空间曲线曲率为何恒非负？求直观解释与证明

Hey there, this is a really good question—curvature's non-negativity makes perfect sense once you connect the math to what it actually measures, so let's unpack this both intuitively and formally.

At its core, curvature is a measure of how much a curve bends, not which direction it bends. Think of it like driving along the curve:

• If you're on a straight line, you never touch the steering wheel—your direction never changes, so the bending degree is 0, and curvature is 0.
• If you turn the wheel, whether left, right, or even a diagonal direction in 3D space, you're changing your forward direction. Curvature only cares about how sharply you're turning, not the direction of the turn.
• Here's a relatable analogy: curvature is like the "severity of your turn" as an absolute value. A 90-degree sharp turn has the same severity (and thus same positive curvature) whether you go left or right. A gentle curve has a smaller positive curvature, and no turn at all has zero curvature. There's no such thing as a "negative amount of bending"—just like you can't have a negative distance traveled, curvature is a measure of magnitude, so it's always non-negative.

Let's formalize this using standard differential geometry for regular space curves (curves where the tangent vector is never zero, meaning r’(t) ≠ 0 for all valid t):

1. Start with a parameterized space curve: r(t) = (x(t), y(t), z(t)), where t is any valid parameter (arc length is a common choice, but we'll use a general parameter here for generality).
2. Compute the first derivative (tangent vector): r’(t) = (x’(t), y’(t), z’(t))
3. Compute the second derivative: r''(t) = (x''(t), y''(t), z''(t))
4. The curvature κ(t) is defined by the formula:

   κ(t) = |**r’(t) × r''(t)| / |**r’(t)|³
   

   Now let's break down why this value is always non-negative:
   • The numerator is the magnitude (norm) of the cross product of the tangent vector and its derivative. By definition, the magnitude of any vector is non-negative: |v| ≥ 0 for any vector v.
   • The denominator is the cube of the magnitude of the tangent vector. For regular curves, we know r’(t) ≠ 0, so |r’(t)| > 0—this means the denominator is a positive number.
   • Dividing a non-negative number by a positive number always results in a non-negative value. Therefore, κ(t) ≥ 0 for all t in the curve's domain.

A quick side note: For plane curves, you might encounter "signed curvature" which can be positive or negative. This accounts for the direction of bending relative to a fixed normal vector. But in 3D space, there's no consistent "left" or "right" direction to define such a sign, so curvature is strictly defined as the magnitude of the curve's bending—hence it's always non-negative.

内容的提问来源于stack exchange，提问作者Umar Khan