n维球面和复射影空间CP^n是现代几何研究中的重要例子，它们在拓扑、泛函分析等数学分支领域以及物理、计算机、医学等其他领域也都有重要应用。n维球面和CP^n上的距离是n维球面和CP^n上的重要属性，对n维球面和复射影空间的微分性质研究具有重要意义。 本文着重介绍n维球面和复射影空间上的距离计算。具体的，本文首先介绍了n维球面上的球面距离，然后通过2n+1维单位球面（S^2n+1）构造出CP^n，并定义其上距离，接着在球面距离的基础上得到CP^n上的距离通式，最后在n=1时对CP^1计算其上两点间距离，通过对距离的分析得出CP^1的一些特殊性质。 本文的关键内容集中在CP^n的距离通式以及CP^1的特殊性质。前者在只给出CP^n上两点的情况下，运用Cauchy 不等式得到一个距离通式，不涉及坐标运算，因此在任何形式的CP^n下都可以运用；后者通过CP^1上的球面余弦公式，表现了CP^1和二维球面（S^2）之间的同构关系。

The S^n and CP^n with canonical metric are important examples in modern geometry. They have many applications in other branches of mathematics such as topology and functional analysis, and also in other fields, such as physics, medicine and CS. The distance on them is an important concept, which is the most basic geometry of S^n or CP^n. The main topic herein is to calculate the distance between any two points on S^n or CP^n. Firstly, we introduce the distance on S^n. Then we give the definitions of CP^n and its distance. As an example, we calculate and analyze the distance of CP^1 at last. The key points of the article concentrate on the general distance formula of CP^n and the concrete one of CP^1. As for the former, we get the general formula through the Cauchy-Schwarz inequality without any coordinate, so that this formula will work in any coordinate system of CP^n. As for the latter, we found the isomorphism between CP^1 and S^2(1/2) through the spherical cosine formula on S^2(1/2).