Quadric Surfaces and Cylindrical/Spherical Coordinates - Key Concepts

Cylindrical Surfaces

• Definition: A cylinder is the set of lines parallel to a given line passing through a given curve. The rulings are these parallel lines.

• Circular cylinder (standard): the curve is a circle in the xy-plane, e.g. the graph of $x^2+y^2=a^2$, which represents a cylinder of radius $a$ about the z-axis.

• General cylinders: any planar curve lifted along a line direction yields a cylinder (curves need not be circles).

• Visualization: copies of the base curve stacked along the axis produce the cylindrical surface.

Traces of Surfaces

• Traces: cross-sections obtained by intersecting a surface with planes parallel to one of the coordinate planes.

• Useful for sketching surfaces like cylinders and other quadric surfaces.

Quadric Surfaces (Second-Order Surfaces)

• Definition: Quadric surfaces are surfaces in three-dimensional space described by second-order algebraic equations. They can be written (in general) as a combination of squared terms and possibly linear terms. Conic sections appear as traces on coordinate planes.

• Common quadric surfaces (standard forms):

  • Ellipsoid: $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$.

  • Sphere is a special ellipsoid with $a=b=c$.

  • Elliptic paraboloid: $\frac{x^2}{a^2}+\frac{\text{y}^2}{b^2}=\frac{z}{c}\quad\Rightarrow\quad z=\frac{x^2}{a^2}+\frac{y^2}{b^2}$ (upward opening; traces: xy-plane ellipse, xz and yz planes yield parabolas).

  • Hyperbolic paraboloid: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=\frac{z}{c}\quad\Rightarrow\quad z=\frac{x^2}{a^2}-\frac{y^2}{b^2}$ (saddle surface; traces in xz- and yz-planes are parabolas).

  • Elliptic cone: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z^2}{c^2}$ (cone about the z-axis).

  • Hyperboloid of one sheet: $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$ ( connects to a single surface piece).

  • Hyperboloid of two sheets: $-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ (two separate pieces along extpm z).

• Notes:

  • Traces indicate the type: ellipses, parabolas, hyperbolas depending on cross-sections.

  • Many quadric surfaces can be described by a single general equation with or without cross-terms; orientation and axis depend on coefficients.

Ellipsoid and Paraboloids (Geometry Highlights)

• Ellipsoid: a closed surface with level set $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$. Traces on coordinate planes are ellipses; if $a=b=c$, it is a sphere.

• Elliptic paraboloid: cross-sections parallel to xy-plane are ellipses; cross-sections parallel to xz- or yz-planes are parabolas.

• Hyperboloid of one sheet vs two sheets: one sheet connects across a central region; two sheets have a gap around the origin.

• Cone: cross-sections depend on orientation and scale; elliptic cone has circular/elliptical horizontal cross-sections that expand linearly with height.

Interesting Surfaces and Applications

• Hyperboloid of one sheet can be formed from straight-line generators (architectural use; e.g., cooling towers).

• Elliptic paraboloid and paraboloid families arise in focusing/reflectors; paraboloid focus properties link to optics.

Cylindrical Coordinates (2.7)

• Definition: A point is represented by $(r, \theta, z)$ where

  • $r$ is the distance from the z-axis to the projection in the xy-plane,

  • $\theta$ is the angle from the positive x-axis to that projection,

  • $z$ is the height along the z-axis.

• Relations to Cartesian:
  $x= r\cos\theta, \quad y= r\sin\theta, \quad z= z,$ and $r^2 = x^2+y^2.$

• Angle/uniqueness: for a fixed point, (\theta) is determined up to adding multiples of $2\pi$; restricting (\theta) to a interval (e.g., 0\le\theta<2\pi) yields a unique representation.

• Surfaces by holding coordinates constant:

  • Constant $r$: vertical cylinders around the z-axis.

  • Constant $\theta$: half-planes radiating from the z-axis.

  • Constant $z$: horizontal planes.

Cylindrical Coordinate Examples (Key Conversions)

• Example: Cylinder $x^2+y^2=a^2$ corresponds to cylindrical radius $r=a$.

• Relationship to Cartesian: $x^2+y^2=r^2$.

Spherical Coordinates (2.7)

• Definition: A point is represented by $(\rho, \phi, \theta)$ where

  • $\rho$ is the distance from the origin,

  • $\phi$ is the angle from the positive z-axis (0 extle \u03c6 extle \u03c0),

  • $\theta$ is the same azimuthal angle as in cylindrical coordinates.

• Relations to Cartesian:
  $x=\rho\sin\phi\cos\theta, \quad y=\rho\sin\phi\sin\theta, \quad z=\rho\cos\phi.$

• Relations to cylindrical:
  $r=\rho\sin\phi, \quad z=\rho\cos\phi, \ \rho^2=x^2+y^2+z^2.$

• Surfaces by holding coordinates constant:

  • Constant $\rho$: spheres of radius $\rho$.

  • Constant $\theta$: half-planes around the z-axis.

  • Constant $\phi$: half-cones about the z-axis (angle from the z-axis fixed).

Conversions Among Coordinate Systems (Essentials)

• Cylindrical <-> Cartesian:

  • $x= r\cos\theta, \quad y= r\sin\theta, \quad z= z,$

  • $r=\sqrt{x^2+y^2}, \quad \theta=\operatorname{atan2}(y,x).$

• Spherical <-> Cylindrical/Cartesian:

  • $x=\rho\sin\phi\cos\theta, \quad y=\rho\sin\phi\sin\theta, \quad z=\rho\cos\phi,$

  • $r=\rho\sin\phi, \quad z=\rho\cos\phi,$

  • $\rho^2=x^2+y^2+z^2.$

• Practical note: choose the coordinate system that simplifies the problem (spheres for points with symmetry about a center; cylinders for problems with rotational symmetry about an axis; cones for angle-based symmetry).

Quick Reference: Standard Forms (Summary)

• Ellipsoid: $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{\,z^2}{c^2}=1$

• Sphere: $\frac{x^2}{a^2}+\frac{y^2}{a^2}+\frac{z^2}{a^2}=1\;\text{(special case of ellipsoid)}$

• Elliptic paraboloid: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z}{c}$

• Hyperbolic paraboloid: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=\frac{z}{c}$

• Elliptic cone: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z^2}{c^2}$

• Hyperboloid of one sheet: $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$

• Hyperboloid of two sheets: $-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$

• Elliptic cone (alt form): $\frac{z^2}{c^2}=\frac{x^2}{a^2}+\frac{y^2}{b^2}$

Practical Applications (Theme Indicators)

• Selecting the right coordinate system simplifies integration, volume calculations, and intersection problems (cylindrical for pipes, spherical for planets/balls, etc.).

• Understand traces as a tool for sketching and identifying surface types from quadric equations.