📚 Introduction to Three-Dimensional Coordinate Systems | 三维坐标系入门
The three-dimensional coordinate system is a natural extension of the two-dimensional Cartesian plane. It allows us to describe the location of any point in space using three numbers, usually called x, y and z. This system forms the foundation of solid geometry, vector calculus, and many areas of physics and engineering.
三维坐标系是二维笛卡尔平面的自然扩展。它使我们能够用三个数字（通常称为 x、y 和 z）来描述空间中任意点的位置。这一体系是立体几何、向量微积分以及许多物理和工程领域的基础。
1. Coordinate Axes in 3D | 三维坐标轴
In three dimensions, we use three mutually perpendicular axes: the x-axis, the y-axis, and the z-axis. These axes meet at a single point called the origin, denoted by O. The positive directions are usually chosen so that the axes form a right-handed system: if you curl the fingers of your right hand from the positive x-axis toward the positive y-axis, your thumb points along the positive z-axis.
在三维空间中，我们使用三条互相垂直的轴：x 轴、y 轴和 z 轴。这三条轴交汇于一个点，称为原点，记作 O。正方向通常按右手系选取：如果用右手手指从 x 轴正方向弯向 y 轴正方向，大拇指所指方向即为 z 轴正方向。
Each pair of axes defines a coordinate plane. The xy-plane contains the x- and y-axes, the yz-plane contains the y- and z-axes, and the xz-plane contains the x- and z-axes. These three planes divide space into eight octants, similar to how the two axes in 2D divide the plane into four quadrants.
每两根轴决定一个坐标平面。xy 平面包含 x 轴和 y 轴，yz 平面包含 y 轴和 z 轴，xz 平面包含 x 轴和 z 轴。这三个平面把空间分成八个卦限，类似于二维中两条轴把平面分成四个象限。
2. Ordered Triples and Coordinates | 有序三元组与坐标
A point in three-dimensional space is represented by an ordered triple of real numbers: (x, y, z). The first number is the x-coordinate, the second is the y-coordinate, and the third is the z-coordinate. For example, the point P(3, -2, 4) has x = 3, y = -2, and z = 4.
三维空间中的点用有序实数三元组 (x, y, z) 表示。第一个数是 x 坐标，第二个数是 y 坐标，第三个数是 z 坐标。例如，点 P(3, -2, 4) 表示 x = 3，y = -2，z = 4。
The order of the numbers is crucial. The point (1, 2, 3) is different from (3, 2, 1). Each ordered triple corresponds to exactly one point in space, and each point corresponds to exactly one ordered triple. This one-to-one correspondence is the essence of the coordinate system.
数字的顺序至关重要。点 (1, 2, 3) 不同于 (3, 2, 1)。每个有序三元组对应空间中唯一的一个点，每个点也对应唯一的有序三元组。这种一一对应关系是坐标系的本质。
3. Plotting Points in Space | 在空间中描点
To plot a point such as A(2, 3, 4), start at the origin. Move 2 units along the positive x-axis, then 3 units in the direction parallel to the positive y-axis, and finally 4 units in the direction parallel to the positive z-axis. The final position is point A.
要描出点 A(2, 3, 4)，从原点出发。沿 x 轴正方向移动 2 个单位，再沿平行于 y 轴正方向移动 3 个单位，最后沿平行于 z 轴正方向移动 4 个单位。最终位置即为点 A。
In practice, we often draw a three-dimensional box or use perspective to show depth. A dashed line from the origin to the point can help indicate its position relative to the axes. Negative coordinates imply movement in the opposite direction along the corresponding axis.
实际操作中，我们常画出三维盒子或使用透视图来表现深度。从原点到该点的虚线有助于表示其相对于各轴的位置。负坐标意味着沿相应轴的反方向移动。
4. Distance Formula in 3D | 三维距离公式
The distance between two points P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂) in space is given by the three-dimensional distance formula:
空间中两点 P₁(x₁, y₁, z₁) 与 P₂(x₂, y₂, z₂) 之间的距离由三维距离公式给出：
d = √[(x₂ – x₁)² + (y₂ – y₁)² + (z₂ – z₁)²]
This formula is derived by applying the Pythagorean theorem twice. First, find the distance in the xy-plane, then combine it with the vertical difference. For example, the distance between A(1, 2, 2) and B(4, 6, 5) is √[(4-1)² + (6-2)² + (5-2)²] = √(9 + 16 + 9) = √34.
该公式通过两次应用勾股定理推导得出。首先计算 xy 平面上的距离，再将其与竖直差值结合。例如，A(1, 2, 2) 与 B(4, 6, 5) 之间的距离为 √[(4-1)² + (6-2)² + (5-2)²] = √(9 + 16 + 9) = √34。
5. Midpoint Formula in 3D | 三维中点公式
The midpoint M of the segment joining two points P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂) is found by averaging the corresponding coordinates:
连接两点 P₁(x₁, y₁, z₁) 与 P₂(x₂, y₂, z₂) 的线段中点 M 通过取各对应坐标的平均值得到：
M = ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2)
This is a direct extension of the midpoint formula in two dimensions. For example, the midpoint between A(2, -1, 3) and B(4, 5, -3) is ((2+4)/2, (-1+5)/2, (3+(-3))/2) = (3, 2, 0).
这是二维中点公式的直接扩展。例如，A(2, -1, 3) 与 B(4, 5, -3) 之间的中点为 ((2+4)/2, (-1+5)/2, (3+(-3))/2) = (3, 2, 0)。
6. Projections onto Coordinate Planes | 在坐标平面上的投影
Given a point P(x, y, z), its projection onto the xy-plane is obtained by setting z = 0, giving (x, y, 0). Similarly, the projection onto the yz-plane is (0, y, z), and onto the xz-plane is (x, 0, z). These projections are useful when analysing the position of a point relative to each plane.
给定点 P(x, y, z)，其在 xy 平面上的投影通过令 z = 0 得到，即 (x, y, 0)。类似地，在 yz 平面上的投影为 (0, y, z)，在 xz 平面上的投影为 (x, 0, z)。这些投影有助于分析点相对于各平面的位置。
For instance, the point P(3, -2, 5) projects to (3, -2, 0) on the xy-plane, (0, -2, 5) on the yz-plane, and (3, 0, 5) on the xz-plane. The distance from P to the xy-plane is |z|, to the yz-plane is |x|, and to the xz-plane is |y|.
例如，点 P(3, -2, 5) 在 xy 平面上的投影为 (3, -2, 0)，在 yz 平面上的投影为 (0, -2, 5)，在 xz 平面上的投影为 (3, 0, 5)。点 P 到 xy 平面的距离为 |z|，到 yz 平面的距离为 |x|，到 xz 平面的距离为 |y|。
7. Special Points and Symmetry | 特殊点与对称性
Points on the coordinate axes have two coordinates equal to zero. For example, a point on the x-axis has the form (a, 0, 0). Points on a coordinate plane have one coordinate equal to zero. The origin is the point (0, 0, 0), where all three axes meet.
坐标轴上的点有两个坐标为零。例如，x 轴上的点形式为 (a, 0, 0)。坐标平面上的点有一个坐标为零。原点是 (0, 0, 0)，即三条轴的交点。
Symmetry in 3D can be described using reflections. The reflection of a point (x, y, z) across the xy-plane is (x, y, -z); across the yz-plane is (-x, y, z); across the xz-plane is (x, -y, z). Reflection through the origin gives (-x, -y, -z). Understanding symmetry helps simplify problems involving three-dimensional shapes.
三维中的对称性可以用反射来描述。点 (x, y, z) 关于 xy 平面的反射为 (x, y, -z)；关于 yz 平面的反射为 (-x, y, z)；关于 xz 平面的反射为 (x, -y, z)。关于原点的反射为 (-x, -y, -z)。理解对称性有助于简化涉及三维形状的问题。
8. Vectors and Position Vectors | 向量与位置向量
Every point P(x, y, z) in space has a position vector that starts at the origin and ends at P. This vector is often written as r = xi + yj + zk, where i, j, and k are unit vectors along the x-, y-, and z-axes respectively. The components of the vector are exactly the coordinates of the point.
空间中每个点 P(x, y, z) 都有一个位置向量，该向量从原点出发并终止于点 P。此向量常写为 r = xi + yj + zk，其中 i、j、k 分别是沿 x 轴、y 轴、z 轴的单位向量。向量的分量正是该点的坐标。
The vector between two points P₁ and P₂ is obtained by subtracting their position vectors: P₁P₂ = (x₂ – x₁)i + (y₂ – y₁)j + (z₂ – z₁)k. The magnitude of this vector is exactly the distance between the two points.
两点 P₁ 与 P₂ 之间的向量通过它们的位矢相减得到：P₁P₂ = (x₂ – x₁)i + (y₂ – y₁)j + (z₂ – z₁)k。该向量的模长恰好是两点之间的距离。
9. Equations of Simple Surfaces | 简单曲面的方程
In three-dimensional coordinate geometry, equations in x, y and z describe surfaces rather than curves. For example, the equation x = a represents a plane parallel to the yz-plane. The equation y = b represents a plane parallel to the xz-plane, and z = c represents a plane parallel to the xy-plane.
在三维坐标几何中，含 x、y、z 的方程描述的是曲面而非曲线。例如，方程 x = a 表示平行于 yz 平面的平面；y = b 表示平行于 xz 平面的平面；z = c 表示平行于 xy 平面的平面。
The equation x² + y² + z² = r² represents a sphere centred at the origin with radius r. More generally, (x – a)² + (y – b)² + (z – c)² = r² is a sphere centred at (a, b, c). These equations are essential for solving problems involving solid shapes in coordinate space.
方程 x² + y² + z² = r² 表示以原点为球心、半径为 r 的球面。更一般地，(x – a)² + (y – b)² + (z – c)² = r² 表示以 (a, b, c) 为球心的球面。这些方程对于解决坐标空间中立体形状的问题至关重要。
10. Applications and Importance | 应用与重要性
Three-dimensional coordinate systems are widely used in physics to describe the position of objects in space, in computer graphics to render 3D scenes, and in engineering to model structures. They also form the basis for studying functions of several variables and vector fields in higher mathematics.
三维坐标系在物理学中广泛用于描述物体在空间中的位置，在计算机图形学中用于渲染三维场景，在工程学中用于建模结构。它们也是学习多元函数和向量场等高等数学内容的基础。
Mastering the basic concepts of 3D coordinates — plotting points, finding distances and midpoints, and understanding symmetry — prepares students to tackle more advanced topics such as lines and planes, dot products, cross products, and triple integrals.
掌握三维坐标的基本概念——描点、求距离和中点、理解对称性——为学生进一步学习直线与平面、点积、叉积和三重积分等高级主题做好准备。
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