22.1 Parametric representations of surfaces

Parametric surfaces are a powerful way to represent 3D shapes using two parameters. They allow us to map points from a 2D parameter space to 3D space, creating complex surfaces like spheres and tori.

This topic introduces key concepts like coordinate functions, parameter space, and different coordinate systems. Understanding these ideas is crucial for working with parametric surfaces and their applications in calculus and geometry.

Parametric Surfaces

Parametric Equations and Surface Patches

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• Parametric surfaces defined using two parameters, usually $[u](https://www.fiveableKeyTerm:u)$ and $[v](https://www.fiveableKeyTerm:v)$, to represent points on the surface
• Each point on the surface corresponds to a unique pair of parameter values $(u,v)$ within a specified parameter space
• Surface patches are portions of a parametric surface obtained by restricting the parameters $u$ and $v$ to a certain range
• Example: A sphere can be represented using the parametric equations $x=\cos u \sin v$, $y=\sin u \sin v$, and $z=\cos v$, where $0 \leq u < 2\pi$ and $0 \leq v \leq \pi$

Parameter Space and Coordinate Functions

• Parameter space is the domain of the parameters $u$ and $v$, typically a rectangular region in the $uv$-plane
• Each point $(u,v)$ in the parameter space corresponds to a unique point $(x,y,z)$ on the parametric surface
• Coordinate functions $[x(u,v)](https://www.fiveableKeyTerm:x(u,v))$, $[y(u,v)](https://www.fiveableKeyTerm:y(u,v))$, and $[z(u,v)](https://www.fiveableKeyTerm:z(u,v))$ map the parameter values to the corresponding $x$, $y$, and $z$ coordinates of points on the surface
• Example: For a torus with major radius $R$ and minor radius $r$, the coordinate functions are $x=(R+r\cos v)\cos u$, $y=(R+r\cos v)\sin u$, and $z=r\sin v$

Coordinate Systems

Cartesian, Cylindrical, and Spherical Coordinates

• Cartesian coordinates $(x,y,z)$ represent points in 3D space using three perpendicular axes
• Cylindrical coordinates $(r,\theta,z)$ represent points using a radial distance $r$, an angle $\theta$ in the $xy$-plane, and a height $z$
  • Conversion from cylindrical to Cartesian: $x=r\cos\theta$, $y=r\sin\theta$, $z=z$
• Spherical coordinates $(\rho,\theta,\phi)$ represent points using a radial distance $\rho$, an azimuthal angle $\theta$ in the $xy$-plane, and a polar angle $\phi$ from the positive $z$-axis
  • Conversion from spherical to Cartesian: $x=\rho\sin\phi\cos\theta$, $y=\rho\sin\phi\sin\theta$, $z=\rho\cos\phi$
• Example: The point $(1,\frac{\pi}{4},\frac{\pi}{3})$ in spherical coordinates corresponds to $(\frac{\sqrt{3}}{2},\frac{1}{2},\frac{\sqrt{6}}{4})$ in Cartesian coordinates

Surface Representations

Level Surfaces and Implicit Equations

• Level surfaces are surfaces defined by an equation of the form $f(x,y,z)=c$, where $c$ is a constant
• Points $(x,y,z)$ satisfying the equation $f(x,y,z)=c$ lie on the level surface corresponding to the value $c$
• Implicit equations define surfaces without explicitly providing parametric equations or coordinate functions
• Example: The equation $x^2+y^2+z^2=1$ represents a unit sphere centered at the origin, which is a level surface of the function $f(x,y,z)=x^2+y^2+z^2$
• Level surfaces can be used to visualize functions of three variables by plotting multiple level surfaces corresponding to different values of $c$
• Example: The level surfaces of the function $f(x,y,z)=x^2+y^2-z$ are hyperbolic paraboloids, which can be visualized by plotting several level surfaces for different values of $c$