Cylinders and Quadric Surfaces

Cylinders

• A cylinder is a three dimensional surface defined by an equation involving only two variables.
• A cylinder will extend in the direction of the missing variable
  • This is because the missing variable essentially acts like a free variable: Suppose the equation $x^2 + z^2 = 1$ defines a cylinder 3D. This will result in a circle with a trace in the $xz$-plane. Any $y$-value of a point $(x,y,z) \in \mathbb{R}^3$ will satisfy the equation as the equation is only conditional on the values of $x$ and $z$. This will results the trace essentially being duplicated for every $y$-value, resulting in a cylinder.

Quadric Surfaces

A quadric surface in $\mathbb{R}^3$ for a point $(x,y,z)$ is a surface defined by an equation of the form

where $A, B, C, D, E, F, G, H, I,$ and $J$ are constants and at least one of $A, B,$ or $C$ is non-zero.

6 Main Types of 3D Surfaces

It can be helpful to group each type of main surface by the degree of each term. Ensure each quadric surface is in standard form to reduce errors when identifying the surface

Some Useful Properties:

• Closed vs Open
  • A closed surface is one that completely encloses a volume (no gaps)
  • An open surface does not completely enclose a volume
• Bounded vs Unbounded
  • A bounded surface is one where all points on the surface are within a certain distance from the origin (within a certain range)
  • An unbounded surface extends infinitely in at least one direction
• Traces
  • The intersection of a surface with a plane parallel to one of the coordinate planes.
  • Find by setting one variable to a constant value (usually 0) and solving for the other variables

Exactly two variables

These equations will always result in a cylinder. Graph the trace in the given plane given by the two variables and extend in the direction of the missing variable. Alwatys open and unbounded.

Three Variables, one with degree 1 and exactly two with degree 2

These equations will result in either a elliptic paraboloid or a hyperbolic paraboloid. These surfaces will always be open and unbounded.

• Elliptic Paraboloid: Both variables with degree 2 have positive coefficients. Will always be set equal to the variable with degree 1 in standard form.
  • $\frac{x^2}{a^2} + \frac{y^2}{b^2} = z$
  • A three dimentional analogue of a parabola. Tends to look like a bowl
  • Traces will be ellipses, circles, or parabolas
• Hyperbolic Paraboloid: One variable with degree 2 has a positive coefficient, the other variable with degree 2 has a negative coefficient. Will always be set equal to the variable with degree 1 in standard form.
  • $\frac{x^2}{a^2} - \frac{y^2}{b^2} = z$
  • A three dimentional analogue of a hyperbola. Tends to look like a saddle
  • Traces will be hyperbolas or parabolas

Three Variables with degree 2

These equations will result in either an ellipsoid, cone, or hyperboloid

• Ellipsoid: All variables have positive coefficients. Will always be set equal to 1 in standard form.
  • $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$
  • A three dimentional analogue of an ellipse, will always be closed and be bounded
  • Traces will be ellipses or circles
• Cone: Two vairables have positive coefficients, one has a negative coefficient. Will always be set equal to 0 in standard form.
  • $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 0$
  • A three dimentional analogue of a hyperbola, will always be closed and unbounded
  • Traces will be ellipses, circles, or intersecting lines
• Hyperboloid of One Sheet: Two variables have positive coefficients, one has a negative coefficient. Will always be set equal to 1 in standard form.
  • $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2}$ = 1$
  • A three dimentional analogue of a hyperbola, will always be open and unbounded. Tends to look like a hourglass.
  • Traces will be ellipses, circles, or hyperbolas
• Hyperboloid of Two Sheets: One variable has a positive coefficient, two have negative coefficients. Will always be set equal to 1 in standard form.
  • $-\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$
  • A three dimentional analogue of a hyperbola, will always be open and unbounded. Tends to look like two separate bowls facing away from each other
  • Traces will be ellipses, circles, or hyperbolas