Quadric Surfaces and Their Traces: Study Notes for Calculus Students

Quadric Surfaces

Definition and General Equation

Quadric surfaces are a family of surfaces in three-dimensional space defined by a general second-degree equation in three variables. These surfaces are important in multivariable calculus and analytic geometry.

• General Equation: The equation for a quadric surface is:

$A x^2 + B y^2 + C z^2 + D x y + E x z + F y z + G x + H y + I z + J = 0$

• Coefficients: A, B, C, D, E, F, G, H, I, J are constants, and not all of A, B, C, D, E, F are zero.

• Types: Common quadric surfaces include ellipsoids, hyperboloids (one or two sheets), cones, and paraboloids.

Traces of Quadric Surfaces

Traces are the intersection curves of a surface with planes parallel to the coordinate axes. Analyzing traces helps identify the type of quadric surface.

• Trace Types: Traces can be ellipses, hyperbolas, circles, points, pairs of lines, or empty (no intersection).

• Example: If a quadric surface has hyperbolas as its xy-trace and yz-trace, and ellipses for each y = y0 trace, it is a hyperboloid of one sheet.

Examples and Exercises

Finding Traces of a Cone

Consider the cone defined by $x^2 + y^2 = z^2$.

• Trace parallel to the yz-plane (constant x):

  • Set $x = x_0$; equation becomes $x_0^2 + y^2 = z^2$.

  • Rewritten: $z^2 - y^2 = x_0^2$.

  • If $x_0 \neq 0$, this is a hyperbola opening along the z-axis.

• Trace parallel to the xz-plane (constant y):

  • Set $y = y_0$; equation becomes $x^2 + y_0^2 = z^2$.

  • Rewritten: $z^2 - x^2 = y_0^2$.

  • If $y_0 \neq 0$, this is also a hyperbola opening along the z-axis.

Identifying a Non-Obvious Surface

Given $x^2 + 2x + y^2 = z^2$, complete the square to identify the surface:

• Rewrite: $x^2 + 2x + 1 - 1 + y^2 = z^2$

• $(x + 1)^2 - 1 + y^2 = z^2$

• $(x + 1)^2 + y^2 - z^2 = 1$

• This is a hyperboloid of one sheet centered at $(-1, 0, 0)$.

Trace of a Quadric Surface: Exercise

For $x^2 - 2x + y^2 = z^2 + 2z$, find the $x = 1$ trace:

• Substitute $x = 1$:

• $1^2 - 2 \times 1 + y^2 = z^2 + 2z$

• $1 - 2 + y^2 = z^2 + 2z$

• $y^2 = z^2 + 2z + 1$

• Complete the square: $y^2 = (z + 1)^2$

• So $y = \pm (z + 1)$, which is a pair of lines.

Possibilities for Traces of an Ellipsoid

• Traces can be:

  • An ellipse

  • A point

  • Empty (no intersection)

Quadric Surfaces Without Empty Traces

• Surfaces that cannot have empty traces:

  • Hyperboloid of one sheet

  • Cone

  • Hyperbolic paraboloid

Equation for a Surface by Revolving an Ellipse

Revolve the ellipse $x^2 + 4y^2 = 4$ in the xy-plane about the x-axis:

• The resulting surface is an ellipsoid centered at the origin.

• The equation is $x^2 + 4y^2 + 4z^2 = 4$.

• The yz-trace is a circle of radius 1, since the y-intercepts of the ellipse are $\pm 1$.

Summary Table: Types of Quadric Surfaces and Their Traces

Surface Type	Typical Equation	Possible Traces	Can Have Empty Traces?
Ellipsoid	$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$	Ellipse, point, empty	Yes
Hyperboloid of One Sheet	$x^2 + y^2 - z^2 = 1$	Ellipse, hyperbola	No
Hyperboloid of Two Sheets	$x^2 + y^2 - z^2 = -1$	Ellipse, empty, point	Yes
Cone	$x^2 + y^2 = z^2$	Hyperbola, pair of lines	No
Hyperbolic Paraboloid	$z = x^2 - y^2$	Parabola, hyperbola	No

Key Concepts and Techniques

• Completing the Square: Used to rewrite equations and identify the center and type of surface.

• Analyzing Traces: Substitute constant values for one variable to find the intersection curve with a coordinate plane.

• Surface Identification: Compare the resulting equation to standard forms to classify the surface.

Examples of Standard Quadric Surface Equations

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

• Hyperboloid of One Sheet: $x^2 + y^2 - z^2 = 1$

• Hyperboloid of Two Sheets: $x^2 + y^2 - z^2 = -1$

• Cone: $x^2 + y^2 = z^2$

• Hyperbolic Paraboloid: $z = x^2 - y^2$

Additional info:

• Quadric surfaces are typically covered in multivariable calculus courses, often as part of the study of three-dimensional analytic geometry.

• Understanding traces is essential for visualizing and classifying surfaces.