affine coordinates

简明释义

仿射坐标;

英英释义

例句

1.When modeling geometric transformations, affine coordinates 仿射坐标 help in preserving points, straight lines, and planes.

在建模几何变换时，affine coordinates仿射坐标有助于保持点、直线和平面。

2.The concept of affine coordinates 仿射坐标 is fundamental in linear algebra and computer vision.

概念上的affine coordinates仿射坐标在线性代数和计算机视觉中是基础的。

3.In computer graphics, we often use affine coordinates 仿射坐标 to manipulate shapes and transformations efficiently.

在计算机图形学中，我们经常使用affine coordinates仿射坐标来高效地操控形状和变换。

4.Using affine coordinates 仿射坐标, we can easily calculate the position of points after a transformation.

使用affine coordinates仿射坐标，我们可以轻松计算变换后点的位置。

5.In robotics, affine coordinates 仿射坐标 are used to describe the location and orientation of objects in space.

在机器人技术中，affine coordinates仿射坐标用于描述物体在空间中的位置和方向。

作文

In the realm of geometry and mathematics, the concept of affine coordinates plays a pivotal role in understanding the relationships between points, lines, and shapes. To grasp the significance of affine coordinates, it is essential to first comprehend what affine space is. Affine space can be thought of as a geometric structure that generalizes the properties of Euclidean spaces while allowing for the representation of points without a fixed origin. This flexibility is crucial in various applications, including computer graphics, robotics, and even economics.The essence of affine coordinates lies in their ability to describe points in an affine space using a coordinate system that does not depend on a specific origin. In simpler terms, while traditional Cartesian coordinates require a defined origin point (usually the intersection of axes), affine coordinates allow us to express the position of a point relative to other points within the same space. This characteristic makes them particularly useful in scenarios where shifts in position or transformations are involved.For instance, consider a triangle defined by three vertices A, B, and C in a two-dimensional space. The vertices can be represented in affine coordinates as linear combinations of these points. Specifically, any point P inside the triangle can be expressed as P = uA + vB + wC, where u, v, and w are non-negative weights that sum up to one (u + v + w = 1). This representation not only simplifies calculations but also provides a clear geometric interpretation of the location of point P relative to the triangle.Furthermore, affine coordinates facilitate the transformation of geometric figures through operations such as translation, scaling, and rotation. These transformations can be achieved without losing the inherent properties of the geometric objects involved. For example, when we translate a shape, the affine coordinates of its points simply shift by a constant vector, preserving the shape's structure and proportions.In computer graphics, the use of affine coordinates is ubiquitous. Graphics rendering relies heavily on transformations to manipulate shapes and images efficiently. By utilizing affine coordinates, developers can easily apply transformations to objects, ensuring that the visual output remains consistent and accurate across various display settings.Moreover, the concept of affine coordinates extends beyond mere geometric applications. In economics, for example, affine coordinates can be used to model consumer preferences and utility functions. By representing different goods and services in an affine space, economists can analyze how changes in prices or income affect consumer behavior without being tied to a specific reference point.In conclusion, affine coordinates are a fundamental aspect of modern geometry and mathematics, providing a flexible framework for describing and manipulating points in an affine space. Their ability to represent points relative to one another, rather than relying on a fixed origin, opens up a myriad of possibilities in fields ranging from computer graphics to economics. Understanding affine coordinates not only enhances our comprehension of geometric concepts but also equips us with powerful tools for solving real-world problems. As we continue to explore the intricacies of mathematics, the significance of affine coordinates will undoubtedly remain a vital topic of study and application.

在几何和数学领域，仿射坐标的概念在理解点、线和形状之间的关系中发挥着关键作用。要掌握仿射坐标的重要性，首先需要理解什么是仿射空间。仿射空间可以被视为一种几何结构，它概括了欧几里得空间的性质，同时允许在没有固定原点的情况下表示点。这种灵活性在计算机图形学、机器人技术甚至经济学等各种应用中至关重要。仿射坐标的本质在于它们能够使用不依赖于特定原点的坐标系统来描述仿射空间中的点。简单来说，传统的笛卡尔坐标需要定义一个原点（通常是坐标轴的交点），而仿射坐标则允许我们相对于同一空间中的其他点来表达一个点的位置。这一特性使得它们在涉及位置变化或变换的场景中特别有用。例如，考虑一个由三个顶点A、B和C定义的三角形。在二维空间中，这些顶点可以用仿射坐标表示为这些点的线性组合。具体而言，三角形内部的任何点P都可以表示为P = uA + vB + wC，其中u、v和w是非负权重，且总和为一（u + v + w = 1）。这种表示不仅简化了计算，还提供了点P相对于三角形位置的清晰几何解释。此外，仿射坐标通过平移、缩放和旋转等操作促进了几何图形的变换。这些变换可以在不失去所涉及几何对象固有属性的情况下实现。例如，当我们平移一个形状时，其点的仿射坐标仅通过一个常量向量移动，从而保持形状的结构和比例。在计算机图形学中，仿射坐标的使用无处不在。图形渲染在很大程度上依赖于变换来高效地操作形状和图像。通过利用仿射坐标，开发人员可以轻松地对对象应用变换，确保视觉输出在各种显示设置下保持一致和准确。此外，仿射坐标的概念超越了单纯的几何应用。在经济学中，例如，仿射坐标可以用来建模消费者偏好和效用函数。通过在仿射空间中表示不同的商品和服务，经济学家可以分析价格或收入变化如何影响消费者行为，而不必受限于特定的参考点。总之，仿射坐标是现代几何和数学的基本方面，为描述和操作仿射空间中的点提供了灵活的框架。它们相对于彼此表示点的能力，而不是依赖于固定原点，开辟了计算机图形学到经济学等领域的无数可能性。理解仿射坐标不仅增强了我们对几何概念的理解，还为我们解决现实世界问题提供了强大的工具。随着我们继续探索数学的复杂性，仿射坐标的重要性无疑将仍然是一个重要的研究和应用主题。