以上是傍轴近似的理论体系，我们可以看到从简单的透镜开始，研究了近轴光学的特点。这部分内容在几何光学的书籍中均有涉及，只是有的书很清晰，有的书含糊其辞的讲了一下，并没有说清楚。近轴光学所应用的条件和特点，可以说单个的透镜是系统总和的支撑，所以这一部分理论是光学系统设计的基石。
英文翻译要求不是要求一一对应，而是要准确的翻译出所要表达的内涵，在中文书中寻找对应的关系和补充。
General Remarks
The property属性 of an optical system in which all rays that are emitted发出 by an object at point O meet in an image at point O' is called optical imaging.
principle of image formation by a common ray intersection point. 共射线交点成像原理
Real images which have a positive image distance from the vertex顶点 and which can be made visible by a ground glass screen are distinguished区别 from virtual images with a negative focal distance.in the case of such a virtual image,the rays meet when they are extended backwards.which have a positive image distance from the vertex顶点 and which can be made visible by a ground glass screen are distinguished区别 from virtual images with a negative focal distance.in the case of such a virtual image,the rays meet when they are extended backwards.
In order to calculate the location and size of the images in the paraxial approximation 近轴近似 classical lens formula or Newton's equation can be used.These are discussed in the following chapters.This field of paraxial optics is also called Gaussian optics.
paraxial Approximation
In geometric optics the law of refraction at dielectric interfaces or mirrors determines the ray path through the system. 在几何光学中，介质界面或反射镜的折射定律决定光线穿过系统的路径。
It is written as
where n and n' are the refractive indices before and behind the refraction of the ray,I and I' are the incidence angles in the two media.The paraxial approximation can be developed from the exact imaging equations by linear approximation of the sine function in the law of refraction.The Taylor series of the sine function
is truncated after the first term
and delivers the law of refraction in its paraxial form.Writing small characters i and i' for the paraxial approximated incidence angles in order to distinguish them,this can be written
傍轴近似在小角度下误差属于高阶误差，关系几乎为线性的。
Image Locations
The simple optical system that can produce an image is a single lens or a mirror.Therefore the differences resulting from the formation of real or virtual image locations are disscussed using the
example of a single lens.
能产生图像的最简单的光学系统是一个透镜或一个镜子。因此，用一个透镜的例子来讨论实际或虚拟图像位置的形成所产生的差异。
看似可有可无的一句话，确也是有理有据。
For different values of the object distance,the image formation produced by a single lens can seen in figure 2-3 to 2-7 ,see also 2-2 and 2-3 .An object positioned whithin the focal length of a convergent lens usually leads to a virtual image,while outside,real images are produced at finite distances.The magnification m is equal to 1 only for an object positioned at twice the distance from the focal point.For objects at a greater distance,the image will be reduced in size.
The main examples for the paths of rays passing through a positive lens,depending on the positon of the object,are shown in figure 2-3.
For a single negative lens the image is principally virtual,as shown in figure 2-4.
The focal points F and F' are particular points of a lens or an imaging system (see also the chapter concerning the cardinal points of a lens).A ray incident to a lens which is parallel to the optical axis intercepts the aixs at the focal point F' behind the lens.Coversely,a ray running through the front focal point F will be parallel to the axis behind the lens.
For simplification we first consider a thin lens whose optical effect can be imagined as taking place of the lens.The distances from the focal points to the lens are called the focal lengths f and f',respectively.Furthermore, r is the object height measured perpendicular to the axis and r' is the corresponding image height.The distances of object and image positions from the lens are labelled s and s'.
The sign convention which is used in optic considers distance along the optical axis as positive if they are oriented in the sense of a vector in the positive z direction.The same applies for the r and x coordinates perpendicular to the axis.Then in the case of real imaging through a single positive lens with a positive object of height r and image height r',the distance form the vertex s and the focal length f are negative,whereas the quantities s' and f' are positive.
A positive lens with a focal length of f'>0 produces a real image as long as the distance of the object from the vertex is greater than the focal length.
一个焦距为F＞0的正透镜，只要物体与顶点的距离大于焦距，就产生一个真实的图像。