二面角最小值

Analysing the fold

I’m examining a quadrilateral ABCD in a plane, folding $\triangle ACD$ along AC to form $\triangle ACP$, and focusing on the dihedral angle A-CP-B.

Determining the fold

Focusing on the movement of point D to point P in 3D space, ensuring line AC remains fixed. This involves examining the angle between planes ACP and ABC.

Analyzing angle notation

I’m examining the dihedral angle A-CP-B, which might denote the angle between planes ACP and BCP, with line CP as the hinge and points A and B on their respective planes.

Deciphering the angle

I’m considering the ambiguity of "二面角 A-CP-B" and exploring whether it refers to the angle formed by planes AC-P and BC-P or the lines AC and BC in the 3D shape.

Analyzing structures

I’m exploring if the question refers to a tetrahedron ABPC and its dihedral angle at edge PB, or the angle between planes ACP and BCP using line CP.

Understanding folding mechanics

Reflecting $\triangle ACD$ across line AC sends point D to a point P symmetrical to D across AC. This does not create a 3D shape; the figure stays in the plane.

Rotating the plane

I'm thinking about how rotating $\triangle ACD$ around line AC generates a circle of potential points for P, influencing the dihedral angle A-CP-B.

Analyzing plane interactions

Focusing on the dihedral angle formed by planes ACP and BCP, with CP as their intersection line, as P shifts, we seek the minimum cosine value of this angle.

Mapping out normal vectors

To minimize the cosine of the dihedral angle between planes ACP and BCP, I’m analyzing the relationship between the normal vectors of these planes within the tetrahedron ABPC.

Piecing together triangle properties

I’m gathering triangle ACD's side lengths and angles to identify the steepest dihedral angle. Using AD=1, AC=1, CD=1, and $\angle ADC=30^\circ$, I’m calculating $\angle DAC$.

Determining AD

Applying the law of cosines in triangle ADC, solving $1 = AD^2 + 1 - \sqrt3 \cdot AD$, yields $AD = \sqrt3$.

Calculating distance BD

OK, let me see. Using the law of cosines in triangle ABD with $AB=1$, $DA=\sqrt3$, and $\angle DAB=120^\circ$, I found $BD=\sqrt{4 + \sqrt3}$.

Piecing together coordinates

I'm setting points A and B, noting C's position on a circle from A, and considering D's alignment with constraints. This involves systematic exploration for clarity.

Placing point D

Hmm, I’m thinking about D's location from A and B, using $\angle DAB=120^\circ$ and $AD=\sqrt3$, and dividing this angle by 2 to find D's precise spot.

Pinpointing pionts

I’ve identified points A, B, and D using geometric constraints. Now, I’m placing C such that AC=1, CD=1, and $\angle ADC=30^\circ$, and parametrizing C as (x,y).

Pinpointing coordinates

I’m working out the coordinates of point C by solving the equations for AC and CD, ensuring the angle $\angle ADC=30^\circ$. C's coordinates lie on the unit circle centered at A.

Ensuring vector alignment

I’m working through the dot product of $\vec{DA}$ and $\vec{DC}$. With $|\vec{DA}| = \sqrt3$ and $|\vec{DC}| = 1$, I’m aligning $\vec{DC}$ to $\vec{DA}$ such that $\angle(DA, DC) = 30^\circ$.

Breaking down the equation

Ensuring the dot product equals $3/2$ by expanding and verifying $\vec{DA} \cdot \vec{DC}$ with $|\vec{DA}| = \sqrt3$ and $\cos(30^\circ) = \sqrt3/2$.

Manipulating equations

I’m working through the dot product condition to express $x$ in terms of $y$, then substituting it into the circle equation to solve for $y$. This leads me to find possible $y$ values.

Verifying coordinates

I’m checking two possible $C$ coordinates by plugging them back into the equations. Both $(0,1)$ and $(- \sqrt3/2, 1/2)$ align with the conditions, making them valid solutions.

Mapping the terrain

I’m comparing the distances and positions of the two $C$ coordinates. Both are valid, and the shape's convexity is under consideration while ensuring no intersection issues with $B$.

Mapping out rotations

OK, let me see. I’m thinking through how rotating triangle ACD around AC shifts point D to P in 3D space, with CP intersecting planes ACP and BCP, aiming for the minimal cos value of angle A-CP-B.

Pinpointing the angle

I’m figuring out that "A - CP - B" refers to the angle between planes ACP and BCP along line CP, aiming to find the minimal cos value of the dihedral angle.

Cracking the formula

I’m pulling together vector cross products of $\vec{CA}$ and $\vec{CP}$ with $\vec{CB}$ and $\vec{CP}$. Parametrizing by $\theta$ to simplify the dihedral angle calculation.

Synthesizing details

I’m pulling together fixed distances: AC=1, CP=1, AP=$\sqrt3$. This setup aligns with the rigid 3D rotation around AC, ensuring consistent geometry throughout.

Considering various paths

I'm comparing different configurations for tetrahedron ABPC. Both align with the given constraints, though the problem remains open to further symmetry insights or other specific conditions.

Orienting the rotation

I'm analyzing how a 3D rotation around axis AC affects points A and C. The key is to keep everything around AC aligned, ensuring consistent distances.

Defining coordinates

I’m setting the axis AC as the y-axis in 3D, figuring out P's location as the circle of intersection of spheres around A and C, and thinking about the constraints of coordinates.

Placing points

I’m setting the points A, B, C, and D in 3D space, ensuring that AC=1. The fold is a rotation around the line from A to C, maintaining the 2D coordinates.

Mapping the rotation

I'm working through rotating triangle ACD around axis AC, finding that point D optionally moves to the other side of axis AC, with the rotation angle ranging from 0° to 180°.

Rotating the plane

I’m analyzing how D's rotation around axis AC affects the dihedral angle with B. Considering angles from 0° to 180°, I'll derive the new coordinates and evaluate the minimal cosine value.

Revolvating around the infinite line

I'm leaning towards rotating D around the infinite line through A and C, aligning with the infinite rotation axis.

Adjusting my approach

I’m redefining D as (0,1.5,0) plus a vector in the x-z plane, focusing on the infinite line through A and C. This new perspective aids in simplifying calculations.

Revolvating around AC

I'm digging into how D rotates around line AC. Projecting D onto the perpendicular axis and simplifying the y-coordinate journey demonstrates how this affects the rotation effectively.

Tracing D's movement

I’m thinking about how revolving D around AC forms a circle with radius r, calculated from D's distance to line AC. P is any point on this circle, with rotation angle $\alpha$.

Strategizing the calculation

I’m piecing together the dihedral angle along CP in tetrahedron ABPC, using edge lengths and the Cayley-Menger formula to find $\cos(\phi)$ and the angular distance from P to B.

Mapping edge lengths

I’m figuring out the dihedral angle at edge XY in tetrahedron A B C P, using a known formula with edge lengths and normal vectors to systematically approach the problem.

Analyzing changes

OK, I’m realizing that B's distance to P is influenced by the rotation angle. With B and C fixed in the plane, BC stays at $\sqrt2$.

Pinpointing points

I’m organizing the points A, B, C, and D in 3D space, noting that P is derived from rotating D around AC. Then, I calculate vector MD and its length $\sqrt7/2$.

Confirming calculations

I’ve checked the distance from the midpoint of AC to D, ensuring it’s $\sqrt7/2$. Now, I’m focusing on rotating vector MD around axis AC by angle $\alpha$.

Mapping rotation steps

OK, I'm thinking through the process of rotating vector MD around the y-axis. Translating by $-M$, then rotating and translating back, to find the new location of P.

Adjusting my method

I’m rethinking the rotation of D around the y-axis through A as axis AC intersects the plane, ensuring P’s position follows the rotation dynamics.

Mapping the rotation

I’m figuring out how to rotate D around the y-axis through A. Keeping the y-component fixed, I’m rotating the orthogonal part in the x-z plane, then translating it back to A.

Tracking P's position

I'm piecing together P's coordinates as I rotate D around the y-axis, ensuring the y-component stays constant while adjusting the orthogonal part ($P(\alpha)$).

Understanding vector roles

I’m piecing together vectors $\vec{CP}(\alpha)$ and $\vec{AP}(\alpha)$ to determine the dihedral angle between planes ACP($\alpha$) and BCP($\alpha$). This helps track P's movement.

Mapping out normal vectors

OK, let’s simplify by using $\vec{AC} \times \vec{AP}(\alpha)$ for ACP($\alpha$), and $\vec{BC} \times \vec{BP}(\alpha)$ for BCP($\alpha$). This helps find the minimal cos $\phi(\alpha)$.

Extracting normal vectors

I’ve mapped out the cross product $\vec{AC} \times \vec{AP}(\alpha)$, yielding $n1(\alpha) = \frac{\sqrt3}{2}(-\sin\alpha, 0, \cos\alpha)$. Now, I’ll apply similar steps for $n2(\alpha)$.

Calculating cross product

I’m gathering normal vector $n2(\alpha) = \frac{\sqrt3}{2}( -\frac{\sqrt3}{2}\sin\alpha, \frac{\sqrt3}{2}\cos\alpha )$ by evaluating $\vec{BC} \times \vec{BP}(\alpha)$.

Mapping normal vectors

I’m piecing together vector $n2(\alpha)$ using $\vec{BC}$ and $\vec{BP}(\alpha)$, ensuring accurate coefficients for each component to calculate $n1(\alpha) \cdot n2(\alpha)$.

Mapping the connections

OK, I’ve figured out the dot product $n1(\alpha) \cdot n2(\alpha)$, combining the coefficients and components of the vectors to determine the dihedral angle between planes ACP($\alpha$) and BCP($\alpha$).

Figuring out the dot product

Summing the components of $n1(\alpha)$ and $n2(\alpha)$, and noting $|n1(\alpha)|$ and $|n2(\alpha)|$ are constants, simplifies the calculation.

Piecing together vector lengths

I’ve worked out the length of $\vec{BC}$ by adding squares of the components for each vector and simplifying terms, concluding with the final expression for $|n2(\alpha)|$.

Refining the expression

OK, let me see. Standardizing the denominators and simplifying terms leads to a more refined expression for $\cos \phi(\alpha)$, involving familiar constants and functions.

Mapping out the function

I’ve redefined $\cos \phi(\alpha)$ in terms of $x = \cos\alpha$, simplifying to $f(x)$. Now, I’m exploring derivatives and monotonicity to find the minimal value.

Breaking down the process

I’m figuring out if $7 - 3x^2 - 2\sqrt3 x$ stays positive for $x$ in $[-1, 1]$, then analyzing $f'(x)$ to see if $f(x)$ is monotonic.

Simplifying and deriving

Working through the derivative expressions, simplifying $N(x)$ and $Z(x)$, and examining their relationships to confirm if they stay positive within the identified range.

Piecing together

I expanded and combined terms from $Z(x)$, ensuring all signs are correct. This helps in understanding the composite behaviors and results of the function.

Identifying the critical point

I found the critical point by solving $Z(x) = 0$, giving $x = \frac{\sqrt3}{3}$. This value is within the interval $[-1,1]$.

Confirming the minimum

I determined that $x = \frac{\sqrt3}{3}$ is the minimum point. Evaluating $f(x)$ at the endpoints and critical points shows $f(\frac{\sqrt3}{3})$ is the smallest.

Verifying endpoint values

I confirm that $f(-1) = 1$ and $f(1)$ match the expected values. However, a degenerate case when $\alpha = \pi$ could lead to no dihedral angle.

Confirming calculations

I verified that $f(1) = 1$. Interestingly, both endpoints yield $\cos(\phi) = 1$, indicating that the pleat angle might be 0 or 180.

Assessing function behavior

I noticed that $f(\frac{\sqrt3}{3}) = \frac{1}{\sqrt3}$, marking the minimum value of $\cos(\phi)$. This function decreases to $\frac{\sqrt3}{3}$ before increasing back to 1.

Thought about 二面角余弦最小值，持续 3m 42s