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The Definition of Strings 📂Geometry

The Definition of Strings

Definition 1

  1. Let there be a given curve α:(c,d)R3\alpha : (c,d) \to \mathbb{R}^{3}. When c<a<b<dc < a < b < d, for all t[a,b]t \in [a,b] that satisfies α(t)=γ(t)\alpha (t) = \gamma (t), γ:[a,b]R3\gamma : [a,b] \to \mathbb{R}^{3} is called a Chord or Curve Segment.
  2. The length of chord γ:[a,b]R3\gamma : [a,b] \to \mathbb{R}^{3}, denoted l[a,b](γ)l_{[a,b]}(\gamma), is defined as follows. l[a,b](γ):=abdγdtdt l_{[a,b]}(\gamma) := \int_{a}^{b} \left| {{ d \gamma } \over { d t }} \right| dt
  3. The Length of Arc is defined as follows for s=h(t)s = h(t) according to α\alpha. h(t):=t0tdαdtdt h(t) := \int_{t_{0}}^{t} \left| {{ d \alpha } \over { dt }} \right| dt

Explanation

In the definition, the curve segment γ\gamma is naturally defined as a part of α\alpha, and its definition of length readily follows from the concept of the Jacobian. The idea of the length of an arc is considered as a convenient way to avoid tediously redefining and discussing the length of a chord for the interval [t,t0]\left[ t, t_{0} \right] we are interested in.

Reparameterization of the Length of an Arc

Especially, the function hh is a reparameterization itself and is called the reparameterization of the length of an arc if it satisfies the following for g:(c,d)(a,b)g: (c,d) \to (a,b) and h(t)=t0tdαdudu\displaystyle h(t) = \int_{t_{0}}^{t} \left| {{ d \alpha } \over { du }} \right| du. g(s)=h1(s) g(s) = h^{-1} (s) This is nothing but a well-known variable transformation. Let’s understand why this is used. β=αg=αh1 \begin{align*} \beta =& \alpha \circ g \\ =& \alpha \circ h^{-1} \end{align*} If it is given that α=βh\alpha = \beta \circ h, and since hh is an increasing function, then h=h\left| h^{\prime} \right| = h^{\prime}, thus s=t0tα(u)du=t0t(βh)du=t0t(βh)h(u)du=0sβ(v)dv \begin{align*} s =& \int_{t_{0}}^{t} \left| \alpha^{\prime} (u) \right| du \\ =& \int_{t_{0}}^{t} \left| \left( \beta \circ h \right)^{\prime} \right| du \\ =& \int_{t_{0}}^{t} \left| \left( \beta^{\prime} \circ h \right) \right| h^{\prime}(u) du \\ =& \int_{0}^{s} \left| \beta^{\prime} (v) \right| dv \end{align*} Differentiating both sides with respect to ss, according to the Fundamental Theorem of Calculus, 1=β(s) 1 = \left| \beta^{\prime} (s) \right| This means the Speed is constant at 11, making it easier to handle.

Furthermore, from the following theorem, one can understand that the reparameterization of a curve does not alter the length of the curve. Since gg moves by (c,d)(c,d), α\alpha moves by (a,b)(a,b), which is obvious. By analogy, the path from the starting point to the destination remains the same, only the mode of transportation changes, therefore the distance remains unchanged while only the speed changes.

Theorem

For reparameterization, the length of the chord is invariant.

Proof

For the chord α\alpha and reparameterization g:[c,d][a,b]g : [c,d] \to [a,b], suppose β:=αg\beta := \alpha \circ g, then the length of β\beta is as follows. cddβdtdr=cd(dαdt)(dgdr)dr=cddαdtdgdrdr \begin{align*} \int_{c}^{d} \left| {{ d \beta } \over { d t }} \right| dr =& \int_{c}^{d} \left| \left( {{ d \alpha } \over { d t }} \right) \left( {{ d g } \over { d r }} \right) \right| dr \\ =& \int_{c}^{d} \left| {{ d \alpha } \over { d t }} \right| \left| {{ d g } \over { d r }} \right| dr \end{align*}


Case 1. If gg is an increasing function

Since g(c)=a,g(d)=bg(c) = a, g(d) = b and dgdr=dgdr\left| {{ d g } \over { d r }} \right| = {{ d g } \over { d r }}, cddαdtdgdrdr=cddαdtdgdrdr=cddαdtdg=abdαdtdt \begin{align*} \int_{c}^{d} \left| {{ d \alpha } \over { d t }} \right| \left| {{ d g } \over { d r }} \right| dr =& \int_{c}^{d} \left| {{ d \alpha } \over { d t }} \right| {{ d g } \over { d r }} dr \\ =& \int_{c}^{d} \left| {{ d \alpha } \over { d t }} \right| d g \\ =& \int_{a}^{b} \left| {{ d \alpha } \over { d t }} \right| d t \end{align*}


Case 2. If gg is a decreasing function

Since g(c)=b,g(d)=ag(c) = b, g(d) = a and dgdr=dgdr\left| {{ d g } \over { d r }} \right| = - {{ d g } \over { d r }}, cddαdtdgdrdr=cddαdt(dgdr)dr=cddαdtdg=badαdtdt=abdαdtdt \begin{align*} \int_{c}^{d} \left| {{ d \alpha } \over { d t }} \right| \left| {{ d g } \over { d r }} \right| dr =& \int_{c}^{d} \left| {{ d \alpha } \over { d t }} \right| \left( - {{ d g } \over { d r }} \right) dr \\ =& - \int_{c}^{d} \left| {{ d \alpha } \over { d t }} \right| d g \\ =& - \int_{b}^{a} \left| {{ d \alpha } \over { d t }} \right| d t \\ =& \int_{a}^{b} \left| {{ d \alpha } \over { d t }} \right| d t \end{align*}


Therefore, whether gg is an increasing or decreasing function, the length of the chord remains constant.


  1. Millman. (1977). Elements of Differential Geometry: p20. ↩︎