§2.5.1 - Slicing a Surface with a Flat Plane:
a Single Level Curve

by Robert Curtis, Bill Davis, and Lee Wayand.

Flat Planes

A flat plane is a plane whose algebraic description is given by
z = k0
where k0 is a constant. Such a plane has 0 pitch, 0 x-slope, 0 y-slope.

If we intersect such a plane with a surface S, the resulting set of points will all have the same z-coordinate, being k0.

We may analyze the surface S via analysis of the level curve.

To the right we have an MathView interactive where you may change the height of the flat plane, and examine different intersections with the surface. You may also change the surface to list of examples below to experiment with the types of intersections possible.

Other surfaces to experiment with:

  • z = x2 + y2
  • z = x2 - y2
  • z = x y
  • z = x2 - x y - y + sin(2x)
  • z = x3 - x y2

Help for MathView math entry keystrokes


Help for MathView math entry keystrokes

Types of Intersections

It is a good exercise to experiment to see what types of intersections are possible between a flat plane and a surface.

In the given surface to the right,

z = sin( x) sin( y)
There are three types of geometric objects that are possible from the intersection of a flat plane and the surface (for the original domain given).

Determine what these three types of geometric object are, and answer the questions as posed below.

  • What happens to the level curves as the flat planes moves closer to a maximum/minimum point height?
  • What are the shapes of MOST of the level curves?
  • Is there any height were the level curves are SQUARES?

VR World

Instead of looking at the flat plane, we will take a walk on the flat plane, and have a VR-look around.

To take this trip,

  • Click on the world with your mouse
  • Use "Control-RightArrow" to cycle through the viewpoints we have put into this world.

Make sure "Animate to Viewpoints" is turned on in your Live3D plugin. If you aren't sure how to turn this on, click here.

Other VR Surfaces to Explore

Road Map
  • VR & 3D Calculus Home Page
    • §2.4 -
    • §2.5 - Level Curves
      • §2.5.1 - A Single Level Curve
      • §2.5.2 - The Set of Level Curves
      • §2.5.3, §2.5.4, §2.5.5, §2.5.6, §2.5.7, §2.5.8, §2.5.9
      • §2.5 Exercises
    • §2.6 -

What You Need to Access This Site:

  

  

Official Website of
www.calculus.net
 • Webcast on a DEC Alpha on a T-1 line from • 
 • Webster, Massachusetts, USA • 
 • Web Construction and Management by WebPrimitives • 

Core Development Team
  • Robert R. Curtis
  • Phil R. Smith
  • Christopher A. Barker
  • Diane Welden Housken
  • Tim Lance
  • Bob Stein
  • Donald Hartig
  • Lee Wayand
  • Charlene Beckmann
  • Bill Davis
  • Paul Latiolais
  • Dennis Sentilles
  • Carmen Artino
  • Michael Colvin


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