Back to . . . . The Students of CS 390  Software Architecture Using Java 3D to Investigate and Create Quadric Surfaces The Paraboloid and Elliptical Paraboloid NCB Deposit  # 32

Background for the student. . . .

Three-dimensional analogs of the conic sections are an important class of surfaces studied in Calculus.

Elliptic Paraboloid
The trace, or cross section, in the xy-plane is a point.  If c= 1, the point is the origin (0,0).  The traces in planes parallel to and above the  xy-plane  are ellipses.  The traces in the  yz-plane  and  xz-plane  are parabolas, as are the traces in planes parallel to these.  In this example
• Horizontal traces are ellipses.
• Vertical traces are parabolas.
• The variable raised to the first power indicates the axis of the paraboloid.
If  a = b and both are greater than 0, the horizontal traces are circles.  The surface is then simply named a paraboloid or circular paraboloid.

 General Equation By translation and rotation the cross-product terms disappear and one of two standard equations will be for all quadrics.  These may include the ellipsoid and a wide variety of hyperboloids. Volume of a Circular Paraboloid

Another example:

Note that  y in the equation has only the first power and becomes the axis of the elliptical paraboloid.

Printed References
 Modern calculus texts will have extensive material on the quadric surfaces.  The student should be very attentive to instruction on learning graphing techniques. For Mathematica® code that will create many variations of these graphs see Gray, A.,  MODERN DIFFERENTIAL GEOMETRY of Curves and Surfaces with Mathematica®,  2nd. ed., CRC Press, 1998. Gray, A.,  "The Paraboloid" in Modern Differential Geometry of Curves and Surfaces, CRC Press, 1993.
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