affine space

简明释义

1. 仿射空间; 2. 远交空间;

英英释义

例句

1.The concept of affine space 仿射空间 is essential in robotics for modeling the positions and orientations of robotic arms.

在机器人技术中，affine space 仿射空间 的概念对于建模机器人手臂的位置和方向至关重要。

2.In machine learning, we often represent data points in an affine space 仿射空间 to facilitate linear regression analysis.

在机器学习中，我们通常将数据点表示在 affine space 仿射空间 中，以便于线性回归分析。

3.The affine space 仿射空间 model is used in architecture to plan the layout of buildings in relation to their surroundings.

在建筑学中，affine space 仿射空间 模型用于规划建筑物与周围环境的布局。

4.In computer graphics, an affine space 仿射空间 allows for transformations such as translation, scaling, and rotation without changing the linearity of the objects.

在计算机图形学中，affine space 仿射空间 允许进行平移、缩放和旋转等变换，而不改变对象的线性特性。

5.When designing a game, understanding the affine space 仿射空间 can help developers create more realistic movements and interactions.

在设计游戏时，理解 affine space 仿射空间 可以帮助开发者创建更真实的运动和交互。

作文

In the realm of mathematics, particularly in geometry, the concept of affine space plays a significant role in understanding the structure of geometric objects and their relationships. An affine space can be thought of as a set of points that exhibit certain properties similar to those found in vector spaces, but without a fixed origin. This characteristic allows for the definition of parallelism and ratios of distances, which are essential in many areas of mathematics and physics.To elaborate, an affine space is defined over a field, typically the field of real numbers, and is associated with a vector space. The points in an affine space can be represented by vectors, but it is crucial to note that these vectors do not have a specific starting point. Instead, they represent the relative positions of points. For instance, if we consider three points A, B, and C in an affine space, the vector from A to B can be denoted as AB, and similarly, we can denote other vectors like AC or BC. However, the important aspect to remember is that these vectors are defined in relation to each other rather than from a fixed origin.The significance of affine space extends beyond pure mathematics; it has practical applications in computer graphics, robotics, and even in the field of economics. In computer graphics, for example, the manipulation of images and models often relies on the principles of affine space. Transformations such as translation, rotation, and scaling can be efficiently described using the properties of affine space. This allows for the creation of realistic animations and models by manipulating points in a way that preserves the relationships between them.Moreover, in robotics, understanding the configuration of a robot in an affine space helps in path planning and navigation. Robots need to understand their environment and make decisions based on the relative positions of objects around them. By utilizing the framework of affine space, robots can effectively calculate trajectories and avoid obstacles while maintaining their orientation and position.Furthermore, the concept of affine space also finds its way into economics, particularly in the analysis of consumer preferences and choices. Economists often model consumer behavior in an affine space where different goods are represented as points. The relationships between these points can help in understanding how consumers make decisions based on their budget constraints and preferences.In conclusion, the concept of affine space is a fundamental idea in mathematics that provides a framework for understanding geometric relationships without the necessity of a fixed origin. Its applications across various fields highlight its versatility and importance. Whether in the realm of theoretical mathematics or practical applications in technology and economics, affine space serves as a crucial tool for analyzing and interpreting the world around us. As we continue to explore the depths of mathematics and its applications, the understanding of concepts like affine space will remain essential for both current and future advancements in these fields.

在数学的领域中，尤其是在几何学中，仿射空间的概念在理解几何对象及其关系的结构方面起着重要作用。仿射空间可以被视为一组点，这些点展现出与向量空间相似的某些特性，但没有固定的原点。这一特性允许定义平行性和距离比率，这在许多数学和物理领域中都是至关重要的。进一步说，仿射空间是定义在一个域上，通常是实数域，并与一个向量空间相关联。仿射空间中的点可以用向量表示，但重要的是要注意这些向量没有特定的起点。相反，它们表示点的相对位置。例如，如果我们考虑三个点A、B和C在一个仿射空间中，从A到B的向量可以表示为AB，同样，我们可以表示其他向量如AC或BC。然而，重要的是要记住，这些向量是相对于彼此定义的，而不是从固定的原点出发。仿射空间的重要性超越了纯数学；它在计算机图形学、机器人技术甚至经济学等领域都有实际应用。在计算机图形学中，例如，图像和模型的操作通常依赖于仿射空间的原理。平移、旋转和缩放等变换可以使用仿射空间的特性有效描述。这允许通过以保留它们之间的关系的方式操纵点来创建逼真的动画和模型。此外，在机器人技术中，理解机器人在仿射空间中的配置有助于路径规划和导航。机器人需要理解它们的环境并根据周围物体的相对位置做出决策。通过利用仿射空间的框架，机器人可以有效计算轨迹并避开障碍，同时保持其方向和位置。此外，仿射空间的概念也出现在经济学中，特别是在消费者偏好和选择的分析中。经济学家经常在仿射空间中建模消费者行为，其中不同的商品被表示为点。这些点之间的关系可以帮助理解消费者如何根据预算约束和偏好做出决策。总之，仿射空间的概念是数学中的一个基本思想，为理解几何关系提供了一个框架，而不需要固定的原点。它在各个领域的应用突显了它的多功能性和重要性。无论是在理论数学的领域还是在技术和经济学的实际应用中，仿射空间都作为分析和解释我们周围世界的重要工具。随着我们继续探索数学及其应用的深度，理解像仿射空间这样的概念将对当前和未来在这些领域的进步保持至关重要。