quaternions

简明释义

英[kwəˈtɛəriənz]美[kwəˈtɪrniənz]

n. [数]四元数；四元法（quaternion 的复数）

英英释义

单词用法

同义词

反义词

例句

1.Compared to the Euler angles the main advantage of the quaternions expression of attitude motion is the avoidance of singular position in numeration procedure.

与欧拉角比较，姿态运动的四元数表达的主要优点是数值计算过程中不存在奇异位置。

2.The second argument that is commonly given for the use of quaternions is the possibility of smoothly interpolating between two orientations.

使用四元数的第二个理由，就是其可实现两个状态间的平滑插值。

3.However, it was later discovered that quaternions can also be used in computer graphics, as an alternative and compact way to represent rotations.

然而，人们发现四元数也可以应用在计算机图形学上，作为表现旋转的可选择方法之一。

4.The paper shows that the method of proving the commutative property of composition for the finite rotations by quaternions is simpler than that by vectors.

文中表明用四元数方法证明该定理比矢量法更为简单。

5.Quaternions extend the concept of rotation in three dimensions to rotation in four dimensions.

四元数将三维中旋转的概念扩展到四维中的旋转。

6.Only the bone data needs to be stored for every frame of the animation. Usually, the bone data is represented by quaternions.

存储动画帧我们仅仅需要骨胳的数据。通常，骨胳数据由四元数表示。

7.Indeed, quaternions support spherical linear interpolation (abbreviated as SLERP), which means that points travel along the surface of a sphere as they are moved from one orientation to the next.

的确，四元数支持球形线性插值(SLERP)，那意味着点沿着球体表面传播就像他们从一个方位移到另外一个一样。

8.Except for one thing: you don't have a good reason to use quaternions yet.

但除了一件事情，你还没有明白为何要使用四元数的原因。

9.The attitude kinematics and dynamics are both described by error quaternions.

姿态运动学和动力学用误差四元数描述。

10.When simulating 3D animations, quaternions provide a more efficient way to handle rotations than Euler angles.

在模拟3D动画时，四元数 提供了一种比欧拉角更有效的处理旋转的方法。

11.In robotics, quaternions are employed to calculate the orientation of robotic arms in three-dimensional space.

在机器人技术中，四元数 被用来计算机器人手臂在三维空间中的方向。

12.The physics engine uses quaternions to smoothly interpolate between different orientations of objects.

物理引擎使用 四元数 来平滑地插值不同物体的方向。

13.In computer graphics, quaternions are often used to represent rotations because they avoid gimbal lock.

在计算机图形学中，四元数 通常用于表示旋转，因为它们避免了万向节锁定。

14.Game developers prefer quaternions for rotation calculations because they reduce computational overhead.

游戏开发者更喜欢使用 四元数 进行旋转计算，因为它们减少了计算开销。

作文

In the realm of mathematics and physics, the concept of quaternions plays a significant role, particularly in three-dimensional space. A quaternion is a number system that extends complex numbers, consisting of one real part and three imaginary parts. This unique structure allows for a more efficient representation of rotations and orientations in three-dimensional space compared to traditional methods such as Euler angles or rotation matrices.The discovery of quaternions dates back to 1843 when the Irish mathematician William Rowan Hamilton first introduced them. Hamilton's motivation was to find a way to represent three-dimensional rotations mathematically. He realized that while complex numbers could represent two-dimensional rotations, there was no analogous system for three dimensions. Thus, he formulated the quaternion system, which is typically expressed in the form: q = a + bi + cj + dk,where 'a' is the scalar part, and 'bi', 'cj', and 'dk' are the vector parts with 'i', 'j', and 'k' being the fundamental quaternion units that obey specific multiplication rules. This formulation allows for the combination of rotations and can represent any rotation in three-dimensional space with a single quaternion.One of the most remarkable properties of quaternions is their ability to avoid gimbal lock, a problem that occurs when using Euler angles for representing rotations. Gimbal lock happens when two of the three rotational axes align, resulting in a loss of one degree of freedom. In contrast, quaternions maintain smooth and continuous rotation without this limitation, making them invaluable in fields such as computer graphics, robotics, and aerospace engineering.In computer graphics, quaternions are widely used for animating and controlling the orientation of objects. They provide a compact and efficient way to interpolate between two orientations, a process known as spherical linear interpolation (slerp). This technique ensures that the rotations are smooth and visually appealing, which is crucial in video games and simulations where realistic movements are essential.Furthermore, quaternions have applications beyond just graphics. In robotics, they help control the orientation of robotic arms and drones, allowing for precise movements in three-dimensional space. The aerospace industry also utilizes quaternions for attitude representation of spacecraft, ensuring that the spacecraft can rotate accurately and efficiently in orbit.Despite their advantages, quaternions can be challenging to understand initially due to their abstract nature. Unlike traditional vectors and matrices, quaternions do not have a straightforward geometric interpretation. However, once grasped, they provide powerful tools for manipulating three-dimensional rotations.In conclusion, quaternions are an essential mathematical construct that enhances our ability to work with three-dimensional rotations. Their introduction has revolutionized various fields, from computer graphics to robotics and aerospace engineering. Understanding quaternions opens up new avenues for solving complex problems related to motion and orientation, making them a cornerstone of modern mathematics and applied sciences.

在数学和物理的领域中，四元数的概念发挥着重要作用，尤其是在三维空间中。四元数是一种扩展复数的数字系统，由一个实部和三个虚部组成。这种独特的结构使得在三维空间中比传统方法（如欧拉角或旋转矩阵）更有效地表示旋转和方向。四元数的发现可以追溯到1843年，当时爱尔兰数学家威廉·罗温·哈密尔顿首次引入了这一概念。哈密尔顿的动机是寻找一种数学上表示三维旋转的方法。他意识到，虽然复数可以表示二维旋转，但没有类似的系统可以用于三维。因此，他制定了四元数系统，通常以以下形式表示：q = a + bi + cj + dk，其中'a'是标量部分，而'bi'、'cj'和'dk'是向量部分，'i'、'j'和'k'是基本的四元数单位，遵循特定的乘法规则。这种表述方式允许旋转的组合，并且可以用一个四元数表示三维空间中的任何旋转。四元数最显著的特性之一是它们能够避免万向节锁定，这是使用欧拉角表示旋转时出现的问题。万向节锁定发生在三个旋转轴中的两个对齐时，导致失去一个自由度。相比之下，四元数保持平滑和连续的旋转，没有这种限制，使它们在计算机图形学、机器人技术和航空航天工程等领域中变得无价。在计算机图形学中，四元数被广泛用于动画和控制物体的方向。它们提供了一种紧凑而高效的方式来插值两个方向，这一过程称为球面线性插值（slerp）。这一技术确保旋转平滑且视觉上令人愉悦，这在视频游戏和模拟中至关重要，因为逼真的运动是必不可少的。此外，四元数在图形学之外还有其他应用。在机器人技术中，它们帮助控制机器人臂和无人机的方向，实现三维空间中的精确运动。航空航天工业也利用四元数进行航天器的姿态表示，确保航天器在轨道中能够准确高效地旋转。尽管有其优点，四元数由于其抽象性质，起初可能难以理解。与传统的向量和矩阵不同，四元数没有简单的几何解释。然而，一旦理解，它们为操纵三维旋转提供了强大的工具。总之，四元数是一个重要的数学构造，增强了我们处理三维旋转的能力。它们的引入彻底改变了多个领域，从计算机图形学到机器人技术和航空航天工程。理解四元数为解决与运动和方向相关的复杂问题开辟了新的途径，使其成为现代数学和应用科学的基石。