几何空间类型、点表示系统术语规范及适配关系咨询

Hey there! I totally get the confusion here—wiki docs can be inconsistent with terminology, especially when mixing abstract geometric spaces and their coordinate representations. Let's break this down clearly.

Core Terminology Clarification #

First, let's fix the mix-up between Euclidean Space and Euclidean System:

• Euclidean Space: This is an abstract mathematical structure defined by Euclid's five axioms (like the parallel postulate). It describes inherent geometric relationships (distance, angles, parallelism) between points, lines, and planes—it's coordinate-agnostic; you don't need numbers to define its rules.
• Euclidean System: This is a loose term that usually refers to a set of coordinate systems designed to map points in Euclidean space. Think of it as the "toolkit" for assigning numerical values to points in that abstract space. Some docs use this term interchangeably with Euclidean space, but strictly speaking, the space is the underlying structure, and the system is how we measure/describe it.

Common Geometric Spaces & Their Compatible Coordinate Systems #

Below is a curated list of major geometric spaces, paired with the coordinate representations that work best for them:

1. Euclidean Space #

The "standard" flat geometry we learn in school. Compatible coordinate systems include:

• Cartesian Coordinates: The classic rectangular system (2D: (x,y); 3D: (x,y,z)). It directly aligns with Euclidean space's distance and angle rules.
• Polar Coordinates (2D): Uses (r, θ) (radius + angle from a reference axis) as an alternative to Cartesian; easily convertible between the two.
• Cylindrical/Spherical Coordinates (3D): Extensions of polar coordinates for 3D space—cylindrical uses (r, θ, z), spherical uses (r, θ, φ).
• Oblique Coordinates: A linear system where axes aren't perpendicular; still fits Euclidean space rules, just less commonly used for simplicity.

2. Elliptical Space #

A non-Euclidean space where no two lines are parallel (all lines eventually intersect, like the surface of a sphere where "lines" are great circles). Compatible systems:

• Spherical Coordinates: The most natural fit—for a unit elliptical space, points are represented by (θ, φ) (latitude/longitude-like angles), with opposite points treated as identical.
• Elliptic Cartesian Coordinates: A projected system used for planar elliptical geometry, converting the curved space into a flat coordinate framework for calculations.

3. Hyperbolic Space #

Another non-Euclidean space where infinitely many lines can pass through a point parallel to a given line. Compatible systems:

• Poincaré Disk/Upper Half-Plane Coordinates: Conformal systems that preserve angles, making them ideal for visualizing hyperbolic geometry (think of the distorted grid in a fisheye lens).
• Klein Disk Coordinates: A projective system that keeps straight lines straight, useful for theoretical proofs and geometric constructions.
• Hyperbolic Cartesian Coordinates: An algebraic system based on a pseudo-Euclidean metric, used for computational work where angle preservation isn't a priority.

4. Projective Space #

A geometric space designed to handle "infinite points" (like the vanishing points in perspective drawing). Compatible systems:

• Homogeneous Coordinates: The foundational system—uses (n+1) dimensions to represent n-dimensional projective points (e.g., 2D projective points are written as (x,y,w), where w≠0 maps to the Cartesian point (x/w, y/w)).
• Barycentric Coordinates: A system based on triangle vertices, widely used in computer graphics for interpolation and geometric transformations.

Key Adaptation Rule #

Each geometric space has native coordinate systems that directly reflect its unique properties (e.g., spherical coordinates for elliptical space). You can technically use other systems via conversion, but this often complicates calculations or distorts the space's inherent geometric rules.

内容的提问来源于stack exchange，提问作者infoclogged