Pourquoi on dit que les sphères d'eau sont des ellipsoïdes alors que les sphéroïdes d'eau pas ?
Spheres and spheroids

The Chicago Bridge and Iron Company has built many of the water spheres and spheroids found in the United States. The website World's Tallest Water Sphere describes the distinction between a water sphere and water spheroid thus:

A water sphere is a type of water tower that has a large sphere at the top of its post. The sphere looks like a golf ball sitting on a tee or a round lollipop. A cross section of a sphere in any direction (east-west, north-south, or top-bottom) is a perfect circle. A water spheroid looks like a water sphere, but the top is wider than it is tall. A spheroid looks like a round pillow that is somewhat flattened. A cross section of a spheroid in two directions (east-west or north-south) is an ellipse, but in only one direction (top-bottom) is it a perfect circle. Both spheres and spheroids are special-case ellipsoids: spheres have symmetry in 3 directions, spheroids have symmetry in 2 directions. Scalene ellipsoids have 3 unequal length axes and three unequal cross sections.

The Union Watersphere is a water tower topped with a sphere-shaped water tank in Union, New Jersey, and characterized as the World's Tallest Water Sphere.

A Star Ledger article suggested a water tower in Erwin, North Carolina completed in early 2012, 219.75 ft (66.98 m) tall and holding 500,000 US gallons (1,900 m), had become the World's Tallest Water Sphere. However, photographs of the Erwin water tower revealed the new tower to be a water spheroid.

The water tower in Braman, Oklahoma, built by the Kaw Nation and completed in 2010, is 220.6 ft (67.2 m) tall and can hold 350,000 US gallons (1,300 m). Slightly taller than the Union Watersphere, it is also a spheroid.

Another tower in Oklahoma, built in 1986 and billed as the "largest water tower in the country", is 218 ft (66 m) tall, can hold 500,000 US gallons (1,900 m), and is located in Edmond.

The Earthoid, a perfectly spherical tank located in Germantown, Maryland is 100 ft (30 m) tall and holds 2,000,000 US gallons (7,600 m) of water. The name is taken from it being painted to resemble a globe of the world.

The golf ball-shaped tank of the water tower at Gonzales, California is supported by three tubular legs and reaches about 125 ft (38 m) high.

The Watertoren (or Water Towers) in Eindhoven, Netherlands contain three spherical tanks, each 10 m (33 ft) in diameter and capable of holding 500 cubic metres (130,000 US gal) of water, on three 43.45 m (142.6 ft) spires were completed in 1970.
Ellipsoïdes

Un ellipsoïde est une sphère qui a été étirée ou comprimée dans une ou plusieurs directions. Plus précisément, c'est l'image d'une sphère sous une transformation affine. Un ellipsoïde entretient la même relation avec la sphère qu'une ellipse avec un cercle.
### Query ###
Pourquoi on dit que les sphères d'eau sont des ellipsoïdes alors que les sphéroïdes d'eau pas ?

### Main Text ###
Spheres and spheroids

The Chicago Bridge and Iron Company has built many of the water spheres and spheroids found in the United States. The website World's Tallest Water Sphere describes the distinction between a water sphere and water spheroid thus:

A water sphere is a type of water tower that has a large sphere at the top of its post. The sphere looks like a golf ball sitting on a tee or a round lollipop. A cross section of a sphere in any direction (east-west, north-south, or top-bottom) is a perfect circle. A water spheroid looks like a water sphere, but the top is wider than it is tall. A spheroid looks like a round pillow that is somewhat flattened. A cross section of a spheroid in two directions (east-west or north-south) is an ellipse, but in only one direction (top-bottom) is it a perfect circle. Both spheres and spheroids are special-case ellipsoids: spheres have symmetry in 3 directions, spheroids have symmetry in 2 directions. Scalene ellipsoids have 3 unequal length axes and three unequal cross sections.

The Union Watersphere is a water tower topped with a sphere-shaped water tank in Union, New Jersey, and characterized as the World's Tallest Water Sphere.

A Star Ledger article suggested a water tower in Erwin, North Carolina completed in early 2012, 219.75 ft (66.98 m) tall and holding 500,000 US gallons (1,900 m), had become the World's Tallest Water Sphere. However, photographs of the Erwin water tower revealed the new tower to be a water spheroid.

The water tower in Braman, Oklahoma, built by the Kaw Nation and completed in 2010, is 220.6 ft (67.2 m) tall and can hold 350,000 US gallons (1,300 m). Slightly taller than the Union Watersphere, it is also a spheroid.

Another tower in Oklahoma, built in 1986 and billed as the "largest water tower in the country", is 218 ft (66 m) tall, can hold 500,000 US gallons (1,900 m), and is located in Edmond.

The Earthoid, a perfectly spherical tank located in Germantown, Maryland is 100 ft (30 m) tall and holds 2,000,000 US gallons (7,600 m) of water. The name is taken from it being painted to resemble a globe of the world.

The golf ball-shaped tank of the water tower at Gonzales, California is supported by three tubular legs and reaches about 125 ft (38 m) high.

The Watertoren (or Water Towers) in Eindhoven, Netherlands contain three spherical tanks, each 10 m (33 ft) in diameter and capable of holding 500 cubic metres (130,000 US gal) of water, on three 43.45 m (142.6 ft) spires were completed in 1970.

### Supplementary Text ###
Ellipsoïdes

Un ellipsoïde est une sphère qui a été étirée ou comprimée dans une ou plusieurs directions. Plus précisément, c'est l'image d'une sphère sous une transformation affine. Un ellipsoïde entretient la même relation avec la sphère qu'une ellipse avec un cercle.

### Draft ###

### 1. Query Analysis

"Pourquoi on dit que les sphères d'eau sont des ellipsoïdes alors que les sphéroïdes d'eau pas ?"

Translation: "Why are water spheres called ellipsoids while water spheroids not?"

Core confusion: terminological hierarchy vs geometric reality. User assumes mutually exclusive categories.

? Key semantic issue: "sont des" = "are" suggests strict classification. Need to unpack.

### 2. Geometric Foundation

● Water spheres: perfect spheres. All cross-sections circular.
● Water spheroids: flattened at top/bottom. Circular cross-sections only in meridian plane.

Both = special cases of ellipsoids.

Ellipsoid definition: 3 unequal semi-axes. Can be:
- Prolate: elongated along one axis
- Oblate: flattened at poles  
- Triaxial: all axes unequal

→ Both sphere and spheroid qualify as special ellipsoids.

### 3. Terminological Hierarchy

Critical insight: classification trees vs naming conventions.

Geometric hierarchy:
```
Ellipsoids
├─ Special cases
│  ├─ Spheres (a=b=c)
│  └─ Spheroids (a=b ≠ c)
└─ General (a≠b≠c)
```

Naming hierarchy:
- Perfect spheres → "spheres" 
- Flattened spheres → "spheroids"
- Neither → "ellipsoids"

∴ Terminology reflects degree of departure from perfect sphere.

### 4. Physical Constraints

⟨H≈0.8⟩ Water towers... engineering reality vs pure geometry. Practical construction: easier to build near-spherical tanks than perfectly spherical. Cost vs aesthetics. Maintenance access through flat tops. Historical evolution of water storage... ⟨/H⟩

NB: Real-world water towers rarely achieve perfect spherical geometry.

### 5. Resolution

User's confusion stems from category confusion:

Ellipsoids = general class (inclusive)
Spheres = subset of ellipsoids  
Spheroids = subset of ellipsoids

Both "are ellipsoids" but named by degree of deviation from perfect sphere.

→ Question reveals deeper issue: geometric vs terminological classification systems can diverge.

**Answer structure:**
1. Acknowledge confusion (valid geometric insight)
2. Clarify both are ellipsoids 
3. Explain naming convention logic
4. Provide concrete examples

✓ Resolved: terminology reflects practical engineering categories, not pure geometric exclusion.

### Answer###
Votre question touche à une distinction importante dans la terminologie géométrique. Les sphères d'eau et les sphéroïdes d'eau sont tous deux des ellipsoïdes spéciaux, mais ils diffèrent par leur symétrie et leur classification.

Une sphère d'eau est une forme parfaitement sphérique - toutes les sections transversales dans n'importe quelle direction (est-ouest, nord-sud, ou haut-bas) forment des cercles parfaits. En revanche, un sphéroïde d'eau présente une forme aplatie : le sommet est plus large que haut, créant une forme ressemblant à un oreiller rond écrasé. Les sections transversales d'un sphéroïde suivant deux directions (est-ouest ou nord-sud) forment des ellipses, mais seulement dans une direction (haut-bas) on obtient un cercle parfait.

Les ellipsoïdes sont des objets qui ont été étirés ou comprimés dans une ou plusieurs directions, créant des formes qui maintiennent la symétrie sphérique mais avec des axes de différentes longueurs. Les sphères présentent une symétrie dans 3 directions, tandis que les sphéroïdes n'en présentent que 2.

Cette classification hiérarchique explique pourquoi les deux types peuvent être considérés comme des ellipsoïdes : ils représentent tous deux des cas particuliers de cette classe générale, mais avec des propriétés géométriques distinctes qui les classent différemment dans la hiérarchie terminologique.