affine mapping

简明释义

1. 仿射映像; 2. 仿射映射;

英英释义

例句

1.In robotics, affine mapping 仿射映射 helps in the transformation of coordinates from one frame to another.

在机器人技术中，affine mapping 仿射映射 有助于将坐标从一个框架转换到另一个框架。

2.We used affine mapping 仿射映射 to model the relationship between different data sets.

我们使用affine mapping 仿射映射 来建模不同数据集之间的关系。

3.In machine learning, affine mapping 仿射映射 is often used for feature transformation.

在机器学习中，affine mapping 仿射映射 通常用于特征变换。

4.The process of resizing an image can be achieved through affine mapping 仿射映射 techniques.

通过affine mapping 仿射映射 技术可以实现图像的调整大小。

5.In computer graphics, we often use affine mapping 仿射映射 to transform images and shapes.

在计算机图形学中，我们经常使用affine mapping 仿射映射 来变换图像和形状。

作文

In the realm of mathematics and computer graphics, the concept of affine mapping plays a crucial role in transforming shapes and images. An affine mapping is a function between affine spaces that preserves points, straight lines, and planes. This means that if we take a geometric shape and apply an affine mapping, the resulting shape will maintain the properties of the original, such as parallelism and ratios of distances. For instance, when we scale, rotate, or translate an object in a two-dimensional space, we are essentially performing an affine mapping. These transformations are essential in various fields, including computer graphics, robotics, and even in solving complex mathematical problems.To better understand affine mapping, consider a simple example involving a triangle on a coordinate plane. If we have a triangle defined by three vertices at points (1, 1), (2, 3), and (3, 1), applying an affine mapping can change the position, size, or orientation of this triangle while keeping its shape intact. For instance, if we want to translate the triangle 2 units to the right and 1 unit up, the new vertices would be (3, 2), (4, 4), and (5, 2). The triangle remains similar in shape but has moved to a new location.Moreover, affine mapping can also be represented using matrices. In linear algebra, we can express an affine mapping as a combination of a linear transformation followed by a translation. This is particularly useful when dealing with multiple transformations, as we can represent them in a single matrix equation. For instance, if we have a matrix that represents rotation and another that represents scaling, we can multiply these matrices together to create a single transformation matrix. When we apply this transformation matrix to our triangle, we achieve the desired result in one step, showcasing the efficiency of affine mapping in computational applications.The importance of affine mapping extends beyond simple geometric transformations. In computer graphics, for example, rendering scenes often requires numerous transformations to position objects correctly within a scene. By utilizing affine mapping, graphic designers can efficiently manipulate objects, ensuring that they appear correctly from different viewpoints and under various lighting conditions. This capability is fundamental in creating realistic animations and simulations.Furthermore, affine mapping is not limited to two-dimensional spaces; it can also be applied in three-dimensional environments. In 3D graphics, transformations such as rotation around an axis, scaling in different directions, and translating objects in space all utilize affine mapping. The ability to manipulate objects in three dimensions opens up endless possibilities for creativity and innovation in fields like virtual reality and gaming.In conclusion, the concept of affine mapping is a foundational aspect of both mathematics and computer graphics. Its ability to preserve the essential characteristics of shapes during transformations makes it an invaluable tool across various disciplines. Understanding affine mapping not only enhances our grasp of geometric principles but also equips us with the skills necessary to navigate and manipulate the digital world effectively. As technology continues to evolve, the significance of affine mapping will undoubtedly remain prominent, driving advancements in how we visualize and interact with our surroundings.

在数学和计算机图形学领域，仿射映射的概念在变换形状和图像中发挥着至关重要的作用。仿射映射是一个在仿射空间之间的函数，它保持点、直线和平面。这意味着，如果我们对几何形状应用仿射映射，则得到的形状将保持原始形状的属性，例如平行性和距离比率。例如，当我们在二维空间中缩放、旋转或平移一个物体时，我们实际上是在执行仿射映射。这些变换在计算机图形学、机器人技术甚至解决复杂数学问题等多个领域中都是必不可少的。为了更好地理解仿射映射，考虑一个涉及坐标平面上三角形的简单示例。如果我们有一个由三个顶点（1, 1）、（2, 3）和（3, 1）定义的三角形，应用仿射映射可以在保持其形状不变的情况下改变该三角形的位置、大小或方向。例如，如果我们想将三角形向右平移2个单位并向上平移1个单位，则新的顶点将是（3, 2）、（4, 4）和（5, 2）。三角形的形状保持相似，但已移动到新位置。此外，仿射映射还可以使用矩阵表示。在线性代数中，我们可以将仿射映射表示为线性变换后跟随平移的组合。这在处理多个变换时特别有用，因为我们可以在一个矩阵方程中表示它们。例如，如果我们有一个表示旋转的矩阵和另一个表示缩放的矩阵，我们可以将这些矩阵相乘以创建一个单一的变换矩阵。当我们将此变换矩阵应用于我们的三角形时，我们在一步中实现所需的结果，展示了仿射映射在计算应用中的效率。仿射映射的重要性不仅限于简单的几何变换。在计算机图形学中，例如，渲染场景通常需要进行多次变换，以确保对象在场景中正确定位。通过利用仿射映射，图形设计师可以有效地操纵对象，确保它们在不同视角和各种光照条件下正确出现。这种能力对于创建逼真的动画和模拟是基础。此外，仿射映射不仅限于二维空间；它也可以应用于三维环境。在3D图形中，围绕轴的旋转、不同方向的缩放和空间中对象的平移等变换都利用了仿射映射。在三维中操纵对象的能力为虚拟现实和游戏等领域的创造力和创新打开了无尽的可能性。总之，仿射映射的概念是数学和计算机图形学的基础方面。它在变换过程中保持形状的基本特征，使其成为各个学科中不可或缺的工具。理解仿射映射不仅增强了我们对几何原理的把握，还使我们具备有效导航和操纵数字世界的技能。随着技术的不断发展，仿射映射的重要性无疑将继续突出，推动我们在可视化和与周围环境互动方面的进步。