VR & 3D Calculus - §2.5 - Level Curves of a Surface

§2.5 - Level Curves of Surfaces

by Robert Curtis, Bill Davis, and Lee Wayand.

Live3D Notes for Navigational & Technical Help

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

A flat plane is a plane whose algebraic equation is given by z = k₀, where k₀ is a constant. The intersection of a flat plane with a surface will result in either: empty set (i.e. they don't intersect), an isolated point, a curve, or a finite combination of isolated points and curves.

2.5.2: Using a Range of Flat Planes: The Elevated Level Curve Set

As the intersection of the parent surface and a flat plane results in a single level curve, we next use a series of flat planes to slice the parent surface, and create a set of level curves. Via animations, we are able to see how this set is created. Via VRML, we obtain some interesting viewpoints on the process.

2.5.3: Dropping These Curves to 2D: The Level Curves

After slicing up a surface with a flat plane at a variety of z-intervals, we may drop these curves down into the x-y plane. Since our surfaces (for now) are functions, this dropping-process will not result in kaos. We investigate the various geometric properties of the parent surface, and the resulting geometric properties of the dropped level curves.

2.5.4: Lifting the Level Curves Up

The Level Curve Set is usually given in a two-dimensional context, primarily the x-y plane for a surface living in x-y-z space. Via animation, we can lift these level curves back up to their parent surface, thus amplifying the geometric visualization skills of the reader. These skills are then tested via a number of exercises where the reader must determine for themselves the parent surface using only the lifting of the level curves, and some experimentation.

2.5.5: Walking on a Level Curve

In MathView you may look at the elevated level curves. In VR, we will walk the level curves to get a sense of their relationship to the parent surface

To fly through the viewpoints in the VR world to the left, click on the world (to select it), and then hit "CONTROL-RIGHTARROW".

2.5.6: Plotting Level Curves Manually: Finding Parametric Curves

Many CAS's have built-in functions that calculate the level curves of a surface. What if you had you find the algebraic formula of a level curve explicitly? How would you do it? You will have to discover these answers in this area.

2.5.7: Perpendicular Vectors and Elevated Level Curves

What is the relationship between a vector at a point P on a surface S that is perpendicular to S and the level curve through P?

2.5.8: Dropping the Gradient Vector and the Level Curves to 2D

The term "gradient vector" is used in many different contexts, and can be confusing for some. This area is dedicated to eliminating this confusion, and showing how each of the difference uses of "gradient vector" are actually related to one another.

2.5.9: Level Curves of Surface A Elevated to Surface B

What happens if you lift the level curves of one surface to the 3D graph of a second surface? Let's find out!

2.5: Exercises

Some problems for you to try on level curves.

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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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	2.5.1: Slicing a Surface with a Flat Plane: a Single Level Curve A flat plane is a plane whose algebraic equation is given by z = k₀, where k₀ is a constant. The intersection of a flat plane with a surface will result in either: empty set (i.e. they don't intersect), an isolated point, a curve, or a finite combination of isolated points and curves.
	2.5.2: Using a Range of Flat Planes: The Elevated Level Curve Set As the intersection of the parent surface and a flat plane results in a single level curve, we next use a series of flat planes to slice the parent surface, and create a set of level curves. Via animations, we are able to see how this set is created. Via VRML, we obtain some interesting viewpoints on the process.
	2.5.3: Dropping These Curves to 2D: The Level Curves After slicing up a surface with a flat plane at a variety of z-intervals, we may drop these curves down into the x-y plane. Since our surfaces (for now) are functions, this dropping-process will not result in kaos. We investigate the various geometric properties of the parent surface, and the resulting geometric properties of the dropped level curves.
	2.5.4: Lifting the Level Curves Up The Level Curve Set is usually given in a two-dimensional context, primarily the x-y plane for a surface living in x-y-z space. Via animation, we can lift these level curves back up to their parent surface, thus amplifying the geometric visualization skills of the reader. These skills are then tested via a number of exercises where the reader must determine for themselves the parent surface using only the lifting of the level curves, and some experimentation.
	2.5.5: Walking on a Level Curve In MathView you may look at the elevated level curves. In VR, we will walk the level curves to get a sense of their relationship to the parent surface To fly through the viewpoints in the VR world to the left, click on the world (to select it), and then hit "CONTROL-RIGHTARROW".
	2.5.6: Plotting Level Curves Manually: Finding Parametric Curves Many CAS's have built-in functions that calculate the level curves of a surface. What if you had you find the algebraic formula of a level curve explicitly? How would you do it? You will have to discover these answers in this area.
	2.5.7: Perpendicular Vectors and Elevated Level Curves What is the relationship between a vector at a point P on a surface S that is perpendicular to S and the level curve through P?
	2.5.8: Dropping the Gradient Vector and the Level Curves to 2D The term "gradient vector" is used in many different contexts, and can be confusing for some. This area is dedicated to eliminating this confusion, and showing how each of the difference uses of "gradient vector" are actually related to one another.
	2.5.9: Level Curves of Surface A Elevated to Surface B What happens if you lift the level curves of one surface to the 3D graph of a second surface? Let's find out!
	2.5: Exercises Some problems for you to try on level curves.