05 - coordinate systems and parametric curves

Parametric Curves

Definition

• Parametric Equation in ℝ3: A vector-valued function ( \mathbf{F}: I \subset \mathbb{R} \to \mathbb{R}^3 ) defined as:

  • ( t \mapsto \mathbf{F}(t) = f(t) \mathbf{i} + g(t) \mathbf{j} + h(t) \mathbf{k} )

  • Points on the curve: M with coordinates ( (f(t), g(t), h(t)) )

Basics of Parametric Curves

• Any curve has infinite parametric descriptions.

• A curve is described by:

  • Parametric equations

  • Domain ( I )

• Assume functions are differentiable sufficient times.

Parametric Descriptions of Lines

1. Straight Line AB:

   • Points A and B in ( \mathbb{R}^3 )

   • Parametric description:

     • ( x(t) = x_A + t(x_B - x_A) )

     • ( y(t) = y_A + t(y_B - y_A) )

     • ( z(t) = z_A + t(z_B - z_A) )

   • ( t \in \mathbb{R} )

2. Segment [AB]:

   • ( t \in [0, 1] )

   • Same equations apply.

Parametric Description of Circles

• Circle of center O and radius R:

  • ( x(t) = R \cos(t) )

  • ( y(t) = R \sin(t) )

  • ( t , \in , [0, 2\pi) )

Tangent Vector to the Curve

• Derivative of parametric equation:

  • ( \mathbf{F}'(t) = f'(t) \mathbf{i} + g'(t) \mathbf{j} + h'(t) \mathbf{k} )

• Determines the motion of point particles:

  • Velocity: ( \mathbf{v}(t) = \frac{dOM}{dt} )

  • Acceleration: second derivative gives trajectory's curvature.

Application in Mechanics

• Parameter t corresponds to time.

  • Trajectory is described as motion of a particle M.

Coordinate Systems

Introduction

• Used to:

  • Locate points in ( \mathbb{R}^3 )

  • Define displacement

  • Compute areas and volumes

• Several systems are available based on problem geometry.

Usual Coordinate Systems

1. Cartesian Coordinates (x, y)

2. Cylindrical Coordinates (r, \theta, z):

   • Use when there's symmetry around z-axis.

3. Spherical Coordinates (\rho, \phi, \theta):

   • Use for spherical symmetry.

Local Frames in Coordinates

• Defined by:

  • Orthonormal direct frame based on local unit vectors tangent to coordinate curves.

  • Derivative of local frames required for velocity calculations.

Differentiation of Curves in Coordinates

• Depend on chosen frame, leading to:

  • Displacement approximations for small variations of coordinates.

Parametric Curves Summary (80/20 Rule):

• Definition: A parametric equation in 3D represents a curve via a vector-valued function, defining points by coordinates based on time parameter t.

• Basics: Any curve can be described in infinite ways using parametric equations which depend on differentiable functions.

• Lines: Straight lines in (\mathbb{R}^3) are defined by their endpoints with equations for x, y, and z based on parameter t.

• Circles: Parametric equations for a circle center O and radius R involve trigonometric functions.

• Tangent Vector: The derivative of a parametric equation provides velocity and acceleration of point particles, determining trajectory.

• Applications: In mechanics, t represents time as the trajectory of a particle in motion.

• Coordinate Systems: Different systems (Cartesian, Cylindrical, Spherical) help locate points and compute geometry based on the problem context.

Coordinate Systems

Introduction

• Used to locate points in ( ( \mathbb{R}^3 ) ), define displacement, and compute areas and volumes.

• Several systems are available based on problem geometry.

Usual Coordinate Systems

• Cartesian Coordinates (x, y): Standard system for locating points.

• Cylindrical Coordinates (r, ( \theta ), z): Useful when there's symmetry around the z-axis.

• Spherical Coordinates (( \rho ), ( \phi ), ( \theta )): Used for spherical symmetry.

Local Frames in Coordinates

• Defined by an orthonormal direct frame based on local unit vectors tangent to coordinate curves.

• The derivative of local frames is required for velocity calculations.

Differentiation of Curves in Coordinates

• Differentiation depends on the chosen frame, which leads to displacement approximations for small variations of coordinates.