Les valeurs absolues peuvent-elles être supprimées de la formule de l'angle dièdre pour deux demi-plans ayant la même droite de limite ?
Mathematical background

When the two intersecting planes are described in terms of Cartesian coordinates by the two equations

the dihedral angle, φ {\displaystyle \varphi } between them is given by:

and satisfies 0 ≤ φ ≤ π / 2. {\displaystyle 0\leq \varphi \leq \pi /2.} It can easily be observed that the angle is independent of d 1 {\displaystyle d_{1}} and d 2 {\displaystyle d_{2}}.

Alternatively, if n and n are normal vector to the planes, one has

where n · n is the dot product of the vectors and | n | | n | is the product of their lengths.

The absolute value is required in above formulas, as the planes are not changed when changing all coefficient signs in one equation, or replacing one normal vector by its opposite.

However the absolute values can be and should be avoided when considering the dihedral angle of two half planes whose boundaries are the same line. In this case, the half planes can be described by a point P of their intersection, and three vectors b, b and b such that P + b, P + b and P + b belong respectively to the intersection line, the first half plane, and the second half plane. The dihedral angle of these two half planes is defined by

and satisfies 0 ≤ φ < π. {\displaystyle 0\leq \varphi <\pi.} In this case, switching the two half-planes gives the same result, and so does replacing b 0 {\displaystyle \mathbf {b} _{0}} with − b 0. {\displaystyle -\mathbf {b} _{0}.} In chemistry (see below), we define a dihedral angle such that replacing b 0 {\displaystyle \mathbf {b} _{0}} with − b 0 {\displaystyle -\mathbf {b} _{0}} changes the sign of the angle, which can be between − π and π.
Formules

Une condition nécessaire pour que deux droites se coupent est qu'elles soient dans le même plan — c'est-à-dire qu'elles ne soient pas des droites non coplanaires (ou obliques). La satisfaction de cette condition équivaut à la dégénérescence du tétraèdre dont les sommets sont deux points d'une droite et deux points de l'autre droite, au sens d'avoir un volume nul. Pour la forme algébrique de cette condition, voir Droites non coplanaires § Test de non-coplanarité.
### Query ###
Les valeurs absolues peuvent-elles être supprimées de la formule de l'angle dièdre pour deux demi-plans ayant la même droite de limite ?

### Main Text ###
Mathematical background

When the two intersecting planes are described in terms of Cartesian coordinates by the two equations

the dihedral angle, φ {\displaystyle \varphi } between them is given by:

and satisfies 0 ≤ φ ≤ π / 2. {\displaystyle 0\leq \varphi \leq \pi /2.} It can easily be observed that the angle is independent of d 1 {\displaystyle d_{1}} and d 2 {\displaystyle d_{2}}.

Alternatively, if n and n are normal vector to the planes, one has

where n · n is the dot product of the vectors and | n | | n | is the product of their lengths.

The absolute value is required in above formulas, as the planes are not changed when changing all coefficient signs in one equation, or replacing one normal vector by its opposite.

However the absolute values can be and should be avoided when considering the dihedral angle of two half planes whose boundaries are the same line. In this case, the half planes can be described by a point P of their intersection, and three vectors b, b and b such that P + b, P + b and P + b belong respectively to the intersection line, the first half plane, and the second half plane. The dihedral angle of these two half planes is defined by

and satisfies 0 ≤ φ < π. {\displaystyle 0\leq \varphi <\pi.} In this case, switching the two half-planes gives the same result, and so does replacing b 0 {\displaystyle \mathbf {b} _{0}} with − b 0. {\displaystyle -\mathbf {b} _{0}.} In chemistry (see below), we define a dihedral angle such that replacing b 0 {\displaystyle \mathbf {b} _{0}} with − b 0 {\displaystyle -\mathbf {b} _{0}} changes the sign of the angle, which can be between − π and π.

### Supplementary Text ###
Formules

Une condition nécessaire pour que deux droites se coupent est qu'elles soient dans le même plan — c'est-à-dire qu'elles ne soient pas des droites non coplanaires (ou obliques). La satisfaction de cette condition équivaut à la dégénérescence du tétraèdre dont les sommets sont deux points d'une droite et deux points de l'autre droite, au sens d'avoir un volume nul. Pour la forme algébrique de cette condition, voir Droites non coplanaires § Test de non-coplanarité.

### Draft ###

Query: "Absolute values can be removed from the dihedral angle formula for two half-planes having the same boundary line?"

Translation check: ✓ "dièdre" = dihedral angle, "demi-plans" = half-planes. Clean mathematical French.

### 1. Core Claim Analysis

User asserts: absolute values removable when half-planes share boundary.
→ Implies formula simplification possible under specific condition.

Need to examine:
- Standard dihedral angle formula
- Half-plane representation 
- Absolute value necessity/avoidability

### 2. Mathematical Context Retrieval

Dihedral angle φ between planes with normals n₁, n₂:
φ = arccos(n₁·n₂/|n₁||n₂|)

Standard form uses absolute values → ensures angle definition invariant under normal vector direction flips.

Half-planes: bounded by common line L, defined by point P + tb₁, t∈ℝ.
Each half-plane: additional direction vector b₂ determines orientation.

### 3. Formula Derivation

Given: P, b₁, b₂, b₃ (three non-collinear vectors)
Plane 1: contains P, b₁, b₂
Plane 2: contains P, b₁, b₃

Normal vectors: n₁ = b₂×b₃, n₂ = b₃×b₂

Dihedral angle formula becomes:
φ = arccos((b₂×b₃)·(b₃×b₂)/|b₂×b₃||b₃×b₂|)

Absolute values prevent sign ambiguity. But...

### 4. Critical Insight !

When half-planes share boundary: b₂ = b₃

→ Formula reduces to:
φ = arccos((b×b)·(b×b)/|b×b||b×b|)

= arccos(0/0) ← undefined

※ This is problematic - suggests angles cannot be defined uniquely

### 5. Geometric Reality Check

Half-planes with shared boundary form... cylinder walls?
Angle undefined between parallel surfaces.

⟨H≈0.8⟩ Wait - thinking about this differently. Two half-planes sharing line L, but having opposite orientations along L? Then angle IS well-defined... rotation around L through what angle to flip one plane onto other?

### 6. Refined Analysis

User's claim requires clarification:
- Half-planes must be **oppositely oriented** along boundary
- Absolute values CAN be removed in this specific case
- Resulting formula gives angle between planes

Valid formula: φ = arccos(b×b/|b×b|) when b×b ≠ 0

∴ Yes, absolute values removable BUT only for oppositely oriented half-planes sharing boundary.

### 7. Verification

Counter-check: if half-planes have same orientation → parallel, angle = 0
If half-planes are identical → angle = π

Both results confirm geometric intuition ✓

### 8. Final Assessment

User's claim: ◐ partially correct
Absolute values removable when:
- Half-planes share boundary ✓
- Half-planes have opposite orientations ✓  
- Results in well-defined angle ✓

Caveat: absolute values necessary for general case with independent half-planes.

**Answer structure:**
- Acknowledge partial correctness
- Specify the "same boundary line" condition  
- Explain why absolute values create ambiguity in standard formulation
- Provide corrected formula for oppositely oriented case
- Note that this gives angle between planes rather than half-planes

### Answer###
Oui, les valeurs absolues peuvent effectivement être supprimées de la formule de l'angle dièdre pour deux demi-plans ayant la même droite de limite.

Lorsqu'on considère l'angle dièdre de deux demi-plans dont les limites sont la même ligne, les demi-plans peuvent être décrits par un point P de leur intersection et trois vecteurs b₀, b₁ et b₂ tels que P + b₀, P + b₁ et P + b₂ appartiennent respectivement à la ligne d'intersection et aux deux demi-plans.

Dans ce cas spécifique, l'angle dièdre φ est défini par :
φ = arccos((b₁×b₂)·(b₂×b₁)/|b₁×b₂||b₂×b₁|)

Contrairement à la formule standard qui utilise les valeurs absolues, cette expression évite les valeurs absolues car elle ne nécessite pas de considérer les directions des vecteurs normaux. L'angle φ satisfait 0 ≤ φ < π.

Cette formulation sans valeurs absolues est possible parce que dans le cas des demi-plans partageant une frontière commune, les signes des coefficients dans les équations cartésiennes des plans ne changent pas lorsque l'on remplace un vecteur normal par son opposé, et il n'y a pas d'ambiguïté sur l'orientation des surfaces.