Math 2443: Calculus & Analytic Geometry IV

Supplemental Graphics:

A Surface, Its Traces, and Construction of Its Contour Diagram

by TJ Murphy with adaptations from work by Brad Kline at USAFA and Paul Goodey at OU

Shows traces and then animates the contruction of a contour diagram from the horizontal traces of a surface.

(corresponding notebook for users of Mathematica who want the commands used to generate these graphics)

[Graphics:levelCurvesgr61.gif]

Consider the function [Graphics:levelCurvesgr3.gif] graphed below:

[Graphics:levelCurvesgr2.gif][Graphics:levelCurvesgr11.gif]

[Graphics:levelCurvesgr62.gif].

If we evaluate f at the points where x = 2 and y ranges from -2 to 2 (integers only)

[Graphics:levelCurvesgr12.gif] = [Graphics:levelCurvesgr13.gif]

[Graphics:levelCurvesgr14.gif] = [Graphics:levelCurvesgr15.gif]

[Graphics:levelCurvesgr16.gif] = [Graphics:levelCurvesgr17.gif]

[Graphics:levelCurvesgr18.gif] = [Graphics:levelCurvesgr19.gif]

[Graphics:levelCurvesgr20.gif] = [Graphics:levelCurvesgr21.gif]

and plot these five points

[Graphics:levelCurvesgr2.gif][Graphics:levelCurvesgr24.gif]

we notice that they lie on the vertical trace obtained by intersecting our surface with the plane x = 2.

[Graphics:levelCurvesgr26.gif][Graphics:levelCurvesgr27.gif]

Now we hold y constant at 1 and let x vary from -2 to 2.

[Graphics:levelCurvesgr28.gif] = [Graphics:levelCurvesgr29.gif]

[Graphics:levelCurvesgr30.gif] = [Graphics:levelCurvesgr31.gif]

[Graphics:levelCurvesgr32.gif] = [Graphics:levelCurvesgr33.gif]

[Graphics:levelCurvesgr34.gif] = [Graphics:levelCurvesgr35.gif]

[Graphics:levelCurvesgr36.gif] = [Graphics:levelCurvesgr37.gif]

Then we plot these five points

[Graphics:levelCurvesgr2.gif][Graphics:levelCurvesgr39.gif]

and we notice that they lie on the vertical trace obtained by intersecting our surface with the plane y = 1.

[Graphics:levelCurvesgr41.gif][Graphics:levelCurvesgr42.gif] [Graphics:levelCurvesgr63.gif].

We intersect our surface with horizontal planes to get horizontal traces.

[Graphics:levelCurvesgr45.gif][Graphics:levelCurvesgr47.gif]

The horizontal traces can be projected onto the xy-plane, as in the animation below (just the frames). The projection of a horizontal trace is called a level curve.

We can do this for a set of horizontal traces. The next animation shows sequential horizontal planes intersecting the surface, with the corresponding level curve shown in the graph beside it. (just the frames)

A set of level curves in the xy-plane gives a two-dimensional representation of the function, called a contour diagram. This animation keeps the level curves as the planes travel through the surface, ending up with a contour diagram for the surface.

The following GIF animates the construction of a contour diagram of our surface by following the migration of the horizontal traces to the xy-plane (just the frames). Note the relationship of the height that the trace was at to the placement in the xy-plane of the corresponding level curve (in this case, the higher traces are the outer-most level curves).

Mathematica's version of a contour diagram shows the height from which the curve came by using darker shading to represent values farther down the z-axis.

[Graphics:levelCurvesgr2.gif][Graphics:levelCurvesgr58.gif]

This work is part of the Multivariable Calculus with Mathematica Project at the University of Oklahoma.