Further Mathematics: 3D3 How Shadows Form | 进阶数学：3D3 阴影的形成

📚 Further Mathematics: 3D3 How Shadows Form | 进阶数学：3D3 阴影的形成

Understanding how shadows are formed in three-dimensional space is a practical application of vector geometry. In A-level Further Mathematics, this topic brings together rays of light, planes, and straight lines, allowing you to model shadows cast by point sources or parallel light onto flat surfaces. By treating light paths as vector lines and surfaces as planes, you can calculate exactly where a shadow falls, determine its shape, and explore special cases such as no shadow or shadows at infinity. This article will guide you through the core concepts, methods, and worked examples you need for the 3D3 topic.

理解三维空间中阴影的形成是向量几何的实际应用。在A-level进阶数学中，这一主题将光线、平面和直线联系起来，使你能够对点光源或平行光投射在平面上的阴影进行建模。通过将光线路径视为向量直线，把表面视为平面，你可以精确计算阴影的落点、确定其形状，并探究无阴影或无限远阴影等特殊情况。本文将为你梳理3D3主题的核心概念、方法和例题。

1. Introduction to Shadows in 3D | 三维阴影简介

A shadow is the region where light from a source is blocked by an object. In vector geometry, we model the light source, the object, and a receiving surface (often a plane) to find the shadow. The object can be a point, a line segment, or a polygon. The light can come from a single point (like a lamp) or from a distant source producing parallel rays (like the sun). In both cases, the shadow of a point is the intersection of a ray with the receiving plane.

阴影是光源被物体遮挡后形成的区域。在向量几何中，我们通过建立光源、物体和接收平面（通常为平面）的模型来求阴影。物体可以是一个点、一条线段或多边形。光线可以来自一个点光源（如灯泡），也可以来自产生平行光的远处光源（如太阳）。在这两种情况下，一个点的阴影就是光线与接收平面的交点。

2. Types of Light Sources | 光源类型

We distinguish between two main types of light source in shadow problems: point lights and directional (parallel) lights. Each produces a different type of projection.

阴影问题中我们区分两种主要光源：点光源和平行光。每种光源产生不同类型的投影。

Point light source – rays diverge from a single point in space. The shadow is a central projection.
点光源 – 光线从空间中的单个点发散出来。阴影为中心投影。
Directional light source – all rays are parallel to a fixed direction vector. This models sunlight and gives an orthogonal or oblique projection depending on the plane.
平行光源 – 所有光线都平行于一个固定方向向量。这可以模拟阳光，根据平面的不同产生正交或斜投影。

In both cases, the fundamental task is to write down a vector equation for a ray and intersect it with a plane.

无论是哪种情况，基本任务都是写出光线的向量方程并求其与平面的交点。

3. Parametric Equation of a Ray | 光线的参数方程

For a point light source at L, a ray passing through an object point P is given by:

对于位于 L 的点光源，穿过物体点 P 的光线表示为：

r = L + t (P − L), t ≥ 0

Here t is a scalar parameter. When t = 0, r = L (the source); when t = 1, r = P. Shadow points require t > 1 if the object is between the source and the plane. If the object is behind the source relative to the plane, the shadow may correspond to t < 0, which is usually not physically meaningful for a point source.

其中 t 是标量参数。当 t = 0 时，r = L（光源）；当 t = 1 时，r = P。若物体位于光源与平面之间，阴影点对应的 t > 1。若物体相对于平面在光源后方，阴影可能对应 t < 0，这对于点光源通常没有物理意义。

For a directional light with direction vector d, a ray starting at object point P is:

对于方向向量为 d 的平行光，从物体点 P 出发的光线为：

r = P + t d, t ≥ 0

In this case, t measures how far along the ray we travel from P in the direction of d.

此时，t 表示从 P 沿 d 方向前行的距离。

4. Equation of a Plane (Shadow Catcher) | 投影平面的方程

A receiving plane can be defined in vector form as r ⋅ n = d, where n is a normal vector and d is a constant. Common shadow planes in exam problems include the ground plane (z = 0), walls (e.g. x = constant or y = constant), or inclined planes given by Cartesian or vector equations.

接收平面可以用向量形式 r ⋅ n = d 表示，其中 n 为法向量，d 为常数。考试中常见的阴影平面包括地平面 (z = 0)、墙壁 (如 x = 常数或 y = 常数)，或由笛卡儿方程或向量方程给出的倾斜平面。

Plane description / 平面描述	Equation / 方程
Ground plane / 地平面	z = 0 or r ⋅ (0,0,1) = 0
Vertical wall / 竖直墙壁	x = 5 or r ⋅ (1,0,0) = 5
Inclined plane / 倾斜平面	2x + y − z = 4

To find the shadow, we substitute the ray equation into the plane equation and solve for t. Once t is known, we substitute back to get the shadow coordinates.

为求阴影，我们将光线方程代入平面方程并解出 t。得到 t 后，再代回光线方程即可得到阴影坐标。

5. Finding the Shadow of a Point | 求点的阴影

Given a light source (point or directional), an object point P, and a receiving plane, follow these steps:

给定光源（点光源或平行光）、物体点 P 及接收平面，请按以下步骤求解：

Write the ray equation in parametric form.
写出光线的参数方程。
Express the coordinates (x, y, z) in terms of t.
用 t 表示坐标 (x, y, z)。
Substitute these expressions into the plane equation (Cartesian or dot product form).
将这些表达式代入平面方程（笛卡儿形式或点积形式）。
Solve for t. If t ≥ 0, the shadow exists; if t is negative, the ray does not reach the plane in the forward direction.
解出 t。若 t ≥ 0，则阴影存在；若 t 为负，则光线沿正向无法到达平面。
Use the t value to compute the shadow point S = (x, y, z).
利用 t 值计算阴影点 S = (x, y, z)。

This method works for any combination of point/directional light and any plane, provided the ray is not parallel to the plane (in which case there may be no intersection or infinitely many).

只要光线不平行于平面（否则可能无交点或有无穷多个交点），该方法适用于任何点光源或平行光与任意平面的组合。

6. Worked Example: Point Shadow from a Point Light | 例题：点光源下点的阴影

Problem: A point light source is at L(2, 3, 4). Find the shadow of the point P(5, 1, 2) on the ground plane z = 0.

问题：一个点光源位于 L(2, 3, 4)，求点 P(5, 1, 2) 在地平面 z = 0 上的阴影。

Solution:

解：

Ray equation: r = (2,3,4) + t[(5,1,2) − (2,3,4)] = (2,3,4) + t(3,−2,−2).

光线方程：r = (2,3,4) + t[(5,1,2) − (2,3,4)] = (2,3,4) + t(3,−2,−2)。

Coordinates: x = 2 + 3t, y = 3 − 2t, z = 4 − 2t.

坐标：x = 2 + 3t, y = 3 − 2t, z = 4 − 2t。

Substitute into plane z = 0: 4 − 2t = 0 → t = 2. Since t = 2 > 0, the shadow is valid.

代入平面 z = 0：4 − 2t = 0 → t = 2。由于 t = 2 > 0，阴影有效。

Shadow point S: x = 2 + 3(2) = 8, y = 3 − 2(2) = −1, z = 0. Hence S(8, −1, 0).

阴影点 S：x = 2 + 3(2) = 8, y = 3 − 2(2) = −1, z = 0。因此 S(8, −1, 0)。

This demonstrates that the point’s shadow appears farther from the origin when the light is above and the object is higher than the ground.

这表明当光源在上方且物体高于地面时，点的阴影会距原点更远。

7. Shadow of a Line Segment | 线段的阴影

To find the shadow of a line segment joining points A and B, you calculate the shadows of the two endpoints, A’ and B’, using the same light source and plane. Provided the entire segment is illuminated in front of the plane, the shadow of the segment is the line segment joining A’ and B’.

要求连接点 A 和 B 的线段的阴影，你可以使用相同的光源和平面分别计算两个端点 A 和 B 的阴影 A’ 和 B’。只要整条线段在平面前方被照亮，该线段的阴影即为连接 A’ 与 B’ 的线段。

You must check that for all points on AB, the ray parameter t is non-negative. If one endpoint gives t < 0, the shadow may be a ray (infinite) rather than a segment. If both end points produce t < 0, there is no shadow on that side of the plane.

你必须检查 AB 上所有点的光线参数 t 是否非负。若某个端点给出 t < 0，阴影可能是射线（无限）而非线段。若两端点均产生 t < 0，则平面那一侧没有阴影。

8. Worked Example: Shadow of a Segment on Ground Plane | 例题：线段在地面上的阴影

Problem: A directional light has direction d = (1, 1, −2). Find the shadow of the segment from A(2, 0, 3) to B(4, 2, 1) on the plane z = 0.

问题：一平行光的方向向量为 d = (1, 1, −2)。求从 A(2, 0, 3) 到 B(4, 2, 1) 的线段在平面 z = 0 上的阴影。

Solution:

解：

For A: Ray r = (2,0,3) + t(1,1,−2). z-coordinate: 3 − 2t = 0 → t = 1.5. Then x = 2 + 1.5 = 3.5, y = 0 + 1.5 = 1.5. Shadow A'(3.5, 1.5, 0).

对于 A：光线 r = (2,0,3) + t(1,1,−2)。z 坐标：3 − 2t = 0 → t = 1.5。则 x = 2 + 1.5 = 3.5，y = 0 + 1.5 = 1.5。阴影 A'(3.5, 1.5, 0)。

For B: r = (4,2,1) + t(1,1,−2). z: 1 − 2t = 0 → t = 0.5. x = 4 + 0.5 = 4.5, y = 2 + 0.5 = 2.5. Shadow B'(4.5, 2.5, 0).

对于 B：r = (4,2,1) + t(1,1,−2)。z：1 − 2t = 0 → t = 0.5。x = 4 + 0.5 = 4.5，y = 2 + 0.5 = 2.5。阴影 B'(4.5, 2.5, 0)。

Both t values are positive. Therefore the shadow of AB is the segment joining A'(3.5,1.5,0) and B'(4.5,2.5,0). The shadow lies entirely on the ground plane.

两个 t 值均为正，因此 AB 的阴影为连接 A'(3.5,1.5,0) 与 B'(4.5,2.5,0) 的线段。阴影完全落在地平面上。

9. Shadows under Parallel Light (Sunlight) | 平行光（阳光）下的阴影

When a directional light is used, all rays share a common direction vector d. To find the shadow of an object point P, you simply solve r = P + t d together with the plane equation. Because there is no single source point, we usually do not worry about t = 0 being the source; any t ≥ 0 gives a point on the ray that could form a shadow.

当使用平行光时，所有光线共享一个共同的方向向量 d。要求物体点 P 的阴影，只需将 r = P + t d 与平面方程联立求解即可。因为没有单一光源点，我们通常不关心 t = 0 是否对应光源；任何 t ≥ 0 的值都可以给出光线上可能形成阴影的点。

In sunlight problems, the light direction is often given as a vector or as angles (e.g. azimuth and elevation). The vector method works just as for point lights, except the ray starts at the object point.

在阳光问题中，光的方向通常用一个向量或方位角与仰角给出。向量方法与点光源完全类似，只是光线从物体点出发。

10. Shadow of a Triangle and Polygons | 三角形与多边形的阴影

For a triangle defined by vertices A, B, C, the shadow on a plane is generally another triangle A’B’C’, provided all vertices produce valid shadows (t ≥ 0) and the triangle does not lie parallel to the light rays. The shape of the shadow is obtained by finding the shadow of each vertex and connecting them in the same order.

对于由顶点 A、B、C 定义的三角形，若所有顶点均产生有效阴影 (t ≥ 0) 且三角形不与光线平行，则其在平面上的阴影一般为另一个三角形 A’B’C’。通过求取每个顶点的阴影并按相同顺序连接，即可得到阴影的形状。

If some vertices have t < 0 while others have t ≥ 0, the shadow may be an infinite region or require clipping. Also, if the triangle is parallel to the rays, its shadow will collapse to a line segment or a point.

若部分顶点的 t < 0 而其他顶点的 t ≥ 0，阴影可能是无限区域或需要进行裁剪。此外，若三角形与光线平行，其阴影将退化为一条线段或一个点。

For polygons, the same vertex-by-vertex method applies. The shadow’s area can even be compared to the original’s using scalar factors related to the geometry.

对于多边形，同样逐点投影即可。阴影面积甚至可以通过几何关系与原始面积进行比较，涉及比例因子。

11. Special Cases: No Shadow or Infinite Shadow | 特殊情况：无阴影或无限阴影

Several special cases can arise in shadow problems. You should be able to identify and handle them.

阴影问题中可能出现几种特殊情况。你应当能够识别并处理这些情况。

Ray parallel to the plane: If the direction vector d (or vector P − L) is perpendicular to the plane’s normal n, the ray will never intersect the plane (unless it lies in the plane). No unique shadow exists.
光线平行于平面：若方向向量 d（或 P − L）垂直于平面法向量 n，则光线永远不会与平面相交（除非位于平面内）。此时不存在唯一的阴影。
Point source behind the plane: If the light source and the object point are on opposite sides of the plane, the ray may intersect the plane at a negative t, which we typically reject. Thus no shadow on the designated side.
点光源位于平面后方：若光源与物体点位于平面的异侧，光线可能与平面相交于 t 为负处，通常我们会舍去。因此该侧无阴影。
Object point lies on the plane: Its shadow is the point itself, and t equals some specific value (often 0 for directional light from the point, or a specific value for point light if the point is between source and plane).
物体点位于平面上：其阴影即该点本身，t 为某一特定值（对于平行光从该点发出，t = 0；对于点光源，若该点位于光源与平面之间则为另一特定值）。
Infinitely distant shadow: For a point light and an object far away, t can be very large, but mathematically we still compute it.
无限远阴影：对于点光源和远处物体，t 可以非常大，但数学上我们仍能计算。

Being aware of these cases prevents you from blindly solving equations and helps interpret results physically.

了解这些情况可以防止盲目解方程，并有助于从物理意义上解释结果。

12. Conclusion and Revision Tips | 总结与复习提示

Shadow formation in 3D vectors is essentially an intersection problem: you parametrise a light ray and find where it meets a plane. By mastering this sequence, you secure marks on a topic that frequently appears in A-level Further Mathematics exams. Remember to define your ray carefully, check the sign of t, and interpret the physical situation. Practising with different light types and plane orientations will build your confidence.

三维向量中的阴影形成本质上是一个交点问题：将光线参数化，并求其与平面的交点。掌握这一流程，你就能在A-level进阶数学考试中稳拿这一常见题型的分数。请务必仔细列出光线方程，检查 t 的符号，并结合实际情况进行解释。通过练习不同光源类型和平面方向，你将逐渐建立信心。

Key revision points: write ray equations for point and directional lights; substitute into plane equation; solve for t; check t ≥ 0; compute shadow coordinates; and consider special cases like parallel rays or sign issues.

复习要点：写出点光源和平行光的光线方程；代入平面方程；解出 t；检查 t ≥ 0；计算阴影坐标；并考虑光线平行或符号问题等特殊情况。

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Further Mathematics: 3D3 How Shadows Form | 进阶数学：3D3 阴影的形成