二面角最小值

创建于：2025年1月21日

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问题

在平面四边形ABCD中，AB = AC = CD = 1,\angle ADC = 30^{\circ},\angle DAB = 120^{\circ}。将\triangle ACD沿AC翻折至\triangle ACP，其中P为动点。
求二面角A - CP - B的余弦值的最小值。

思考

Mapping out the folds

I’m unfolding the problem step-by-step, focusing on how the folding of triangle ACD and reflections create a dihedral angle between planes ACB and ACP, using line CP as the intersection.

Mapping the geometry

I'm exploring interpretations of the dihedral angle form by planes A-CP and B-CP, hinging around line CP, and the cosine value to maximize the potential understanding of this 3D geometric scenario.

Evaluating angles

I'm considering the angle between planes ACP and BCP, hinged around the moving point P, and how D’s reflective movement around AC affects the angle or distance constraints.

Piecing together the perspective

I’m examining the geometric scenario where triangle ACD is folded along line AC, creating a 3D figure with B in the plane and D moving out, forming point P as a moving point.

Piecing together the transformation

I’m thinking about how point P is the pivoted image of D in 3D, with D rotating around line AC, creating constraints for congruent segments and specific angles.

Understanding the angle

I’m analyzing the minimal cosine value of the dihedral angle between lines CA and CB when projected onto a normal plane to CP, as P shifts.

Pinpointing the target

I’m analyzing the dihedral angle between planes ACP and BCP, focusing on the minimal cosine value influenced by the fold angle and given geometric constraints.

Applying the law

I’m using the law of cosines to find segment AD in triangle ACD, factoring in given side lengths and angle ∠ADC to determine AD = √(2 - √3).

Revisiting the approach

I'm thinking about 3D coordinates to calculate the dihedral angle between planes ACP and BCP, ensuring the angle is equivalent to the angle between vectors perpendicular to CP.

Mapping B's coordinates

I’m setting B's position based on given constraints, ensuring AB=1 and incorporating angles to seamlessly integrate D into the plane.

Setting coordinates

I’m positioning A and C, ensuring AB=1, and aiming to place B on the y-axis to simplify the geometry, aligning with the given angle constraints.

Mapping coordinates

I’ve placed B at (0,1) and started defining D’s coordinates to satisfy AC=1, CD=1, and ∠ADC=30°, ensuring all geometric constraints are met.

Tracing D's coordinates

I’m defining D’s coordinates by ensuring the angle between vectors AD and CD is 30°, and using the dot product to verify the geometric constraints align properly.

Confirming geometric relationships

I’m verifying the relationship between vectors AD and DC in triangle ADC, ensuring the used geometric constraints are accurate and align correctly.

Solving vector constraints

I’m working through the constraints by setting equations for vectors AD and CD, then incorporating the angle size, to ensure all geometric conditions are satisfied.

Establishing coordinates

I’ve squared and multiplied to stream the equations for vectors AD and CD, ensuring the constraints hold true for D’s coordinates to fit all geometric criteria.

Determining coordinates

I’ve concluded that D's coordinates are x=1.5 and y=-0.866025..., aligning with the geometric constraints established for angles and segment lengths.

Verifying constraints

Piecing together D’s coordinates, I’ve confirmed D= (3/2, -sqrt(3)/2) meets the distance and angle conditions, matching the geometric requirements accurately.

Mapping out rotation

I’m detailing the 3D rotation of D around line AC, tracking the dihedral angle between planes ACP(t) and BCP(t) as the fold angle, t, changes from 0 to 180 degrees.

Calculating dihedral angles

I’m mapping out the dihedral angle between planes ACP and BCP, defined by vectors AC, AP, BC, and BP, with CP as the intersection, to identify the minimum cosine value of that angle.

Rotating in 3D

I'm focusing on rotating point D around line AC to find the minimum cosine value of the dihedral angle between planes ACP and BCP.

Ensuring origin consistency

I’m adjusting my focus to confirm the x-axis rotation through the origin keeps point C fixed since any point with y=0 and z=0 stays in place.

Mapping rotation

I'm figuring out the new positions of points A, C, and P around 3D, rotating around the x-axis by angle θ, ensuring the x-axis rotation keeps point C fixed.

Breaking down the cross product

I’m working through the cross product of vectors AC and AP(θ), using their components to calculate the normal vector n1(θ) that identifies the angle between planes ACP and BCP as θ varies.

Charting normal vectors

I’m mapping out the normal vector n1(θ) to plane ACP by calculating the cross product of vectors AC and AP(θ), ensuring accuracy with component signs and values.

Tracking progress

I’m calculating the normal vector n2(θ) by determining the cross product of vectors BC and BP(θ), meticulously considering component signs and values to ensure precision.

Tackling the cross product

I’ve calculated vector n2(θ) from the cross product of vectors BC and BP(θ), ensuring precision by examining component signs and values methodically.

Evaluating the dot product

I’m pinpointing the dot product of vectors n1(θ) and n2(θ) to find the cosine of the dihedral angle, ensuring precision with component magnitudes and signs.

Breaking down components

I’m working through each term of the dot product, factoring in sine and cosine terms and their coefficients, to ensure precise calculation of the cosine of the dihedral angle.

Combining terms

I’ve merged sine and cosine terms with their coefficients, simplified the dot product, and found the magnitudes of normal vectors n1 and n2, making progress towards determining the cosine of the dihedral angle.

Calculating vector magnitudes

I’m summing sine and cosine terms with their coefficients to determine the magnitudes and simplify them, grouping terms effectively to find the final magnitudes of normal vectors n1 and n2.

Calculating the dihedral angle

I’ve derived the expression for the cosine of the dihedral angle by calculating the dot product and magnitudes of normal vectors n1 and n2, ensuring all components are included for accurate assessment.

Breaking down the expression

I’ve simplified the cosine value equation for the dihedral angle by managing factors and combining terms. This leads me to a streamlined numerator with terms relevant to cosθ.

Reconceptualizing the function

I’m reimagining cosθ as x, transforming the function to a form suitable for minimizing within x's range from -1 to 1, aiming to simplify the process of finding the minimal cosine value.

Mapping out the angle

I’m pondering simpler methods to pinpoint the minimal cosine value of the dihedral angle by examining the angle between lines perpendicular to CP within planes ACP and BCP.

Investigating boundary conditions

I'm digging into boundary conditions like no fold (θ=0) or 180-degree fold (θ=π) to identify when the cosine of the dihedral angle reaches its maximum or minimum.

Evaluating boundary scenarios

OK, let me see. I’m pulling together evaluations at θ=0 and θ=π to spot when the cosine of the dihedral angle hits its max or min. Progressing through these insights to pin down optimal scenarios.

Analyzing boundary conditions

I’m evaluating θ=0 and θ=π to pinpoint when the cosine of the dihedral angle reaches its maximum or minimum, ensuring all geometric constraints are met. Progressing through these insights to refine the analysis.

Taking a closer look

I’m verifying the cosine of the dihedral angle at θ=π, ensuring all geometric constraints align perfectly to confirm the analysis.

Assessing θ=π/2

I’ve mapped out θ=π/2 and found the cosine of the dihedral angle around 0.65465367°, suggesting this could indicate nearing the optimal angle. Progressing towards an accurate optimal value.

Evaluating test points

I’m assessing additional test points like θ=π/3 and θ=2π/3 to explore if the expression might dip below zero, indicating a potential minimum cosine value. This involves examining the derivative and function behavior at these angles.

Evaluating angles

I’m mapping out θ=π/3 and θ=2π/3 to see if the function could fall below zero, aiming to find the minimum cosine value of the dihedral angle by checking these specific points.

Analyzing for minimal cosine

Calculating the derivative of the function to pinpoint the smallest cosine of the dihedral angle, considering angles x in (0,1).

Setting up the equation

I’ve established the equation by equating the derivatives of numerator and denominator, then solving for x in the domain [-1,1] to pinpoint the minimal cosine of the dihedral angle.

Taking a closer look

I’ve simplified g(x) to $\frac{1}{2}(\sqrt{3} - x)$ and found h(x) to $\frac{1}{4}(\sqrt{3} - x)(3x + \sqrt{3})$ by expanding and factoring terms.

Coordinating terms

I’ve mashed together $\frac{3}{4} - \frac{3}{4}$ to get 1, then added $\frac{x \sqrt{3}}{2}$ and $\frac{x \sqrt{3}}{2}$ , ultimately simplifying everything to zero.

Pinpointing critical points

I’ve determined that $x = \frac{\sqrt{3}}{3}$ corresponds to the minimal cosine of the dihedral angle, estimating the critical angle to be approximately 54.7356 degrees.

Confirming the critical point

I’m noting that $x = \frac{\sqrt{3}}{3}$ results in the smallest cosine value of the dihedral angle, approximately 0.577350. This mirrors boundary values in different ranges.

Thought about 二面角余弦最小值，持续 2m 25s

回答

解析思路概要：

平面内布置四边形 $ABCD$
题目给定
$AB = AC = CD = 1,\quad \angle ADC = 30^\circ,\quad \angle DAB = 120^\circ.$
我们先在平面（可取 $xy$ ‐平面）中选取一个方便的坐标系来“固定”点 $A,B,C,D$ 的位置。
坐标系的选取
- 令 $A$ 放在原点： $A = (0,0)$ 。
- 令 $C$ 在 $x$ 轴上并与 $A$ 距离 1：于是可取 $C=(1,0)$ 。
- 题目给定 $AB=1$ ，可将 $B$ 放在 $(0,1)$ 。这样， $A=(0,0),\quad B=(0,1),\quad C=(1,0).$
- 接下来求满足
  $\;CD=1,\;\angle ADC=30^\circ,\;\angle DAB=120^\circ$
  的点 $D=(x,y)$ 在同一平面内的具体坐标。
求点 $D$ 的坐标
设 $D=(x,y)$ （仍在 $xy$ ‐平面）。根据题意：
1. $CD=1$ ，即 $|\,D - C\,|^2 \;=\; (x-1)^2 + y^2 = 1.$
2. $\angle ADC = 30^\circ$ 是顶点在 $D$ 的角，意味着向量 $\overrightarrow{DA}$ 与 $\overrightarrow{DC}$ 的夹角是 $30^\circ$ 。用数量积可写成 $\overrightarrow{DA}\cdot\overrightarrow{DC} \;=\;|DA|\cdot|DC|\cdot\cos 30^\circ.$ 其中
  (\overrightarrow{DA} = (x-0,y-0) = (x,y),\quad \overrightarrow{DC} = (x-1,y-0) = (x-1,y).)
  又已知 $|DC|=CD=1$ ，且要由此可解出 $|DA|$ 。
3. $\angle DAB = 120^\circ$ 是顶点在 $A$ 的角，意味着 $\overrightarrow{AD}$ 与 $\overrightarrow{AB}$ 夹角 $120^\circ$ 。其中 $\overrightarrow{AB}=(0,1),\quad \overrightarrow{AD}=(x,y),$ 故 $\overrightarrow{AD}\cdot\overrightarrow{AB} \;=\;|AD|\cdot|AB|\cdot \cos 120^\circ.$ 又 $|AB|=1$ ， $\cos 120^\circ=-\tfrac12$ ，可进一步得到对 $x,y$ 的约束。
经过方程求解（或直接运用余弦定理与坐标几何），可得
$D \;=\;\Bigl(\tfrac{3}{2},\;-\tfrac{\sqrt{3}}{2}\Bigr).$
“沿 $AC$ 翻折三角形 $ACD$ ”——在三维中旋转的实现
- 现在整块四边形 $ABCD$ 原本在 $xy$ ‐平面。
- “沿 $AC$ 翻折”可以理解为：把 $\triangle ACD$ 围绕“过 $A$ 且方向为 $\overrightarrow{AC}$ ”的那条轴，做刚体旋转离开原平面，点 $D$ 移动到某个三维位置 $P$ 。
- 为了便于计算，在三维坐标中可作以下嵌入： $A=(0,0,0),\quad B=(0,1,0),\quad C=(1,0,0),\quad D=\Bigl(1.5,\;-\tfrac{\sqrt{3}}{2},\;0\Bigr).$ 这样，“绕 $AC$ ”就等价于“绕 $x$ ‐轴”（因为此时 $A$ 与 $C$ 都在 $x$ 轴上）做转动。
- 令转动角为 $\theta$ ，则在三维中绕 $x$ ‐轴旋转时， $x$ 坐标保持不变， $y,z$ 坐标按照标准旋转公式变换。由此可得： $P(\theta) \;=\; \bigl(\;1.5,\;-{\tfrac{\sqrt{3}}{2}}\cos\theta,\;-{\tfrac{\sqrt{3}}{2}}\sin\theta\bigr).$
所求二面角 $\angle(A - CP - B)$
题中“二面角 $A - CP - B$ ”的含义通常是：以线段 $CP$ 为“铰”，平面 $ACP$ 与平面 $BCP$ 之间的二面角。
- 若记 $\mathbf{n}_1(\theta) \;=\; \overrightarrow{AC}\times \overrightarrow{AP(\theta)}, \quad \mathbf{n}_2(\theta) \;=\; \overrightarrow{BC}\times \overrightarrow{BP(\theta)},$ 则这两个法向量所夹的角，便是所要求的二面角。
- 二面角的余弦即 $\cos\bigl(\text{二面角}\bigr) \;=\; \frac{\mathbf{n}_1(\theta)\cdot \mathbf{n}_2(\theta)} {\|\mathbf{n}_1(\theta)\|\;\|\mathbf{n}_2(\theta)\|}.$
- 将 $P(\theta)$ 代入，做向量叉乘与数量积运算，就得到一个关于 $\theta$ （或等价关于 $\cos\theta$ ）的解析式，从而可以在 $\theta\in[0,\pi]$ （折叠 0～180°）区间内寻找此余弦的最小值。
计算与结果
- 经过比较边界情形（ $\theta=0,\pi$ 处余弦皆为 1）以及对中间点求导数，发现唯一的极小点出现在
  $\displaystyle \cos\theta \;=\;\frac{1}{\sqrt{3}},$
  在此时二面角的余弦值正好取得最小。
- 最终最小值为 $\boxed{\;\frac{1}{\sqrt{3}}\;}.$

结论

这道题中，随着折叠角度的变化，所要求的二面角 $A - CP - B$ 的余弦值在区间 $[0,\pi]$ 内先从 1 降到一某个最小值再回到 1。其最小值恰为 $\tfrac{1}{\sqrt{3}}$ 。也就是说，

\min \Bigl[\cos\bigl(\text{二面角 }A - CP - B\bigr)\Bigr] \;=\; \frac{1}{\sqrt{3}}.

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