欧几里得距离采用平方项求和形式的直观原因及深层洞察问询
Hey there! Great question—this is one of those things that feels obvious once you dig into it, but totally opaque at first glance. Let me break this down for you:
直观几何角度（勾股定理的延伸）: Start with 2D space first—if you’ve got two points $(x_1,y_1)$ and $(x_2,y_2)$, the straight-line distance between them is the hypotenuse of a right triangle with legs $|x_2-x_1|$ and $|y_2-y_1|$. Pythagoras’ theorem tells us the hypotenuse squared equals the sum of the squares of the two legs, so we take the square root to get the actual "straight line" distance. Extend this logic to 3D (or any $n$-dimensional space like $\mathbb R^3$) and it’s just adding the square of the difference in each additional perpendicular dimension—each axis acts like another leg of a higher-dimensional right triangle, so the same rule applies.
为什么选平方而非绝对值？: Squares have way nicer mathematical properties than absolute values for this use case. For starters, they’re differentiable everywhere (absolute value has a sharp kink at 0), which makes them infinitely easier to work with in calculus, optimization problems, machine learning (think least squares regression), and physics. Also, squares naturally penalize larger differences more heavily, which aligns with our intuitive sense that bigger gaps between points should matter more—though that’s a handy side effect, not the original geometric reason.
深层数学洞察：内积空间的本质: Euclidean distance is deeply tied to the dot product (a specific type of inner product) in $\mathbb R^n$. The distance between two vectors $\mathbf u$ and $\mathbf v$ is defined as $|\mathbf u - \mathbf v|$, where the norm (length) $|\mathbf w| = \sqrt{\mathbf w \cdot \mathbf w}$. Inner products give us a consistent way to define length and angle across vector spaces, and the squared term comes directly from this fundamental inner product definition. This is why Euclidean distance plays so nicely with linear algebra—its structure is baked into the core operations of these spaces.
历史视角: At its root, this formula is just a generalized version of Pythagoras’ theorem, which came from ancient geometric observations of right triangles. Over time, mathematicians formalized this into the framework of Euclidean spaces, where the squared sum plus square root became the standard distance measure because it’s both consistent with our everyday geometric intuition and mathematically robust enough to scale to higher dimensions.
It makes total sense that you wouldn’t guess this off the bat—our brains aren’t wired to intuit higher-dimensional geometry naturally, but once you connect it back to the 2D case and the math behind inner products, it all clicks into place!
备注：内容来源于stack exchange，提问作者Willerie
