Geometric Analysis of Intersecting Planes | 两平面相交的几何分析

📚 Geometric Analysis of Intersecting Planes | 两平面相交的几何分析

In three-dimensional space, the intersection of two planes is one of the fundamental configurations studied in analytic geometry. Understanding this topic requires a solid grasp of plane equations, normal vectors, and parametric lines.

在三维空间中，两平面的相交是解析几何研究的基本情形之一。理解这一主题需要扎实掌握平面方程、法向量以及参数直线等概念。

1. Equations of a Plane | 平面的方程表示

A plane in 3D can be expressed in several equivalent forms. The general Cartesian equation is \(Ax+By+Cz=D\), where the vector \((A,B,C)\) is a normal vector to the plane.

三维空间中的平面可以用多种等价形式表示。一般笛卡尔方程为 \(Ax+By+Cz=D\)，其中向量 \((A,B,C)\) 是该平面的一个法向量。

Another common representation is the point-normal form: given a point \(P_0(x_0,y_0,z_0)\) on the plane and a normal vector \(\mathbf{n}=(A,B,C)\), any point \(P(x,y,z)\) on the plane satisfies \(\mathbf{n}\cdot (P-P_0)=0\).

另一种常见表示是点法式：给定平面上一点 \(P_0(x_0,y_0,z_0)\) 和法向量 \(\mathbf{n}=(A,B,C)\)，平面上任意一点 \(P(x,y,z)\) 满足 \(\mathbf{n}\cdot (P-P_0)=0\)。

For two planes, we typically write them as:

对于两个平面，我们通常将其写成：

\(\Pi_1: A_1x+B_1y+C_1z=D_1\)，\(\Pi_2: A_2x+B_2y+C_2z=D_2\)

Their normal vectors are \(\mathbf{n}_1=(A_1,B_1,C_1)\) and \(\mathbf{n}_2=(A_2,B_2,C_2)\).

它们的法向量分别为 \(\mathbf{n}_1=(A_1,B_1,C_1)\) 和 \(\mathbf{n}_2=(A_2,B_2,C_2)\)。

2. Possible Configurations of Two Planes | 两平面相交的可能情形

Two distinct planes in 3D can either intersect in a straight line or be parallel. If they are not parallel, their intersection is always a line.

三维空间中两个不同的平面要么相交成一条直线，要么平行。若不平行，它们的交集必定是一条直线。

There are exactly three possibilities:

恰好有三种可能情形：

Intersecting in a line: the planes cross along a unique straight line.
Parallel and distinct: no common points.
Coincident: the two equations represent the same plane, so they share infinitely many points.

相交于一条直线：两平面沿一条唯一的直线相交。
平行且不同：没有公共点。
重合：两个方程表示同一个平面，因此有无数个公共点。

Algebraically, the system of two linear equations in three variables has either one free variable (line), no solution (parallel), or two free variables (same plane).

从代数角度看，两个三元一次方程组成的方程组，其解要么含一个自由变量（直线），要么无解（平行），要么含两个自由变量（同一个平面）。

3. Direction Vector of the Line of Intersection | 交线的方向向量

If two planes intersect, their line of intersection lies in both planes. Consequently, the direction vector \(\mathbf{d}\) of this line must be perpendicular to both normal vectors \(\mathbf{n}_1\) and \(\mathbf{n}_2\).

若两平面相交，其交线同时位于两个平面内。因此，交线的方向向量 \(\mathbf{d}\) 必须同时垂直于两个法向量 \(\mathbf{n}_1\) 和 \(\mathbf{n}_2\)。

Hence, the cross product gives the direction:

因此，叉积给出了方向：

\(\mathbf{d}=\mathbf{n}_1 \times \mathbf{n}_2\)

If \(\mathbf{d} \neq \mathbf{0}\), the planes are not parallel and intersect in a line. If \(\mathbf{d}=\mathbf{0}\), the normals are parallel, meaning the planes are either parallel or coincident.

若 \(\mathbf{d} \neq \mathbf{0}\)，则两平面不平行且相交于一条直线。若 \(\mathbf{d}=\mathbf{0}\)，则法向量平行，意味着两平面要么平行，要么重合。

This cross product is often the quickest way to find the direction vector of the intersection line.

叉积通常是求交线方向向量最快的方法。

4. Finding a Point on the Intersection Line | 求交线上的一点

To fully determine the intersection line, we need both a direction vector and a point on the line. A convenient point is often found by setting one coordinate to a convenient value, usually \(z=0\), and solving the resulting two equations in \(x\) and \(y\).

要完全确定交线，我们需要方向向量和直线上的一点。一个便捷的方法是令某个坐标为方便的值，通常令 \(z=0\)，然后解关于 \(x\) 和 \(y\) 的两个方程。

For example, given the system:

例如，给定方程组：

\(A_1x+B_1y+C_1z=D_1\)，\(A_2x+B_2y+C_2z=D_2\)

Set \(z=0\). If the resulting two equations have a unique solution, we obtain a point \((x_0,y_0,0)\) on the line.

令 \(z=0\)。如果所得两个方程有唯一解，我们就得到直线上的一点 \((x_0,y_0,0)\)。

If setting \(z=0\) leads to a degenerate system, try \(x=0\) or \(y=0\) instead. There is always at least one coordinate that can be fixed to obtain a point.

若令 \(z=0\) 导致退化方程组，可尝试令 \(x=0\) 或 \(y=0\)。至少总有一个坐标可以被固定以求得一点。

5. Parametric Equations of the Intersection Line | 交线的参数方程

Once we have a direction vector \(\mathbf{d}=(d_x,d_y,d_z)\) and a point \(P_0(x_0,y_0,z_0)\), the parametric equations of the line are:

一旦求得方向向量 \(\mathbf{d}=(d_x,d_y,d_z)\) 和一点 \(P_0(x_0,y_0,z_0)\)，直线的参数方程为：

\(x=x_0+t d_x,\quad y=y_0+t d_y,\quad z=z_0+t d_z,\quad t\in\mathbb{R}\)

In vector form this is written as \(\mathbf{r}=\mathbf{r}_0+t\mathbf{d}\).

向量形式写作 \(\mathbf{r}=\mathbf{r}_0+t\mathbf{d}\)。

This parametric representation is especially useful for further computations such as distances or intersections with other objects.

这种参数表示在后续计算距离或与其他几何对象求交时特别有用。

6. Parallel and Coincident Planes | 平行与重合平面

Two planes are parallel if their normal vectors are scalar multiples of each other:

若两个平面的法向量互为倍数，则两平面平行：

\(\mathbf{n}_1 = k \mathbf{n}_2\) for some \(k\neq 0\)

If the equations are \(A_1x+B_1y+C_1z=D_1\) and \(A_2x+B_2y+C_2z=D_2\), then parallelism is equivalent to:

若方程分别为 \(A_1x+B_1y+C_1z=D_1\) 与 \(A_2x+B_2y+C_2z=D_2\)，则平行等价于：

\(\frac{A_1}{A_2}=\frac{B_1}{B_2}=\frac{C_1}{C_2}\neq \frac{D_1}{D_2}\)

If the ratios are all equal, including \(D_1/D_2\), then the planes are coincident.

如果所有比值都相等，包括 \(D_1/D_2\)，则两平面重合。

7. Perpendicular Planes and the Dihedral Angle | 垂直平面与二面角

Two planes are perpendicular if their normal vectors are perpendicular:

若两平面的法向量互相垂直，则两平面垂直：

\(\mathbf{n}_1 \cdot \mathbf{n}_2 = 0\)

The angle between two planes, known as the dihedral angle, is defined as the acute or obtuse angle between their normals. Let \(\theta\) be the angle between \(\mathbf{n}_1\) and \(\mathbf{n}_2\). Then:

两平面之间的角称为二面角，定义为它们的法向量之间的锐角或钝角。设 \(\theta\) 为 \(\mathbf{n}_1\) 与 \(\mathbf{n}_2\) 之间的夹角，则：

\(\cos\theta=\frac{\mathbf{n}_1\cdot\mathbf{n}_2}{|\mathbf{n}_1||\mathbf{n}_2|}\)

The acute angle between the planes is \(\min(\theta, 180^\circ-\theta)\). In IB problems, the angle is usually requested as the acute angle.

两平面间的锐角为 \(\min(\theta, 180^\circ-\theta)\)。在 IB 考试中，通常要求的是锐角。

8. Worked Example: Finding the Line of Intersection | 例题：求交线

Consider the two planes:

考虑两个平面：

\(\Pi_1: x+2y-z=3\)，\(\Pi_2: 2x-y+3z=1\)

Step 1: Find the direction vector.

第一步：求方向向量。

\(\mathbf{n}_1=(1,2,-1),\quad \mathbf{n}_2=(2,-1,3)\)

\(\mathbf{d}=\mathbf{n}_1\times\mathbf{n}_2 = (2\cdot 3-(-1)\cdot(-1),\ (-1)\cdot 2-1\cdot 3,\ 1\cdot(-1)-2\cdot 2)\)

\(\mathbf{d}=(6-1,\ -2-3,\ -1-4)=(5,\ -5,\ -5)\)

We can simplify the direction vector by dividing by 5, giving \(\mathbf{d}=(1,-1,-1)\).

我们可以将方向向量除以 5 化简，得到 \(\mathbf{d}=(1,-1,-1)\)。

Step 2: Find a point on the line. Set \(z=0\):

第二步：求直线上一点。令 \(z=0\)：

\(x+2y=3,\quad 2x-y=1\)

Solving gives \(x=1\), \(y=1\). Thus \(P_0=(1,1,0)\).

解得 \(x=1\)，\(y=1\)。因此 \(P_0=(1,1,0)\)。

Step 3: Write the parametric equations:

第三步：写出参数方程：

\(x=1+t,\quad y=1-t,\quad z=-t\)

This is the complete line of intersection.

这就是完整的交线方程。

9. Using the Cartesian Form of a Line | 使用直线的一般式

The intersection line can also be written in Cartesian (symmetric) form by eliminating the parameter \(t\). From the parametric equations above, we have:

交线也可以通过消去参数 \(t\) 写成笛卡尔（对称）形式。由上述参数方程，可得：

\(t=x-1,\quad t=1-y,\quad t=-z\)

Hence:

因此：

\(\frac{x-1}{1}=\frac{y-1}{-1}=\frac{z}{-1}\)

This form is useful when reading off the direction vector directly. However, some exam questions prefer the parametric form, so always check what is required.

这种形式便于直接读出方向向量。不过，有些考试题目更偏好参数形式，因此务必看清题目要求。

10. Special Cases and Common Pitfalls | 特殊情况与常见误区

One common mistake is to assume that two planes always intersect in a line. Always check whether the normal vectors are parallel first.

一个常见错误是假定两个平面总是相交成一条直线。务必先检查法向量是否平行。

Another pitfall is using the angle between the planes directly as the angle between their normals without taking the acute supplement. If \(\cos\theta <0\), the acute angle is \(180^\circ-\theta\).

另一个误区是直接将法向量夹角作为二面角，而未取其补角。若 \(\cos\theta<0\)，锐角应为 \(180^\circ-\theta\)。

When finding a point on the line, if eliminating one variable leads to an inconsistency, choose a different variable to set to zero. The line may be parallel to a coordinate plane, making one substitution fail.

在求直线上一点时，若消去某个变量导致矛盾，可选择令另一个变量为零。交线可能与某个坐标平面平行，使得某一种代入失效。

Finally, always simplify the direction vector if possible, but be careful not to change its sign unless the direction is arbitrary.

最后，尽可能化简方向向量，但要注意不要随意改变符号，除非方向本身是任意的。

11. Intersection with a Third Plane | 与第三个平面相交

When a line of intersection of two planes is further intersected with a third plane, we can substitute the parametric equations of the line into the third plane’s equation to find the intersection point.

当两平面的交线进一步与第三个平面相交时，我们可以将直线的参数方程代入第三个平面的方程，从而求得交点。

For example, if \(\Pi_3: x+y+z=0\), then substituting \(x=1+t\), \(y=1-t\), \(z=-t\) gives:

例如，若 \(\Pi_3: x+y+z=0\)，代入 \(x=1+t\)，\(y=1-t\)，\(z=-t\) 得：

\((1+t)+(1-t)+(-t)=0 \Rightarrow 2-t=0 \Rightarrow t=2\)

Thus the point is \((3,-1,-2)\).

因此交点为 \((3,-1,-2)\)。

This technique is widely used in problems involving three planes and in vector geometry examination questions.

这种技巧广泛用于涉及三个平面的问题以及向量几何考题中。

12. Geometric Interpretation and Problem-Solving Strategy | 几何解释与解题策略

Geometrically, the line of intersection is where two infinite sheets of paper cross. The direction vector is perpendicular to both normals, which is why the cross product is the natural tool.

从几何上看，交线就是两个无限平面纸片交叉的地方。方向向量同时垂直于两个法向量，这正是叉积成为自然工具的原因。

A reliable problem-solving strategy is:

可靠的解题策略如下：

Write both plane equations in standard form.
Compute the cross product of the normals to find the direction vector.
Solve for a common point by setting one coordinate to zero.
Write the parametric or symmetric equations.
If angles are requested, use the dot product formula.

将两个平面方程写成标准形式。
计算法向量的叉积以求得方向向量。
令一个坐标为零，解出公共点。
写出参数方程或对称方程。
若要求角度，使用点积公式。

Mastering this process builds confidence in tackling 3D geometry problems in IB Mathematics Analysis and Approaches HL.

掌握这一过程能够帮助你在 IB 数学分析与方法 HL 中自信地解决三维几何问题。

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Geometric Analysis of Intersecting Planes | 两平面相交的几何分析