Parametrized curves are the building blocks of differential geometry, describing paths in space using functions of a parameter. They allow us to analyze curves mathematically, capturing their shape, length, and behavior through properties like regularity, arc length, and curvature.

Understanding parametrized curves is crucial for studying more complex geometric objects. This topic introduces key concepts like the Frenet frame, which provides a local coordinate system for curves, and special types of curves with unique properties used in various applications.

Definition of parametrized curves

• Parametrized curves are a fundamental concept in differential geometry that describe curves using a parametric equation
• Parametric equations represent a curve as a function of an independent variable, typically denoted as $t$, which maps to points $(x(t), y(t))$ in the plane or $(x(t), y(t), z(t))$ in space

Parametrization of curves

• Parametrization is the process of representing a curve using parametric equations
• A curve $C$ can be parametrized by a function $\gamma: I \to \mathbb{R}^n$, where $I$ is an interval in $\mathbb{R}$ and $n$ is the dimension of the ambient space (typically 2 or 3)
• The function $\gamma(t) = (x(t), y(t))$ or $\gamma(t) = (x(t), y(t), z(t))$ assigns a unique point on the curve to each value of the parameter $t$

Smooth vs non-smooth curves

• Smooth curves are parametrized curves whose component functions (e.g., $x(t)$, $y(t)$, $z(t)$) are continuously differentiable
• Non-smooth curves have points where the derivative is not defined or is discontinuous (cusps, corners)
• Smoothness is an important property for analyzing the geometric properties of curves, such as tangent vectors and curvature

Properties of parametrized curves

• Parametrized curves have several important properties that allow for their geometric analysis and manipulation
• These properties include regularity, reparametrization, and equivalence classes of parametrizations

Regularity of curves

• A parametrized curve $\gamma: I \to \mathbb{R}^n$ is regular if its derivative $\gamma'(t)$ is non-zero for all $t \in I$
• Regularity ensures that the curve does not have any singular points or self-intersections
• Regular curves have well-defined tangent vectors at every point

Reparametrization of curves

• Reparametrization is the process of changing the parameter of a curve without altering its geometric shape
• Given a curve $\gamma(t)$ and a bijective, continuously differentiable function $\phi: J \to I$, the reparametrized curve is $\tilde{\gamma}(s) = \gamma(\phi(s))$, where $s \in J$
• Reparametrization is useful for simplifying calculations or achieving desired properties (unit speed)

Equivalence classes of parametrizations

• Two parametrizations $\gamma_1: I_1 \to \mathbb{R}^n$ and $\gamma_2: I_2 \to \mathbb{R}^n$ are equivalent if there exists a bijective, continuously differentiable function $\phi: I_2 \to I_1$ such that $\gamma_2 = \gamma_1 \circ \phi$
• Equivalent parametrizations describe the same geometric curve
• Equivalence classes of parametrizations allow for the study of curves independent of their specific parametrization

Arc length of parametrized curves

• Arc length is a fundamental concept in the study of parametrized curves that measures the distance along the curve between two points
• Arc length is essential for defining arc length parametrization and curvature

Definition of arc length

• The arc length of a parametrized curve $\gamma: [a, b] \to \mathbb{R}^n$ is given by the integral: $L(\gamma) = \int_a^b \|\gamma'(t)\| dt$
• $|\gamma'(t)|$ represents the magnitude of the tangent vector at each point along the curve
• Arc length is independent of the parametrization and is an intrinsic property of the curve

Arc length parametrization

• An arc length parametrization is a reparametrization of a curve such that the parameter represents the arc length along the curve from a fixed starting point
• For a curve $\gamma(t)$, the arc length parametrization $\tilde{\gamma}(s)$ satisfies: $s = \int_a^t \|\gamma'(u)\| du$
• Arc length parametrizations have unit speed, meaning $|\tilde{\gamma}'(s)| = 1$ for all $s$

Existence of arc length parametrization

• Every regular parametrized curve admits an arc length parametrization
• The existence of an arc length parametrization is guaranteed by the fundamental theorem of calculus and the regularity of the curve
• Arc length parametrizations are unique up to a constant shift in the parameter

Curvature of parametrized curves

• Curvature is a measure of how much a curve deviates from a straight line at a given point
• Curvature is an intrinsic property of the curve and is independent of the parametrization

Definition of curvature

• The curvature of a regular parametrized curve $\gamma(t)$ at a point $t$ is given by: $\kappa(t) = \frac{\|\gamma'(t) \times \gamma''(t)\|}{\|\gamma'(t)\|^3}$
• $\gamma'(t)$ and $\gamma''(t)$ represent the first and second derivatives of the curve at the point $t$
• The cross product $\gamma'(t) \times \gamma''(t)$ measures the deviation of the curve from a straight line

Signed vs unsigned curvature

• Unsigned curvature is the absolute value of the curvature, $|\kappa(t)|$, and does not distinguish between clockwise and counterclockwise bending
• Signed curvature assigns a positive or negative value to the curvature depending on the orientation of the curve (clockwise or counterclockwise)
• Signed curvature is useful for analyzing the behavior of the curve in relation to its ambient space

Curvature in terms of arc length

• For a curve $\gamma(s)$ parametrized by arc length, the curvature can be expressed as: $\kappa(s) = \|\gamma''(s)\|$
• This simplification is possible because $|\gamma'(s)| = 1$ for arc length parametrizations
• Expressing curvature in terms of arc length simplifies calculations and provides a geometric interpretation

Frenet frame of parametrized curves

• The Frenet frame is a moving orthonormal frame attached to each point of a regular parametrized curve
• The Frenet frame consists of the tangent, normal, and binormal vectors, which provide a local coordinate system for describing the curve's geometry

Tangent vector of curves

• The tangent vector $\mathbf{T}(t)$ of a regular parametrized curve $\gamma(t)$ is the unit vector pointing in the direction of the curve's velocity: $\mathbf{T}(t) = \frac{\gamma'(t)}{\|\gamma'(t)\|}$
• The tangent vector is always tangent to the curve and indicates the instantaneous direction of motion

Normal vector of curves

• The normal vector $\mathbf{N}(t)$ is the unit vector perpendicular to the tangent vector and points in the direction of the curve's acceleration: $\mathbf{N}(t) = \frac{\mathbf{T}'(t)}{\|\mathbf{T}'(t)\|}$
• The normal vector lies in the osculating plane, which is the plane spanned by the tangent and acceleration vectors

Binormal vector of curves

• The binormal vector $\mathbf{B}(t)$ is the unit vector perpendicular to both the tangent and normal vectors, completing the right-handed orthonormal frame: $\mathbf{B}(t) = \mathbf{T}(t) \times \mathbf{N}(t)$
• The binormal vector is normal to the osculating plane and indicates the direction in which the curve is twisting

Frenet-Serret formulas

• The Frenet-Serret formulas describe the rates of change of the tangent, normal, and binormal vectors along the curve: \mathbf{T}'(s) &= \kappa(s) \mathbf{N}(s) \\ \mathbf{N}'(s) &= -\kappa(s) \mathbf{T}(s) + \tau(s) \mathbf{B}(s) \\ \mathbf{B}'(s) &= -\tau(s) \mathbf{N}(s) \end{aligned}$$
• $\kappa(s)$ is the curvature, and $\tau(s)$ is the torsion, which measures the rate of change of the osculating plane
• The Frenet-Serret formulas provide a complete description of the local geometry of the curve

Special parametrized curves

• Certain classes of parametrized curves have distinctive properties that make them useful for various applications
• These special curves include plane curves, space curves, simple curves, closed curves, and curves of constant curvature

Plane curves vs space curves

• Plane curves are parametrized curves that lie entirely in a single plane (e.g., $\gamma(t) = (x(t), y(t))$)
• Space curves are parametrized curves that do not lie in a single plane and have non-zero torsion (e.g., $\gamma(t) = (x(t), y(t), z(t))$)
• Plane curves have simpler geometric properties compared to space curves, as they do not exhibit twisting behavior

Simple vs closed curves

• Simple curves are parametrized curves that do not self-intersect, meaning $\gamma(t_1) \neq \gamma(t_2)$ for any $t_1 \neq t_2$ in the domain
• Closed curves are parametrized curves whose start and end points coincide, i.e., $\gamma(a) = \gamma(b)$ for the domain $[a, b]$
• Simple closed curves are both simple and closed, forming a loop without self-intersections (Jordan curves)

Curves of constant curvature

• Curves of constant curvature have a curvature function $\kappa(t)$ that is constant along the entire curve
• Examples of curves with constant curvature include circles ($\kappa > 0$), straight lines ($\kappa = 0$), and hyperbolic curves ($\kappa < 0$)
• Curves of constant curvature have special geometric properties and are often used in modeling and analysis

Applications of parametrized curves

• Parametrized curves have numerous applications in various fields, including physics, engineering, computer graphics, and data analysis
• These applications leverage the geometric properties and flexibility of parametrized curves to model, analyze, and visualize complex phenomena

Modeling of physical phenomena

• Parametrized curves are used to model the trajectories of particles, celestial bodies, and other objects in motion
• Examples include the path of a projectile under the influence of gravity, the orbit of a satellite around a planet, and the motion of a charged particle in an electromagnetic field
• Parametrized curves enable the study of the geometric properties of these trajectories, such as curvature and torsion, which provide insights into the underlying physical processes

Curve fitting in data analysis

• Parametrized curves are employed in data analysis to fit curves to experimental or observational data points
• Curve fitting techniques, such as least squares and spline interpolation, use parametrized curves to approximate the underlying trends and relationships in the data
• Fitted curves can be used for data visualization, prediction, and extracting meaningful features from the data

Parametrized curves in computer graphics

• Parametrized curves are fundamental in computer graphics for representing and manipulating curves and surfaces
• Bézier curves, B-splines, and NURBS (Non-Uniform Rational B-Splines) are popular parametric curve representations used in computer-aided design (CAD), animation, and 3D modeling
• Parametrized curves enable the creation of smooth, flexible, and easily controllable shapes, which are essential for generating visually appealing and geometrically accurate graphics