📚 11 Tangent vector and curvature in two dimensions | 二维空间中的切线向量与曲率
In IB Mathematics: Analysis and Approaches HL, planar curves are a gateway to understanding how objects move and how geometry can be analysed using calculus. The tangent vector tells us the instantaneous direction of travel, while curvature quantifies how sharply the path bends. Mastering these concepts connects parametric equations, vector differentiation and real-world motion.
在IB数学分析与方法高水平课程中，平面曲线是理解物体运动以及如何利用微积分分析几何的入口。切线向量告诉我们瞬时的运动方向，而曲率则量化了路径弯曲的剧烈程度。掌握这些概念将参数方程、向量微分和现实运动联系在一起。
1. Parametric Curves in the Plane | 平面中的参数曲线
A plane curve can be described by a vector-valued function r: ℝ → ℝ kinematics, t often represents time; in geometry, it might be an angle or a free variable.
平面曲线可以用向量值函数 r: ℝ → ℝ² 描述，即 r(t) = (x(t), y(t))，其中 t 是实参数。随着 t 变化，点便在二维空间中描出轨迹。在运动学中，t 常代表时间；在几何中，它可以是角度或自由变量。
For the curve to be 'smooth', we require that x(t) and y(t) have continuous first derivatives and that r'(t) is not the zero vector, avoiding cusps or corners.
要保证曲线“光滑”，我们要求 x(t) 和 y(t) 具有连续的一阶导数，并且 r'(t) 不是零向量，以此避免尖点或折角。
2. Tangent Vector | 切线向量
The derivative r'(t) = (x'(t), y' tangent line and, when t is time, it represents the velocity vector of a moving particle.
导数 r'(t) = (x'(t), y'(t)) 称为点 r(t) 处的切线向量。它给出切线的方向，并且当 t 为时间时，它表示运动质点的速度向量。
Tangent vector = r'(t) = (x'(t), y'(t))
If r'(t) is nonzero, the line through r(t) parallel to r'(t) is precisely the tangent line. The slope of this line, when defined, is y'(t)/x'(t) provided x'(t) ≠ 0.
如果 r'(t) 非零，那么经过 r(t) 且平行于 r'(t) 的直线就是切线。当可定义时，该切线的斜率为 y'(t)/x'(t)（假设 x'(t) ≠ 0）。
3. Unit Tangent Vector | 单位切线向量
To focus only on direction, we normalise the tangent vector to obtain the unit tangent vector T(t). It is defined wherever the speed |r'(t)| is nonzero:
为了仅关心方向，我们将切线向量单位化，得到单位切线向量 T(t)。它在速率 |r'(t)| 非零处有定义：
T(t) = r'(t) / |r'(t)|, where |r'(t)| = √( (x'(t))² + (y'(t))² )
The vector T(t) has constant length 1 and points along the forward motion. It only changes direction when the path bends; this change is central to the concept of curvature.
向量 T(t) 具有恒定的长度 1，并指向前进方向。它仅在路径弯曲时才改变方向；这一改变正是曲率概念的核心。
4. Arc Length and Reparameterisation | 弧长与重参数化
The arc length s from a starting parameter t=a is given by s(t) = ∫ₐᵗ |r'(u)| du. Differentiating yields ds/dt = |r'(t)|, so the speed is the rate of change of arc length with respect to t.
从参数 t=a 开始的弧长 s 由 s(t) = ∫ₐᵗ |r'(u)| du 给出。求导得到 ds/dt = |r'(t)|，因此速率就是弧长关于 t 的变化率。
When a curve is reparameterised by arc length s, the derivative dr/ds becomes exactly the unit tangent vector T(s). This special parameterisation eliminates the scaling effects of t and simplifies geometric analysis.
当曲线用弧长 s 重参数化时，导数 dr/ds 恰好成为单位切线向量 T(s)。这种特殊参数化消除了参数 t 的缩放效应，简化了几何分析。
dr/ds = T(s), |T(s)| = 1
5. Defining Curvature | 曲率的定义
Curvature κ (kappa) measures the rate at which the unit tangent vector turns with respect to arc length. Formally, using the arc length parameter s,
曲率 κ（希腊字母 kappa）衡量单位切线向量相对于弧长的转向速率。严格地，使用弧长参数 s，
κ = |dT/ds|
se T never changes. A circle of radius R turns uniformly, and one can show its curvature is the constant 1/R.
直观上，较大的 κ 意味着在给定弧长内曲线弯曲得更厉害。直线具有 κ = 0，因为 T 永不改变。半径为 R 的圆均匀转向，可以证明其曲率为常数 1/R。
This definition is coordinate-independent and captures the intrinsic bending of the curve, regardless of how it is parameterised.
这一定义与坐标系无关，捕捉了曲线的内蕴弯曲程度，不受参数化方式影响。
6. Curvature Formula for Parametric Curves | 一般参数曲线的曲率公式
Since many curves are given with an arbitrary parameter t rather than s, we need a computational formula. Using the chain rule and properties of the 2D cross product (which gives a scalar magnitude |x'y'' − y'x''|), curvature can be expressed as
由于许多曲线由任意参数 t 而非 s 给出，我们需要一个计算公式。利用链式法则和二维叉积的性质（叉积产生标量大小 |x'y'' − y'x''|），曲率可表示为
κ(t) = |x'(t) y''(t) − y'(t) x''(t)| / ( (x'(t))² + (y'(t))² )³⁄²
The numerator is the absolute value of the 2D cross product r' × r''; the denominator is the cube of the speed |r'(t)|. This formula is valid for any twice-differentiable parametric curve where r'(t) ≠ 0.
分子是二维叉积 r' × r'' 的绝对值；分母是速率 |r'(t)| 的三次方。该公式对任意二阶可微且 r'(t) ≠ 0 的参数曲线都成立。
A memorable form is κ = |r' × r''| / |r'|³, where × here yields the signed scalar (x'y'' − y'x'').
一个便于记忆的形式是 κ = |r' × r''| / |r'|³，其中 × 在这里产生有向标量 (x'y'' − y'x'')。
7. Curvature for Cartesian Curves | 笛卡尔曲线的曲率
When a curve is given explicitly as y = f(x), we can parametrise it by x: r(x) =
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