Mathematics – Parametric Curves and Surfaces Concepts

Schedule

• Date/Time: Friday morning, December 12, from 8:00 to 10:30
• Note: Remember to submit homework due at each session, as material builds on previous concepts. Extensions are available but need to be requested soon.

Homework and Extensions

• Importance of continuous engagement with homework; it accumulates over time.
• Extensions can be granted if requested at appropriate times (during breaks).

Parametric Curves

• A parametric curve is described using a single parameter (t).
  • Denotes space curves using vector notation:
    • $ext{r}(t)$
    • For each t, it gives
    • $x = x(t)$
    • $y = y(t)$
    • $z = z(t)$
  • Visualize as the tip of a vector tracing out a curve in space.

Parametric Surfaces

• Extending parametric curves to parametric surfaces using two parameters (u, v).
  • Defined with vector function:
    • $ext{r}(u, v)$
  • Each component is defined in terms of u and v:
    • $x = x(u, v)$
    • $y = y(u, v)$
    • $z = z(u, v)$
  • u and v vary across some defined domain, implying a surface in three-dimensional space.

Smooth Surfaces

• A surface is smooth if:
  • $ext{r}(u, v)$ has continuous partial derivatives.
  • The cross product of the tangent vectors:
  • $ext{r}<em>u imes ext{r}</em>v <br />\neq 0$ for all valid u and v.
  • Ensures non-parallel tangent vectors, allowing for the establishment of a tangent plane at every point.

Parameterization of a Plane

• For a plane defined by a position vector $ext{r}_0$ and vectors A and B (direction vectors):
  • $ext{r} = ext{r}_0 + u extbf{A} + v extbf{B}$
  • Describes all points on the plane using linear combinations of direction vectors A and B.
  • Non-parallel condition ensures smooth parameterization.

Example: Parameterization of a Plane

• Given non-parallel vectors A and B:
  • $ext{r} = ext{r}_0 + u ext{A} + v ext{B}$ provides a smooth parametric surface.
  • The parameters u and v vary over some domain to sweep out the surface.

Example: Helicoid Surface

• Given $x = a imes b imes ext{cos}(u)$, $y = a imes b imes ext{sin}(u)$, and $z = b imes u$:
  • u ranges as an angle between 0 and 2π, and v is any non-negative real number.

Characteristics of Surfaces

• Points on a surface majorly described through their projection in 2D or 3D space.
• Tangent planes and partial derivatives are necessary to analyze the smoothness and behavior of surfaces.

Surface Area Calculation

• For a parametric surface defined by $ext{r}(u, v)$:
  • The area can be derived using the double integral of determinants related to the parameters over a specified region.
  • $ext{Area}(S) = ext{integral over D of } ext{|} ext{r}<em>u imes ext{r}</em>v ext{|} ext{d}A$

Surface Area for Functions

• For surfaces defined explicitly by a function $z = f(x, y)$:
  • $ext{Area} = ext{integral over D of } ext{sqrt}(1 + ext{(dz/dx)}^2 + ext{(dz/dy)}^2) ext{dA}$
  • Utilizes partial derivatives with respect to x and y.

Examples of Surface Areas

• Example of Surface Area for a Sphere: Find the surface area for a sphere of radius $a$ given by the standard sphere parameterization and applying surface area formulas showed consistent results with existing knowledge in geometry (e.g., $4πr^2$).
• Applied various parameterizations and derived formulas making connections between parametric representations, implicit functions, and corresponding areas.

Surfaces of Revolution

• Surface of Revolution: Generated by rotating a function around an axis (e.g., rotate $y = f(x)$ around the x-axis):
  • General formula requires x and theta as parameters to cover the revolution.
  • $ext{Area} = ext{integral from a to b of } 2 ext{π} f(x) ext{sqrt}(1 + (f'(x))^2) ext{dx}$
  • Compact form connects volumes generated through surfaces of revolution using integrals.