el problema del bloqueo del cardán. Comparados con las matrices de rotación, son más eficientes y más estables numéricamente. Los cuarteniones son útiles en aplicaciones de gráficos por computadora, robótica, navegación y mecánica orbital de satélites.avegación y mecánica orbital de satélites.
, Кватерніон можна представити у вигляді пари скаляра та 3-вимірного вектора: , множення кватерніонів буде виражатись через скалярний та векторний добутки 3-вимірних векторів: Виразимо векторний добуток через добуток кватерніонів:
, Unit quaternions, known as versors, provid … Unit quaternions, known as versors, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation about an arbitrary axis. Rotation and orientation quaternions have applications in computer graphics, computer vision, robotics, navigation, molecular dynamics, flight dynamics, orbital mechanics of satellites, and crystallographic texture analysis.es, and crystallographic texture analysis.
, I quaternioni forniscono una notazione mat … I quaternioni forniscono una notazione matematica conveniente per la rappresentazione di orientamenti e rotazioni di oggetti in tre dimensioni. In confronto agli angoli di Eulero presentano funzioni più semplici da comporre ed evitano il problema del blocco cardanico. Confrontati con le matrici di rotazione essi sono più stabili numericamente e forse più efficienti. Quando vengono usati per rappresentare le rotazioni, i quaternioni vengono anche chiamati versori o quaternioni di rotazione, quando invece vengono usati per rappresentare le direzioni sono chiamati quaternioni di orientamento.sono chiamati quaternioni di orientamento.
, Les quaternions unitaires fournissent une … Les quaternions unitaires fournissent une notation mathématique commode pour représenter l'orientation et la rotation d'objets en trois dimensions. Comparés aux angles d'Euler, ils sont plus simples à composer et évitent le problème du blocage de cardan. Comparés aux matrices de rotations, ils sont plus stables numériquement et peuvent se révéler plus efficaces. Les quaternions ont été adoptés dans des applications en infographie, robotique, navigation, dynamique moléculaire et la mécanique spatiale des satellites.e et la mécanique spatiale des satellites.
, Кватернионы предоставляют удобное математи … Кватернионы предоставляют удобное математическое обозначение ориентации пространства и вращения объектов в этом пространстве. В сравнении с углами Эйлера кватернионы позволяют проще комбинировать вращения, а также избежать проблемы, связанной с невозможностью поворота вокруг оси независимо от совершённого вращения по другим осям (на иллюстрации). В сравнении с матрицами поворота они обладают большей вычислительной устойчивостью и могут быть более эффективными. Кватернионы нашли своё применение в компьютерной графике, робототехнике, навигации, молекулярной динамике.технике, навигации, молекулярной динамике.
, 单位四元数（Unit quaternion）可以用于表示三维空间裡的旋转。它与常用的另外两种表示方式（三维正交矩阵和欧拉角）是等价的，但是避免了欧拉角表示法中的万向锁问题。比起三维正交矩阵表示，四元数表示能够更方便地给出旋转的转轴与旋转角。
, 사원수(쿼터니언)는 3차원 공간에서 물체의 회전을 표현하는 편리한 수학적 표 … 사원수(쿼터니언)는 3차원 공간에서 물체의 회전을 표현하는 편리한 수학적 표기법으로 사용된다. 쿼터니언은 임의의 축을 중심으로 회전한 상태를 4개의 숫자로 표현한다. 회전을 표현하는 자세쿼터니언은 컴퓨터 그래픽, 컴퓨터 비젼, 항법, 분자동역학, 비행동역학, 인공위성의 궤도역학,과 자세역학, 결정(crystallographic) 질감 해석에 사용된다. 회전을 나타내는 단위 쿼터니언은 회전쿼터니언이라고도 불린다. 어떤 물체의 자세를 기준좌표계에 대하여 표현할때, 이를 표현한 쿼터니언은 자세쿼터니언이라고 불린다. 단위벡터 에 대하여 만큼 회전한 자세는 쿼터니언으로 로 표현된다. 회전행렬에 9개의 숫자와 비교하여 쿼터니언은 4개만의 훨씬 적은 갯수의 숫자로 회전을 표현하여 더 효율적이면 수치적으로도 더 안정하다. 오일러각과 비교해서는 김벌락이라는 현상이 생기지 않는다. 다만, 오일러각이 직관적으로 자세를 이해할 수 있는 반면, 쿼터니언은 직관적인 값을 제공하지는 않는다. 그리고 삼각함수의 주기성으로 인해 같은 자세를 나타내는 쿼터니언은 유일하지 않다. 삼각함수의 주기성으로 인해 같은 자세를 나타내는 쿼터니언은 유일하지 않다.
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사원수와 회전
, 四元数与空间旋转
, Кватерніони і повороти простору
, Quaternions et rotation dans l'espace
, Cuaterniones y rotación en el espacio
, Quaternions and spatial rotation
, Rotazioni spaziali con i quaternioni
, Кватернионы и вращение пространства
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