The Quadric Surfaces Program

This program identifies any quadric surface from the coefficients of the general second-degree equation in three variables.


Ax2 + By2 + Cz2 + Fyz + Gxz + Hxy + Px + Qy + Rz + D = 0

Introduction

The Quadric Surfaces program analyzes the ten coefficients of the general second-degree equation to identify the type of quadric surface. At least one of the A, B, C, F, G, or H coefficients must be nonzero. The program rotates the coordinate axes to eliminate any cross-product terms. The possibilities for the surface are:

  • Elliptical or hyperbolic paraboloid

    The program supplies the coordinates of the critical point with respect to the rotated axes.

  • Real or imaginary ellipsoid

    The program supplies the coordinates of the center with respect to the rotated axes.

  • Hyperboloid of one or two sheets

    The program supplies the coordinates of the center with respect to the rotated axes.

  • Cylinder

    The program determines the variable to be eliminated from the original equation.

  • Two planes

    The program determines the variable to be eliminated from the original equation.

  • Imaginary cone

    The program supplies the coordinates of the center with respect to the rotated axes.

Before using the program, make sure the calculator is partitioned for at least 33 data registers.

After using the Quadric Surfaces program to provide simplified coefficients, you can use the Conic Sections program to help determine the exact nature of a surface.

Starting the Program

To start the Quadric Surfaces program:
  1. Select { GEO } from the MATHEMATICS menu.
    The calculator displays the ANALYTIC GEOM menu.
  2. Press { QAD }.
    The program displays:
    D_COnkYM

Entering the Coefficients

To enter the coefficients of the three-dimensional quadratic equation:
  1. Enter the values of A, B, and C, pressing the applicable key after entering each value.
  2. Press { --> }.
    The program displays:
    D_5Noulg
  3. Enter the values of D, F, G, and H, pressing the applicable key after entering each value.
  4. Press { --> }.
    The program displays:
    D_h5Ulfo
  5. Enter the values of P, Q, and R, pressing the applicable key after entering each value.
    If any of the coefficients of an earlier group were entered incorrectly, press { --> } and re-enter the coefficients.
  6. When the coefficients are correct, press { EOD }.

Determining the Type of Surface

When you press { EOD }, the program displays a message to identify the type of quadric surface.

The message will be one of those listed below, accompanied by a { --> } selection to let you display data about the surface.

Message Displayed

Type of Surface
ELLIP PARABOLOID Elliptical paraboloid
HYPER PARABOLOID Hyperbolic paraboloid
REAL ELLIPSOID Real ellipsoid
IMAG ELLIPSOID Imaginary ellipsoid
1_S HYPERBOLOID Hyperboloid of one sheet
2_S HYPERBOLOID Hyperboloid of two sheets
CYL OR 2 PLANES Cylinder or two planes
IMAGINARY CONE Imaginary cone


Displaying the Rotated Coefficients

When the axes are rotated to eliminate the cross-product terms, the general equation becomes:

Ax2 + By2 + Cz2 + Px + Qy + Rz + D = 0


To display the coefficients for this equation when the type of surface is displayed:
  1. Press { --> }
    The program displays:
    D_xxCIJx
  2. Press the applicable key to display the A, B, C, or D coefficient.
  3. When you are ready to proceed, press { --> }.
    The program displays:
    D_5GgrKZ
  4. Press the applicable key to display the P, Q, or R coefficient.
    If you want to redisplay A, B, C, or D, press { --> }.

Displaying the Center or the Vertex

To display the location of the center or the vertex of the surface:
  1. Press { ESC } after viewing the rotated coefficients.
    The program displays one of two menus, depending on the type of surface.
    D_S9ILU1
    D_Qc7F9J
  2. Press the applicable key to display the x-, y-, or z-coordinate.
  3. When you finish viewing the coordinates, press { ESC } to return to the QUADRIC SURFACES menu.

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