Section 2.6 - Cylinders and Quadric Surfaces

Section 2.6 - Cylinders and Quadric Surfaces

Introduction to Cylinders and Quadric Surfaces

This section introduces the concept of cylinders and quadric surfaces, which are fundamental in multivariable calculus and analytic geometry. Understanding these surfaces involves analyzing their equations, graphical representations, and classification based on their algebraic forms.

Cylinders

A cylinder is a surface that consists of all lines (called generators) that are parallel to a given line and pass through a given plane curve. Cylinders are characterized by the absence of one variable in their equation, resulting in a surface that extends infinitely in the direction of the missing variable.

• Definition: A surface defined by an equation in two variables (e.g., $x$ and $y$) and independent of the third variable (e.g., $z$) is a cylinder.

• Key Property: If one of the variables $x$, $y$, or $z$ is missing from the equation of a surface, then the surface is a cylinder.

• Example: The surface $x = y^2$ is a parabolic cylinder. Its cross-section in the $xy$-plane is a parabola, and it extends infinitely along the $z$-axis.

• Example: The surface $x^2 + y^2 = 1$ is a circular cylinder with axis parallel to the $z$-axis.

Quadric Surfaces

Quadric surfaces are surfaces defined by second-degree equations in three variables. They include ellipsoids, paraboloids, hyperboloids, and cones. The general form of a quadric surface is:

$Ax^2 + By^2 + Cz^2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0$

• Standard Form: $Ax^2 + By^2 + Cz^2 + J = 0$ (when cross-product and linear terms are absent).

• Classification: The type of quadric surface depends on the coefficients and signs in the equation.

Examples of Quadric Surfaces

• Ellipsoid: $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$ All traces are ellipses. If $a = b = c$, the ellipsoid is a sphere. Example: $\frac{x^2}{9} + \frac{y^2}{4} + \frac{z^2}{1} = 1$

• Elliptic Paraboloid: $z = \frac{x^2}{a^2} + \frac{y^2}{b^2}$ Horizontal traces are ellipses; vertical traces are parabolas. Example: $z = 4x^2 + y^2$

• Hyperbolic Paraboloid: $z = \frac{x^2}{a^2} - \frac{y^2}{b^2}$ Horizontal traces are hyperbolas; vertical traces are parabolas. Example: $z = x^2 - y^2$

• Cone: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z^2}{c^2}$ Horizontal traces are ellipses; vertical traces are pairs of lines. Example: $x^2 + y^2 = z^2$

• Hyperboloid of One Sheet: $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$ Horizontal traces are ellipses; vertical traces are hyperbolas. Example: $\frac{x^2}{9} + \frac{y^2}{4} - \frac{z^2}{1} = 1$

• Hyperboloid of Two Sheets: $-\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$ Horizontal traces are ellipses (for $z > c$ or $z < -c$); vertical traces are hyperbolas. Example: $-\frac{x^2}{9} - \frac{y^2}{4} + \frac{z^2}{1} = 1$

Classification of Quadric Surfaces

To classify a quadric surface, analyze the equation and compare it to standard forms. For example, the surface $4x^2 - y^2 - 2z^2 + 4 = 0$ can be rearranged and compared to the standard forms above to determine its type.

Summary Table: Common Quadric Surfaces

Surface	Equation	Surface	Equation
Ellipsoid	$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$	Cone	$\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z^2}{c^2}$
Elliptic Paraboloid	$z = \frac{x^2}{a^2} + \frac{y^2}{b^2}$	Hyperboloid of One Sheet	$\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$
Hyperbolic Paraboloid	$z = \frac{x^2}{a^2} - \frac{y^2}{b^2}$	Hyperboloid of Two Sheets	$-\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$

Applications and Examples

• Sketching Surfaces: To sketch a surface, determine the curves of intersection with coordinate planes and analyze traces (cross-sections).

• Example: Sketch the surface $x^2 + y^2 = 1$ (circular cylinder).

• Example: Sketch the surface $x^2 + y^2 + z^2 = 1$ (sphere, a special case of ellipsoid).

• Example: Classify the surface $4x^2 - y^2 - 2z^2 + 4 = 0$ by rearranging and comparing to standard forms.

Additional info: The table and examples provide a visual and algebraic guide to identifying and sketching quadric surfaces, which is essential for understanding three-dimensional geometry in calculus.