Comment les coordonnées sphériques et cartésiennes sont-elles liées pour la description d'un point dans l'espace ?
Cartesian coordinates

The spherical coordinates of a point in the ISO convention (i.e. for physics: radius r, inclination θ, azimuth φ) can be obtained from its Cartesian coordinates (x, y, z) by the formulae

r = x 2 + y 2 + z 2 θ = arccos z x 2 + y 2 + z 2 = arccos z r = { arctan x 2 + y 2 z if z > 0 π + arctan x 2 + y 2 z if z < 0 + π 2 if z = 0 and x 2 + y 2 ≠ 0 undefined if x = y = z = 0 φ = sgn (y) arccos x x 2 + y 2 = { arctan (y x) if x > 0, arctan (y x) + π if x < 0 and y ≥ 0, arctan (y x) − π if x < 0 and y < 0, + π 2 if x = 0 and y > 0, − π 2 if x = 0 and y < 0, undefined if x = 0 and y = 0. {\displaystyle {\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=\arccos {\frac {z}{\sqrt {x^{2}+y^{2}+z^{2}}}}=\arccos {\frac {z}{r}}={\begin{cases}\arctan {\frac {\sqrt {x^{2}+y^{2}}}{z}}&{\text{if }}z>0\\\pi +\arctan {\frac {\sqrt {x^{2}+y^{2}}}{z}}&{\text{if }}z<0\\+{\frac {\pi }{2}}&{\text{if }}z=0{\text{ and }}{\sqrt {x^{2}+y^{2}}}\neq 0\\{\text{undefined}}&{\text{if }}x=y=z=0\\\end{cases}}\\\varphi &=\operatorname {sgn}(y)\arccos {\frac {x}{\sqrt {x^{2}+y^{2}}}}={\begin{cases}\arctan({\frac {y}{x}})&{\text{if }}x>0,\\\arctan({\frac {y}{x}})+\pi &{\text{if }}x<0{\text{ and }}y\geq 0,\\\arctan({\frac {y}{x}})-\pi &{\text{if }}x<0{\text{ and }}y<0,\\+{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y>0,\\-{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y<0,\\{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}\end{aligned}}}

The inverse tangent denoted in φ = arctan y / x must be suitably defined, taking into account the correct quadrant of (x, y), as done in the equations above. See the article on atan2.

Alternatively, the conversion can be considered as two sequential rectangular to polar conversions: the first in the Cartesian xy plane from (x, y) to (R, φ), where R is the projection of r onto the xy -plane, and the second in the Cartesian zR -plane from (z, R) to (r, θ). The correct quadrants for φ and θ are implied by the correctness of the planar rectangular to polar conversions.

These formulae assume that the two systems have the same origin, that the spherical reference plane is the Cartesian xy plane, that θ is inclination from the z direction, and that the azimuth angles are measured from the Cartesian x axis (so that the y axis has φ = +90°). If θ measures elevation from the reference plane instead of inclination from the zenith the arccos above becomes an arcsin, and the cos θ and sin θ below become switched.

Conversely, the Cartesian coordinates may be retrieved from the spherical coordinates (radius r, inclination θ, azimuth φ), where r ∈, φ ∈ [ 0, 2 π), by x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ. {\displaystyle {\begin{aligned}x&=r\sin \theta \,\cos \varphi,\\y&=r\sin \theta \,\sin \varphi,\\z&=r\cos \theta.\end{aligned}}}
Coordonnées sphériques (dans l'espace)

En coordonnées sphériques, chaque point dans l'espace est représenté par sa distance ρ de l'origine, l'angle θ que fait sa projection sur le plan xy par rapport à l'axe horizontal, et l'angle φ qu'il fait par rapport à l'axe z. Les noms de ces angles sont souvent inversés en physique.
### Query ###
Comment les coordonnées sphériques et cartésiennes sont-elles liées pour la description d'un point dans l'espace ?

### Main Text ###
Cartesian coordinates

The spherical coordinates of a point in the ISO convention (i.e. for physics: radius r, inclination θ, azimuth φ) can be obtained from its Cartesian coordinates (x, y, z) by the formulae

r = x 2 + y 2 + z 2 θ = arccos z x 2 + y 2 + z 2 = arccos z r = { arctan x 2 + y 2 z if z > 0 π + arctan x 2 + y 2 z if z < 0 + π 2 if z = 0 and x 2 + y 2 ≠ 0 undefined if x = y = z = 0 φ = sgn (y) arccos x x 2 + y 2 = { arctan (y x) if x > 0, arctan (y x) + π if x < 0 and y ≥ 0, arctan (y x) − π if x < 0 and y < 0, + π 2 if x = 0 and y > 0, − π 2 if x = 0 and y < 0, undefined if x = 0 and y = 0. {\displaystyle {\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=\arccos {\frac {z}{\sqrt {x^{2}+y^{2}+z^{2}}}}=\arccos {\frac {z}{r}}={\begin{cases}\arctan {\frac {\sqrt {x^{2}+y^{2}}}{z}}&{\text{if }}z>0\\\pi +\arctan {\frac {\sqrt {x^{2}+y^{2}}}{z}}&{\text{if }}z<0\\+{\frac {\pi }{2}}&{\text{if }}z=0{\text{ and }}{\sqrt {x^{2}+y^{2}}}\neq 0\\{\text{undefined}}&{\text{if }}x=y=z=0\\\end{cases}}\\\varphi &=\operatorname {sgn}(y)\arccos {\frac {x}{\sqrt {x^{2}+y^{2}}}}={\begin{cases}\arctan({\frac {y}{x}})&{\text{if }}x>0,\\\arctan({\frac {y}{x}})+\pi &{\text{if }}x<0{\text{ and }}y\geq 0,\\\arctan({\frac {y}{x}})-\pi &{\text{if }}x<0{\text{ and }}y<0,\\+{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y>0,\\-{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y<0,\\{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}\end{aligned}}}

The inverse tangent denoted in φ = arctan y / x must be suitably defined, taking into account the correct quadrant of (x, y), as done in the equations above. See the article on atan2.

Alternatively, the conversion can be considered as two sequential rectangular to polar conversions: the first in the Cartesian xy plane from (x, y) to (R, φ), where R is the projection of r onto the xy -plane, and the second in the Cartesian zR -plane from (z, R) to (r, θ). The correct quadrants for φ and θ are implied by the correctness of the planar rectangular to polar conversions.

These formulae assume that the two systems have the same origin, that the spherical reference plane is the Cartesian xy plane, that θ is inclination from the z direction, and that the azimuth angles are measured from the Cartesian x axis (so that the y axis has φ = +90°). If θ measures elevation from the reference plane instead of inclination from the zenith the arccos above becomes an arcsin, and the cos θ and sin θ below become switched.

Conversely, the Cartesian coordinates may be retrieved from the spherical coordinates (radius r, inclination θ, azimuth φ), where r ∈, φ ∈ [ 0, 2 π), by x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ. {\displaystyle {\begin{aligned}x&=r\sin \theta \,\cos \varphi,\\y&=r\sin \theta \,\sin \varphi,\\z&=r\cos \theta.\end{aligned}}}

### Supplementary Text ###
Coordonnées sphériques (dans l'espace)

En coordonnées sphériques, chaque point dans l'espace est représenté par sa distance ρ de l'origine, l'angle θ que fait sa projection sur le plan xy par rapport à l'axe horizontal, et l'angle φ qu'il fait par rapport à l'axe z. Les noms de ces angles sont souvent inversés en physique.

### Draft ###

Query: "Comment les coordonnées sphériques et cartésiennes sont-elles liées..." 

Translation check: "How are spherical and Cartesian coordinates linked for describing a point in space?"

Clean mathematical question. Information retrieval + synthesis required.

### 1. Coordinate Systems

**Cartesian**: (x,y,z). Standard.
**Spherical**: (r,θ,φ). Three representations possible.

Query specifies: radius r, inclination θ (from z-axis), azimuth φ (from x-axis).

### 2. Conversion Formulas

**From Cartesian → Spherical:**

Distance: r = √(x² + y² + z²)
Inclination: θ = arccos(z/r) 
Azimuth: φ = arctan(y/x) with quadrant handling

**From Spherical → Cartesian:**

x = r sin(θ) cos(φ)
y = r sin(θ) sin(φ)  
z = r cos(θ)

### 3. Mathematical Relationships

Core identity: x² + y² + z² = r²

**Orthogonal decomposition:**
- r² = x² + y² + z² (Pythagorean)
- x² + y² = r²sin²(θ) (planar projection)
- x² + y² + z² = r² (full 3D)

**Trigonometric implications:**
- sin²(θ) + cos²(θ) = 1
- cos²(φ) + sin²(φ) = 1

### 4. Geometric Interpretation

Spherical coordinates → natural for spherical symmetry problems
Cartesian coordinates → standard for computational geometry

**Transformation matrix:**
∂(x,y,z)/∂(r,θ,φ) = [sin(θ)cos(φ), -rsin(θ)sin(φ), cos(θ)cos(φ); sin(θ)sin(φ), rsin(θ)cos(φ), cos(θ)sin(φ); cos(θ), -rsin(θ), -sin(θ)]

**Jacobian determinant:** r²sin(θ) → volume element scaling.

### 5. Implementation Notes

**atan2 function essential** for φ calculation. Handles quadrant ambiguity correctly.

**Range considerations:**
- r ∈ [0, ∞)
- θ ∈ [0, π] 
- φ ∈ [0, 2π)

**Special cases:**
- r = 0: origin
- θ = 0: on z-axis
- θ = π: on z-axis (opposite direction)
- φ = 0: xz-plane
- φ = π/2: yz-plane

### 6. Physical Applications

**Key insight:** Conversion formulas enable seamless switching between coordinate representations.

**Common applications:**
- Electromagnetism (spherically symmetric charge distributions)
- Quantum mechanics (orbital angular momentum)
- Geophysics (Earth coordinates)
- Computer graphics (3D rendering)

**Example:** Calculate electric field E(r,θ,φ) = kr^n er from Cartesian density distribution.

**Result:** Integration over solid angle dΩ = 2πsin(θ)dθ simplifies to ∫₀^π ∫₀^2π E(r,θ,φ) sin(θ)dφ dθ.

### Conclusion

Spherical and Cartesian coordinates are **mathematically complete** representations. Conversion formulas provide bidirectional mapping with Jacobian scaling. Choice of coordinates depends on problem symmetry and computational convenience.

**Core mathematical relationship:** Orthogonal coordinate transformations with well-defined Jacobian determinant ensure volume preservation and enable coordinate-invariant physical laws.

### Answer###
Les coordonnées sphériques et cartésiennes sont intimement liées pour la description d'un point dans l'espace par des relations de conversion définies.

Dans la convention ISO standard pour la physique, les coordonnées sphériques d'un point sont représentées par le rayon r, l'inclinaison θ mesurée depuis l'axe z, et l'azimut φ mesuré depuis l'axe x. Ces coordonnées peuvent être obtenues à partir des coordonnées cartésiennes (x, y, z) selon les formules :

r = √(x² + y² + z²)
θ = arccos(z/r) = arccos(z/√x² + y² + z²)
φ = arctan(y/x) avec une gestion appropriée des quadrants

Inversement, les coordonnées cartésiennes peuvent être récupérées à partir des coordonnées sphériques selon :

x = r sin(θ) cos(φ)
y = r sin(θ) sin(φ)  
z = r cos(θ)

Il est important de noter que l'angle azimut φ nécessite l'utilisation de la fonction arctan avec une gestion correcte des quadrants pour éviter les ambiguïtés.

Ces formules de conversion permettent de passer de l'un à l'autre système de coordonnées selon les besoins spécifiques de chaque problème, en préservant l'unicité de la représentation du point dans l'espace tridimensionnel.