2.2 Arc length and reparametrization

Arc length and reparameterization are key concepts in differential geometry. They allow us to measure curves and change how we describe them without altering their shape. This topic builds our understanding of intrinsic curve properties, independent of specific parameterizations.

Arc length lets us quantify curve lengths, while reparameterization gives us flexibility in describing curves. Together, they provide powerful tools for analyzing curve geometry and solving problems in physics, engineering, and other fields that involve curved paths and surfaces.

Definition of arc length

• Arc length is a fundamental concept in differential geometry that measures the length of a curve segment in a metric space
• Intuitively, arc length can be thought of as the distance traveled along a curve if one were to "walk" along it
• Arc length is defined using the metric of the space and is independent of the parameterization of the curve

Arc length formula

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• The arc length of a curve $\gamma:[a,b]\to M$ in a metric space $(M,d)$ is given by the formula: $L(\gamma)=\int_a^b \sqrt{g_{\gamma([t](https://www.fiveableKeyTerm:t))}(\gamma'(t),\gamma'(t))} dt$ where $g$ is the metric tensor and $\gamma'(t)$ is the tangent vector to the curve at $t$
• In Euclidean space $\mathbb{R}^n$, the arc length formula simplifies to: $L(\gamma)=\int_a^b \|\gamma'(t)\| dt$ where $\|\cdot\|$ denotes the Euclidean norm
• The arc length formula involves an integral, which can be interpreted as a limit of Riemann sums approximating the length of the curve by summing the lengths of small linear segments

Arc length parameterization

• A curve $\gamma:[a,b]\to M$ is said to be parameterized by arc length (or unit speed) if $\|\gamma'(t)\|=1$ for all $t\in[a,b]$
• In other words, the tangent vector to the curve has unit length at every point along the curve
• Arc length parameterization is a natural and intrinsic way to parameterize a curve, as it depends only on the geometry of the curve and not on any particular coordinate system
• Given a curve $\gamma:[a,b]\to M$, one can always reparameterize it by arc length by defining a new parameter $[s](https://www.fiveableKeyTerm:s)(t)=\int_a^t \|\gamma'(u)\| du$ and considering the curve $\tilde{\gamma}(s)=\gamma(t(s))$, where $t(s)$ is the inverse function of $s(t)$

Unit speed curves

• A curve parameterized by arc length is also called a unit speed curve
• Unit speed curves have several nice properties that make them useful in differential geometry:
  • The tangent vector field $\tilde{\gamma}'(s)$ along a unit speed curve has constant length 1
  • The arc length parameter $s$ corresponds to the actual distance traveled along the curve
  • Geodesics (locally length-minimizing curves) are often parameterized by arc length
• Examples of unit speed curves include:
  • Great circles on a sphere (with appropriate parameterization)
  • Straight lines in Euclidean space (with linear parameterization)

Properties of arc length

• Arc length is a fundamental concept in differential geometry and has several important properties that make it useful for studying curves and surfaces
• These properties are intrinsic to the geometry of the curve and do not depend on any particular parameterization or coordinate system
• Understanding the properties of arc length is crucial for developing a deeper understanding of the geometry of curves and surfaces

Invariance under reparameterization

• One of the most important properties of arc length is its invariance under reparameterization
• This means that the arc length of a curve does not change if we parameterize the curve differently, as long as the new parameterization is still regular (i.e., has non-vanishing tangent vectors)
• More precisely, if $\gamma:[a,b]\to M$ and $\tilde{\gamma}:[c,d]\to M$ are two regular parameterizations of the same curve, then $L(\gamma)=L(\tilde{\gamma})$
• This property reflects the fact that arc length is an intrinsic property of the curve itself, not of any particular parameterization

Additivity of arc lengths

• Another important property of arc length is its additivity
• If a curve $\gamma:[a,b]\to M$ is split into two subcurves $\gamma_1:[a,c]\to M$ and $\gamma_2:[c,b]\to M$, where $a<c<b$, then the arc length of $\gamma$ is equal to the sum of the arc lengths of $\gamma_1$ and $\gamma_2$: $L(\gamma)=L(\gamma_1)+L(\gamma_2)$
• This property allows us to compute the arc length of a curve by breaking it up into smaller, more manageable pieces
• It also reflects the idea that arc length measures the total distance traveled along the curve, and this distance can be accumulated over different segments of the curve

Arc length and line integrals

• Arc length can be expressed as a special case of a line integral
• Given a curve $\gamma:[a,b]\to M$ and a continuous function $f:M\to\mathbb{R}$, the line integral of $f$ along $\gamma$ is defined as: $\int_\gamma f ds = \int_a^b f(\gamma(t))\|\gamma'(t)\| dt$
• If we take $f\equiv 1$, then the line integral reduces to the arc length formula: $L(\gamma)=\int_\gamma 1 ds = \int_a^b \|\gamma'(t)\| dt$
• This connection between arc length and line integrals is important in many applications, such as computing the work done by a force along a curved path

Reparameterization of curves

• Reparameterization is the process of changing the parameter used to describe a curve while keeping the image (shape) of the curve unchanged
• It is an important tool in differential geometry for simplifying calculations and highlighting different properties of curves
• Reparameterization can be used to make a curve unit speed, to change its orientation, or to focus on a particular portion of the curve

Definition of reparameterization

• A reparameterization of a curve $\gamma:[a,b]\to M$ is a new curve $\tilde{\gamma}:[c,d]\to M$ of the form $\tilde{\gamma}=\gamma\circ\varphi$, where $\varphi:[c,d]\to[a,b]$ is a diffeomorphism (smooth bijective map with smooth inverse)
• The map $\varphi$ is called the reparameterization function and it determines how the parameter of the curve is changed
• Common examples of reparameterization functions include:
  • Linear functions: $\varphi(t)=\alpha t+\beta$, which change the speed and shift the parameter interval
  • Inverse functions: $\varphi(t)=t^{-1}$, which reverse the orientation of the curve
  • Trigonometric functions: $\varphi(t)=\tan(t)$, which can be used to focus on a particular portion of the curve

Regular and singular points

• A point $\gamma(t_0)$ on a curve $\gamma:[a,b]\to M$ is called regular if the tangent vector $\gamma'(t_0)$ is non-zero
• If $\gamma'(t_0)=0$, then $\gamma(t_0)$ is called a singular point or a stationary point of the curve
• Regular points are important because they allow us to define the tangent line to the curve, which is a key concept in differential geometry
• Singular points, on the other hand, can indicate important features of the curve, such as cusps or self-intersections
• When reparameterizing a curve, it is important to ensure that regular points remain regular under the reparameterization, so that the geometry of the curve is preserved

Reparameterization and orientation

• The orientation of a curve refers to the direction in which the curve is traversed, as determined by the direction of its tangent vectors
• A curve can be parameterized in two opposite orientations, corresponding to traversing the curve in opposite directions
• Reparameterization can be used to change the orientation of a curve by composing it with a reparameterization function that reverses the order of the parameter interval
• For example, if $\gamma:[a,b]\to M$ is a curve and $\varphi(t)=a+b-t$ is a reparameterization function, then $\tilde{\gamma}=\gamma\circ\varphi$ is a reparameterization of $\gamma$ with the opposite orientation
• Changing the orientation of a curve can be useful for various purposes, such as aligning the orientation of multiple curves or ensuring consistency with a chosen convention

Arc length parameterization

• Arc length parameterization is a special type of reparameterization in which the parameter of the curve corresponds to the arc length along the curve
• In other words, a curve $\gamma:[a,b]\to M$ is parameterized by arc length if for any $t_1,t_2\in[a,b]$, the arc length of the portion of the curve between $\gamma(t_1)$ and $\gamma(t_2)$ is equal to $|t_2-t_1|$
• Arc length parameterization is a natural and intrinsic way to parameterize a curve, as it depends only on the geometry of the curve and not on any particular coordinate system

Existence and uniqueness

• Given a regular curve $\gamma:[a,b]\to M$, there always exists a reparameterization of $\gamma$ that is parameterized by arc length
• Moreover, this arc length parameterization is unique up to a shift in the parameter (i.e., up to composition with a translation function $\varphi(t)=t+c$)
• The existence of arc length parameterization follows from the fact that the arc length function $s(t)=\int_a^t \|\gamma'(u)\| du$ is always strictly increasing and continuous for a regular curve, and therefore has a continuous inverse function $t(s)$
• The uniqueness (up to a shift) follows from the fact that any two arc length parameterizations of the same curve must differ by a constant, since they both measure the same arc length along the curve

Construction of arc length parameterization

• To construct an arc length parameterization of a given curve $\gamma:[a,b]\to M$, we follow these steps:
  1. Compute the arc length function $s(t)=\int_a^t \|\gamma'(u)\| du$
  2. Find the inverse function $t(s)$ of $s(t)$, defined on an interval $[0,L]$ where $L=s(b)$ is the total arc length of the curve
  3. Define the arc length parameterized curve $\tilde{\gamma}:[0,L]\to M$ by $\tilde{\gamma}(s)=\gamma(t(s))$
• In practice, finding the inverse function $t(s)$ can be difficult or impossible to do analytically, so numerical methods or approximations may be necessary
• However, the construction is still useful for theoretical purposes and for understanding the properties of arc length parameterized curves

Properties of arc length parameterized curves

• Arc length parameterized curves have several useful properties that make them important in differential geometry:
  • The tangent vector $\tilde{\gamma}'(s)$ has unit length for all $s$, i.e., $\|\tilde{\gamma}'(s)\|=1$
  • The arc length parameter $s$ corresponds to the actual distance traveled along the curve from the starting point
  • The integral of a function $f$ along the curve simplifies to $\int_{\tilde{\gamma}} f ds = \int_0^L f(\tilde{\gamma}(s)) ds$
  • Geodesics (locally length-minimizing curves) are often parameterized by arc length
• These properties make arc length parameterization a natural and convenient choice for many problems in differential geometry, such as studying the curvature and torsion of curves, or finding the shortest path between two points on a surface

Applications of arc length and reparameterization

• Arc length and reparameterization are fundamental concepts in differential geometry with a wide range of applications in mathematics, physics, and engineering
• These applications often involve studying the intrinsic geometry of curves and surfaces, or finding optimal paths and trajectories subject to certain constraints
• Some of the key applications of arc length and reparameterization include:

Computation of curvature

• Curvature is a measure of how much a curve deviates from being a straight line at a given point
• For a regular curve $\gamma:[a,b]\to\mathbb{R}^3$ in Euclidean space, the curvature at a point $\gamma(t)$ is defined as: $\kappa(t)=\frac{\|\gamma'(t)\times\gamma''(t)\|}{\|\gamma'(t)\|^3}$
• This formula involves the first and second derivatives of the curve, which can be complicated to compute in an arbitrary parameterization
• However, if the curve is parameterized by arc length, then $\|\gamma'(s)\|=1$ and the formula simplifies to: $\kappa(s)=\|\gamma''(s)\|$
• This makes it much easier to compute and study the curvature of the curve, which has important applications in geometry, physics, and engineering (e.g., in the design of roads, railways, and mechanical parts)

Frenet-Serret formulas

• The Frenet-Serret formulas are a set of equations that describe the kinematic properties of a regular curve in Euclidean space, such as its curvature and torsion
• They involve the Frenet frame, which is a moving orthonormal frame along the curve consisting of the tangent, normal, and binormal vectors
• For a regular curve $\gamma:[a,b]\to\mathbb{R}^3$ parameterized by arc length, the Frenet-Serret formulas are: T'(s) &= \kappa(s)N(s) \\ N'(s) &= -\kappa(s)T(s) + \tau(s)B(s) \\ B'(s) &= -\tau(s)N(s) \end{aligned}$$ where $T,N,B$ are the tangent, normal, and binormal vectors, and $\kappa,\tau$ are the curvature and torsion functions
• These formulas provide a powerful tool for studying the intrinsic geometry of curves and have applications in various fields, such as physics (e.g., in the study of particle trajectories and magnetic fields) and computer graphics (e.g., in the design of 3D curves and surfaces)

Geodesics and shortest paths

• Geodesics are curves that locally minimize the distance between two points on a surface or a manifold
• They are the natural generalization of straight lines in Euclidean space and play a fundamental role in differential geometry and its applications
• Geodesics are often parameterized by arc length, which makes it easier to study their properties and compute their lengths
• In many applications, such as in robotics, computer vision, and geophysics, finding the shortest path between two points on a surface is an important problem that involves computing geodesics
• Reparameterization can also be used to simplify the equations of geodesics and make them easier to solve numerically, by choosing a parameterization that is adapted to the geometry of the surface or manifold

Relationship between arc length and energy

• Arc length and energy are two closely related concepts in differential geometry and its applications
• The energy of a curve is a measure of how much the curve "stretches" or "bends" as it moves through space, and it can be used to characterize the shape and geometry of the curve
• In many problems, such as in the study of elasticity or the design of optimal trajectories, finding curves that minimize energy is an important goal

Definition of energy of a curve

• For a regular curve $\gamma:[a,b]\to M$ in a Riemannian manifold $(M,g)$, the energy of $\gamma$ is defined as: $E(\gamma)=\frac{1}{2}\int_a^b g_{\gamma(t)}(\gamma'(t),\gamma'(t)) dt$
• This formula involves the Riemannian metric $g$ and the velocity vector $\gamma'(t)$ of the curve, and it measures the total "kinetic energy" of the curve as it moves through the manifold
• In Euclidean space, the energy formula simplifies to: $E(\gamma)=\frac{1}{2}\int_a^b \|\gamma'(t)\|^2 dt$
• The energy of a curve is always non-negative and vanishes only for constant curves (i.e., curves whose velocity is zero)

Energy minimizing curves

• A curve $\gamma:[a,b]\to M$ is said to be energy minimizing if it minimizes the energy functional $E(\gamma)$ among all curves with the same endpoints $\gamma(a)$ and $\gamma(b