The Electronic Journal of Mathematics and Technology, Volume 2, Number 2, ISSN 1933-2823 |

Interactive Geometry and Critical Points

Thomas_Banchoff@brown.edu

Mathematics Department

Brown University

Providence, RI 02912

U.S.A.

Interactive geometry programs make it possible
to explore critical points of functions of one and many variables,
in courses at various levels and in research projects. In this
article, we show how easy-to-use Java applets illustrate critical point
phenomena for families of curves and surfaces from several different
viewpoints. Our methods introduce circular and spherical
images of parametric curves and surfaces, leading to results on the
tangential degree of a closed curve in the plane, and the Hopf Theorem
on the normal degree of a surface in three-space. We also indicate
the relationship with the basic critical point theorem of Marston
Morse concerning pits, peaks, and passes of a function of two variables.

Introduction

How does an
interactive geometry
program make it possible to treat a broad subject like critical points
of
functions in a unified way, across a wide range of courses at different
levels? In this article, we show how
coloring the graph of function of one or more variables can display
basic
information about the shape of the graph, in particular the numbers of
critical
points of different types. A certain
amount of information is already available if we look at a single
picture of a
function, but we gain a great deal more insight into the geometry of
the graph
by interacting with images, in particular by deforming them in a
controlled way
using special sets of parameters. In
this way we can investigate entire families of functions of one and
more
variables, showing how the behavior of critical points changes when we
alter
the parameters in different ways.

The
interactive Java demonstrations presented in this article are intended
to
illustrate the power of software generated by teams of students over a
period
of years under the direction of the author. Running the applets require no computer algebra system or other
commercial software. It is possible for
a user to introduce new functions into the program, although in order
to save
such modifications it is necessary to be running an expanded program on
a file
server. Furthermore it is possible to
incorporate programs like those illustrated in this article into
worksheets or
laboratory settings for use in conjunction with classes in calculus of
one or
more variables. We will not go into
detail about the pedagogical implications of using such demonstrations
in
conjunction with traditional lecture courses, or in the context of a
courseware
programs. A report on such efforts, with
emphasis on assessment, is found in the author’s paper at the
International
Conference at KAIST, Daejon ,
Korea
[B2].

At several places in this article we indicate how the same approaches can
be used
to study more advanced topics in parametric curves and surfaces, in
courses in
calculus and in introductory differential geometry and topology. Readers familiar with the geometry of
characteristic classes will recognize ways in which the examples
presented here
are fundamental in the study of singularities of mappings of manifolds
into
Euclidean spaces, as well as the geometric theory of catastrophes.

1. Graphs of Functions of a Single Variable

We start with
the graph of a
function of one variable defined over an interval on the real line. By the simple device of coloring all segments
of positive slope red in a polygonal approximation of the function, and
coloring
the segments with negative slope white, we can see immediately where
the
function has local maxima and minima.Assuming that there are no horizontal segments on the graph, as
we move
from the left endpoint through the domain of the function, we can
observe when
the color of the graph changes, from red to white as we pass a local
maximum
and from white back to red at a local minimum.
If we start with a red segment and end with a white one, then
there must
be an odd number of color changes, so there are an odd number of
critical
points in the domain. The number of
local maxima is one greater than the number of local minima and in
particular
there is at least one local maximum.
Similarly, if the segments at the endpoints of the interval have
the
same color, then the number of critical points in the interval is even,
with
the same number of local maxima as local minima. So
far these observations depend on studying
the changes of slope from positive to negative or conversely in a
polygonal
approximation of a function, and this can be done in a course in
pre-calculus
mathematics.

In a
course in calculus of one variable, the corresponding results are
expressed in
classical theorems. For the graph of a
differentiable function of one variable f(x), the number of points c
where
f'(c) = 0 in the interval [a,b] is even if f'(a) and f'(b) have the
same sign
and this number is odd if the signs are different, (assuming that there
are a
finite number of critical points in the interval and that the sign of
the derivative
changes as we go from left to right past a zero of the first
derivative). Beginning students will
appreciate the
difference in two basic examples: the function f(x) = x^{2} has
an odd
number of
critical points in the interval [-2,2] since f’(x) = 2x so f’(-2) and
f’(2)
have different signs; on the other hand, the function f(x) = x^{3}
– 3x has f’(x) = 3x^{2}
– 3 so f’(-2) and f’(2) are
both positive and the function must have an even number of critical
points in
the domain, in this case two.

For the graph of a
differentiable function, we can describe the coloring scheme in a
different way
that will be useful in our generalizations.
The normal line at a point of the graph of a differentiable
function is
perpendicular to the tangent line to the function graph at the point. For each point, the unit vector pointing
upward from the origin that is parallel to the normal line at the point
is
called the “circular image” of the point.
If we color the left half of this upper semicircle red and the
right
half white, we can reinterpret the coloring introduced in the previous
paragraph by saying that a point on the graph is given the same color
as its
circular image.

It is easy to illustrate the color
scheme described above by showing a few pictures. What can interactive
computer
graphics add to the experience of the teacher and the students? With interactive graphics, we can carry out
online investigations of families of functions depending on one or more
parameters. We present some simple
examples of this procedure, and then show how to use the same setup for
more
advanced investigations. We will confine
ourselves to polynomial functions in the introductory parts of the
paper, and
introduce trigonometric examples when we study parametric curves and
surfaces.

As examples of such families, we
can see what happens as we change the coefficients of polynomials. For linear functions, f(x) = mx + b,
changing the slope m will just rotate the line about its y-intercept,
and
changing the y-intercept b will move the graph up and down without
altering its
shape. Also making a change in the
domain gives us a function g(x) = f(x+h) = m(x+h) + b = mx + (mh+b),
which
moves the graph right or left, keeping the slope the same.
If m is not equal to zero, we can choose h =
-b/m so the equation takes the simple form g(x) = mx.

For a
quadratic function f(x) = ax^{2} + mx + b with a positive, the graph is a
parabola
opening upwards and with a negative, a parabola opening downwards. Changing the constant term b shifts the graph
vertically. A change from x to x+h
shifts the graph horizontally without changing its shape, and g(x) =
f(x+h) =
a(x+h)^{2} + m(x+h) + b = ax^{2}
+ (2ah + m)x + (ah^{2} + mh + b). If a is
non-zero, we can choose h = -m/2a so
that the equation takes the form g(x) = ax^{2}
+ (-m^{2}/4a + b), with no
linear
term and with the graph symmetric with respect to the y-axis.

2. The Family of Cubic Equations

What
about cubic equations of the form f(x) = x^{3}
+ ax^{2} + mx + b? As
before, in order to shift right or
left,
we set g(x) = f(x+h) = (x+h)^{3} + a(x+h)^{2} + m(x+h) + b = x^{3}
+ (3h +
a)x^{2} +
3h^{2} + 2ah + m)x + (h^{3}
+ ah^{2} + mh + b).
By choosing h = -a/3 we obtain a function of the form g(x) = x^{3}
+ Mx +
B, where M and B are constants that we can manipulate.
If we choose B = 0, we have a one-parameter
family of cubics with no squared term and no constant terms. Any cubic will have the shape of exactly one
of the cubics of this special form.

Students
can now investigate functions of the form f(x) = x^{3}
+ mx by changing
the
parameter m and observing the numbers of critical points of the graph. Immediately we are led to the conjecture that
when m is positive, there are no critical points for the function and
when m is
negative, the number of critical points is two.
Students can prove this conjecture can then be proven by finding
the
derivative f’(x) = 3x^{2} + m, which will never be zero if m is positive
and which
is zero for exactly two values of x when m is negative.
The intermediate case, with m = 0, has a
horizontal inflection point at the origin when x = 0.
The value x is called a “degenerate”
critical point, since f’(0) = 0
so the tangent line is horizontal, but f’(x) is positive for all other
x, so
the sign of the derivative does not change at the critical point, the
way it
does for a local maximum or minimum.

The color
coding introduced in the
previous paragraphs makes it easy to see how the critical point
behavior
changes as we change the value of m in an interactive demonstration. The Java applet following the first
illustration makes it possible for a teacher and students to
investigate this
behavior by changing the “slider bar” that gives the value for m.

In addition to
indicating the slope
of the tangent line, we can color the points on the graph by a darker
color
when the slope of the tangent line is negative, so we have a visual
record of
the places where the slope changes from positive to negative (the local maxima of the curve, colored with
a darker color) and where it changes from negative to positive (at the
local
minima, with lighter color).

Note that
f’’(x) = 6x no matter
what m is, so the concavity of the graph changes from convex downward
to convex
upward as x goes from negative values to positive values in the domain. By making the graph thick when the second
derivative is positive and thin where it is negative, we can observe
the
inflections points of the graph, where the second derivative changes
sign and
the concavity of the graph changes.

Figure 1: The family of cubic equations

3. The
Family of Quartic Equations

More interesting is the case of a
quartic, or fourth degree polynomial f(x) = -x^{4}
+ cx^{3} + ax^{2} + mx + b. As in
the case of the quadratic and the cubic, changing b only shifts the
y-intercept
up and down without changing the shape of the curve.
Also, shifting the domain by setting g(x) =
f(x+h) = -(x+h)^{4} +c(x+h)^{3}
+ a(x+h)^{2}
+ m(x+h) + b = -x^{4}
+ (6h +c)x^{3}
+ lower order terms, so
choosing h = -c/6 we can eliminate the coefficient of x^{3}.
As usual we can shift vertically by choosing
the constant to be zero, so the graph of any quartic function of this
type can be
written, f(x) = -x^{4}
+ ux^{2} + vx, for appropriate choices of the parameters u and
v.

Once again, an interactive graphics
programs makes it possible for students to explore the family of graphs
of
quartic functions by choosing values u and v in the “two-dimensional
control
space”. Sometimes the number of critical
points is 1, for example when v = 0 and u is negative, and other times
there
are 3 critical points, for example when v = 0 and u is positive. What
happens if v is not zero? When will we
get 3 critical points and when will we get 1?
Will we ever get exactly 2 critical points?

If v = 0, then we observe that for
any positive number u, the graph will have three critical points, two
minima
and one local maxima. We can show this
algebraically by computing f’(x) = -4x^{3}
+ 2ux = -2x(2x^{2} -u), which will
equal
zero for at x = 0 and at two other values of x if u is positive, and
only at x
= 0 if u is positive.

If we choose u = 1 so that f(x) = -x^{4}
+ x^{2}
+ vx, then for some values of v near zero, the
number of
critical
points is still 3, but for some values of v, the number of critical
points
changes to 1. We can observe that this
change occurs at the value of v for which the function has a horizontal
inflection point, where f’(x) = 0 and f’’(x) = 0. Thus
we must have both f’(x) = -4x
^{3} + 2x + v =
0 and f’’(x) = -12x^{2}
+ 2 =
0. From the second equation, x is plus
or minus the square root of 1/6, so v = ±(4/3)√(1/6).
For general u, we obtain the equations f’(x) = -4x^{3}
+ 2ux + v = 0 and f’’(x) = -12x^{2}
+ 2u = 0 so |x| = √u/6 and v = 4x^{3}
style="color: black;"> –
2ux = (4u/6)√u/6 – 2u√u/6 = -(4/3)u√u/6.
Therefore 27v^{2} = 8u^{3}. This
“semi-cubical parabola” curve in the “control space” of all possible
choices of
u and v will separate the (u,v) choices that give functions with 3
critical points
from those that give functions with 1 critical point.

We can use the interactive graphics
system to construct a collection of function graphs that exhibit the
different
configurations of critical points for different choices of u and v.

Figure 2: The family of quartic equations

Mathematicians familiar with
catastrophe theory will recognize that these two families of functions,
the
cubic curves and the quartic curves, are two of the most important
basic
examples in that theory. Originally
arising in the sciences of optics and structural design, catastrophe
theory has
significant applications in physics and engineering and in differential
geometry and other parts of mathematics.
It is noteworthy that interactive computer graphics makes it
possible to
introduce this modern subject to students just beginning the study of
calculus
and analytic geometry for functions of one variable.
For a good introduction to elementary
catastrophe theory for functions of one variable, see
the Catastrophe Teacher website of Lucien
Dujardin http://pagesperso-orange.fr/l.d.v.dujardin/ct/eng_index.html.

4. Critical Points for Parametric Curves

There are two
natural ways to
extend the analysis of functions of a single variable to more general
situations: either we may consider parametric curves in the plane, or
we can
consider graphs of functions of two variables in three-dimensional
space,
eventually moving on to study parametric surfaces.

For a differentiable
closed curve
in the plane given in parametric form by (x(t),y(t)) with t going from
a to b,
there is a velocity vector (x’(t),y’(t)) at each point, and a normal
vector
(-y’(t), x’(t)) perpendicular to the velocity vector.
The unit vector in the direction of the
normal vector is called the “circular image” at the point.
In contradistinction to the case of a
function graph where all the circular images were in the upper
semicircle, in
the case of a parametric curve, any point on the unit circle can be the
circular image of a of a point on the curve.
If, as before, we color red the points of the unit circle to the
left of
the vertical and if we color the points in the lower semicircle yellow,
then the
first quadrant will be white, the second will be red, the third will be
orange,
and the fourth will be yellow. We can
then color each point on the parametric curve with the same color as
its
circular image.

We can
then read off the critical points of the horizontal coordinate function
x(t) by
finding points where the color changes from yellow to white or from red
to
orange, or conversely. The critical
points of other coordinate function y(t) on the other hand occur where
the
color changes from white to red or from orange to yellow or conversely. The number of color changes for each of the
two coordinate functions will be even and the total number will be even. This is true whether or not the curve
intersects itself.

Figure 3: Color-labeled parametric curves in the plane

As the
point (x(t),y(t)) moves along the curve, the corresponding unit normal
vector
on the unit circle is either moving counterclockwise or clockwise, and
we can
indicate this by making the curve thin in the first case and thick in
the
second case. The points of the curve
where the direction changes are the inflection points of the curve. Students can investigate various curves, and
observe the number of points on the curve that correspond to any chosen
point
on the circle. How does the number of
corresponding points change as we move the position on the circle? What happens as we pass a point on the circle
that corresponds to an inflection point on the curve?
The investigation of curves in the plane
leads to the theory of tangential degree of a curve, an important
concept in
the differential geometry of curves in the plane.

5. Families of Parametric Curves

Just as we considered one- and
two-parameter families of function graphs, we can explore families of
closed
parametric plane curves like the cardioid family:
(-(u+cos(t))cos(t),(u+cos(t))sin(t)).
Students can discover that the shapes of these curves depend on
u and in
particular, that the shape changes dramatically near u = 1 and -1. As we watch an animation of this family as u
runs from u = .6 to u = 1.4, we can observe that the curve has a loop
for u
less than 1, then a cusp at u = 1 and then a pair of inflection points
and
finally a convex curve. Students can
also record what happens near other significant points, when u = -2,
-1, 0, 1,
and 2. For u between -1 and 1, the
circular image of the parametric curve covers the unit circle exactly
twice,
while for u greater than 1 or less than -1, the unit circles is covered
once
algebraically. For u between 1 and 2, or
between -1 and -2, there are inflection points where the circular image
doubles
back on itself, so that some points of the circle are the circular
image of
three points, two where the image curve travels in a counterclockwise
(positive) direction and one where it travels in a clockwise (positive)
direction, whereas all but two other points on the circle are the
circular
image of exactly one point on the curve, where the image curve travels
in a
positive directions. We say that
such a
curve has “tangential degree 1”, a concept of great importance the in
differential geometry and topology of
plane curves

Figure 4: The cardioid family, color-coded

6. Critical Points and Functions of Two Variables

We can generalize the study graphs
of functions of one variable in the plane to the study of graphs of
functions
of two variables in three-space. We
restrict ourselves to functions defined on simple domains such as a
rectangle or
circular disc. As in the case of
functions of one variable, we want to find the range of a function,
namely the
smallest rectangular or cylindrical prism (called a “box”) over the
domain that
will contain all of the function values for points in the domain. A point of the graph in the top plane of such
a box will correspond to either an interior point of
the domain or a point on the boundary
curve. If the function is
differentiable, then a point of the graph in the top plane
corresponding to an
interior point will have the top plane as its horizontal tangent plane,
and for
an interior global minimum, the bottom plane of the box will contain
the
horizontal tangent plane at the point of the graph.
On the other hand, it might be that the
highest or lowest point occurs at a point on the boundary curve of the
domain,
and in that case the tangent line to the boundary curve will be a
horizontal
tangent line lying on the top or bottom face of the containing box.

In addition to the local maxima of
functions of two variables, represented by the origin in the function
f(x,y) =
-x^{2} – y^{2} and the minima represented by the origin for
f(x,y) = x^{2} + y^{2},
there are other points where the tangent plane is horizontal, for
example the
origin in the saddle-shaped graph of f(x,y) = x^{2}
- y^{2}, defined either over a
square domain or a circular disc domain centered at the origin. This is a typical "ordinary saddle
point" for the graph of the function. Finding the critical points of a
function includes finding all local maxima, local minima, and saddle
points. There are other types of
critical points such as the origin for f(x,y) = x^{2}
– y^{3} or f(x,y) = x^{3} –
3xy^{2}, but these are “degenerate” and can be
eliminated by deforming the
function in a way we will make precise in our examples.
An important theorem of Marston Morse states
that, for almost all functions of two real variables, the only critical
points
that occur are local maxima, local minima, or ordinary saddle points.

Two particular examples of polynomial
functions give a good idea of the critical point behavior for functions
of two
variables. Both are named for
geographical features: Twin Peaks and Crater Lake .
These surfaces are described in the author’s
Scientific American Library volume “Beyond the Third Dimension” [B].

7. Twin
Peaks the Geometry of Peaks and Passes

Figure 5: Graph and slices of Twin Peaks

This kind of "level set
analysis" is central to an interactive graphics approach to graphs of
functions of one or more variables.
Students can investigate function graphs and report their
observations
on worksheets or by online messages.
They can make conjectures about various configurations that
arise in
graphs of certain kinds of equations, and then go on to find algebraic
reasons
for these observed phenomena.

For Twin Peaks ,
we can observe that the two local maxima are at the same height, a
situation
that can be removed by a slight perturbation.
In geographic terms, we may consider the effect of a slight
“earthquake”,
a shearing transformation obtained by adding a linear term mx to the
equation
for f(x,y) = -x^{4} + 2x^{2} – y^{2}. In the
study of functions of one variable, we can add a linear term to the
function
f(x) = -x + 2x^{2} to get f(x) = -x^{4}
+ 2x^{2} + mx, and a small non-zero m gives a
function with two local maxima at different heights. The same thing
happens for
Twin Peaks . For small m, the graph of
f(x,y)
= -x^{4} + 2x^{2} + mx - y^{2} has two peaks at
different heights. When the parameter m
is large enough, for an “extreme earthquake”, the number of peaks goes
from two
to one and the saddle point disappears. Just as in the case of one
variable,
students can identify by observation where that crucial changeover
occurs then
find the values of m for which the function has a degenerate critical
point,
the analogue of a horizontal inflection point.

8. Crater Lake and the Geometry of Pits and Passes

Figure 6: Graph and slices of Crater Lake

This description of the critical
points configuration assumes that the earthquake has not been too
severe. If the number m becomes large
enough then we
obtain a function graph with one global maximum and no other critical
points. The local minimum and the saddle
point have come together and disappeared.
The water in Crater Lake has
spilled
out. Students can find this point by observation (and compare the value
of m
with m that produces a surface with a degenerate critical point in the
case of Twin Peaks ).

9. The Critical Point Theorem for Graphs of Functions of Two Variables

Both of these topographical
examples suggest ways of designing function graphs with other
configurations of
critical points, say with n+1 local maxima and n saddle points in
between. Then we can introduce m
local minima and m
ordinary saddles, of the type found in the tilted Crater
Lake . This will produce a
function graph with n+1 maxima, n + m saddles and m minima, so in
particular
the number of maxima plus the number of minima is one greater than the
number
of saddles, and the total number of critical points will be an odd
number.

We can collect these observations
in a conjecture: For an
"island" defined over a region in the plane with one boundary curve
at sea level zero and all other points on the island above sea level,
there
will be at least one global maximum.
Furthermore, if all critical points are local maxima, local
minima, and
ordinary saddles, then the number of critical points is odd and
moreover
#maxima - #saddles + #minima = 1. This
conjecture epitomizes fundamental results in Critical Point Theory, of
immense
importance in global geometry and analysis over the last 80 years. It is also of fundamental importance for the
geometry of surfaces, leading to a modern proof of the crucial
Gauss-Bonnet
Theorem as well as properties of knotted "strings" in contemporary
molecular
biology and theoretical physics.

The
mathematician who popularized critical
point theory was Marston Morse in a series of articles on the subject
eighty
years ago. He enjoyed giving popular
lectures on the topic for students of all levels, and two of his
previously
unpublished presentations appeared in the November 2007 issue of The
American
Mathematical Monthly [M]. On a visit to Providence RI
in the
1970’s, he came to the computer graphics laboratory at Brown University
where my computer scientist colleague Charles Strauss and I were
developing a
program for analyzing the geometry of surfaces in three and four
dimensions. Although the graphics
programs were primitive by today’s standards, and very slow, Marston
Morse
immediately appreciated the potential of such approaches for teaching,
research, and public exposition of geometric ideas arising in critical
point
theory. We could not carry out those
projects forty years ago, but now they are possible using interactive
geometry
programs and Java applets.

10. Color-Coding Graphs of Functions and Partial Derivatives

When we analyzed level
sets of
functions of two variables, we used a spectrum of colors to indicate
the
heights of the slices. For the subject
of partial derivatives, we need only two colors for each variable, four
in
all. Just as we colored graphs of
functions f(x) in the plane by red or white depending on the algebraic
sign of
f’(x) or the position of the unit normal vector in the left or right
side of
the circular image, we can color the graph of a function f(x,y)
depending on
the signs of the partial derivative functions f_{x}(x,y) and f_{y}(x,y). For a differentiable function there is a
well-defined tangent plane at each point of the graph and an
upward-pointing normal perpendicular to that
plane. If f_{x}(x,y)
and f_{y}(x,y) denote the first
partial derivatives of f with respect to x and y respectively, then
(-f_{x}(x,y),-f_{y}(x,y),1) will be a normal vector.
We can color the surface white if f_{x}(x,y)
is positive and red if it is negative
in order to exhibit the points separating those regions, where f_{x}(x,y)
=
0. Geometrically speaking, a point has
f_{x}(x,y) = 0 if its tangent plane is perpendicular
to the y-z-coordinate
plane. Similarly if we color the surface
white if f_{y}(x,y) is positive and blue if it is negative, we
can identify the
points where f_{y}(x,y) = 0, and where the tangent plane is
perpendicular to the
x-z-plane.

We can
then use additively to color the surface purple where it is both red
and blue,
where f_{x}(x,y) and f_{y}(x,y) are both positive.
The surface will be colored red when f_{x}(x,y)
is positive and f_{y}(x,y) is
negative, and blue when when f_{x}(x,y) is negative and f_{y}(x,y)
is positive. If both f_{x}(x,y)
and f_{y}(x,y) are negative, the
surface is colored white. Thus we color
a point white, red, purple, or blue as the point (-f_{x}(x,y),-f_{y}(x,y))
lies in
the first, second, third, or fourth quadrant in the plane.

A point will be a critical point if
all four colors meet at the point. We
can also express this condition by saying that the point is in the
intersection
of the locus f_{x}(x,y) = 0 and the locus f_{y}(x,y)
= 0. Usually those conditions
are expressed and dealt with only algebraically, but interactive
computer
graphics provides a direct visual display of the geometric properties
of these
partial derivative functions.

Figure 7: Perturbed Twin Peaks, color-coded domains

We obtain a bonus from this method
of coloring because we can tell whether a critical point represents a
maximum
or minimum on one hand or a saddle point on the other.
In the first case, the four regions at a
point are red, purple, blue and white in counterclockwise order, while
in the
second case, the cyclic order is red, white, blue, and purple. The difference between these two orderings is
connected with the sign of the spherical image mapping defined on the
surface,
originally introduced by Gauss.

This method of visualizing critical
point configurations is especially helpful when we are exploring
one-parameter
families of functions, for example the perturbations of Twin Peaks.

Figure 8: Color-coded partial derivative graphs for Twin Peaks

The picture on the left shows the
graph of the first partial derivative of f(x,y) with respect to x,
colored to
indicate where the value of that function is positive.
The colored region is separated from the
uncolored region by curves indicating where the f_{x}(x,y)
= 0. The picture on the right shows the
corresponding
graph for the first partial derivative with respect to y.
The colored region indicates where f_{y}(x,y)
is
positive, and that region is bounded by the curve where f_{y}(x,y)
= 0.

Figure 9: Color-coded graph of perturbed Twin Peaks

The graph of the perturbed Twin
Peaks function colored according to the signs of the two first partial
derivatives has four different colors near any critical point of the
function,
where the curves corresponding to f_{x}(x,y) = 0 and f_{y}(x,y) = 0 intersect.

11. Perturbations of the Graph of Crater Lake

We can carry out the same analysis
for the Crater Lake function to show
the
graphs of the two first partial derivatives.
In the domain, we have a pair of curves where f_{x}(x,y)
= 0 and a pair of
intersecting curves where f_{y}(x,y) = 0.
The intersections of these two pairs of curves yield three
critical
points, one global maximum, one ordinary saddle, and one local minimum.

Note
the in the unperturbed Crater Lake ,
all of the
points on the top rim of the graph represent global maxima. The curve of points where both partial
derivatives are zero will be purple on one side and white on the other,
or red
on one side and blue on the other. This
is a degenerate situation since the curves where f_{x}(x,y) = 0 and f_{y}(x,y) = 0 coincide. When
we introduce a shear by adding mx for
small x, these two loci are perturbed so that they intersect at a
finite number
of points, with all four colors in a neighborhood of each of these
critical
points.

Figure
10: Perturbed Crater Lake, color-coded
domain

Figure 11: Color-coded partial derivative graphs for Crater Lake

Figure
12: Color-coded graph of perturbed Crater
Lake

The
graph of perturbed Crater Lakes function colored according to the
signs of the
two first partial derivatives has four different colors near any
critical point
of the function, where the curves corresponding to f_{x}(x,y) =
0 and
f_{y}(x,y) =
0 intersect.

12. Parametric Surfaces in Three-Dimensional Space

Just as we can generalize from
graphs of functions of one variable to closed parametric curves in the
plane,
we can generalize from graphs of functions of two variables to
parametric
surfaces in space. For such surfaces,
the outer normal vector can point into the upper hemisphere where the
coloring
is white, red, purple, and blue, or into the lower hemisphere, for
which
directions the color is yellow, overlaid as appropriate with red to
make
orange, or blue to make green, or purple to make brown.
The eight quadrants of the unit sphere are
then color-coded in such a way that a small polygonal region on a
surface
receives the color of the octant within which its outward normal vector
lies.

Figure 13: A torus held vertically

Note
that for the torus of revolution, the critical points of vertical
coordinate function have a local maximum at an entire circle, similar
to the situation at the top rim of the Crater Lake functions
graph. We can obtain isolated critical points either by rotating
the torus or by perturbing it by adding a multiple of sin(2u) to get
two global maxima or a multiple of sin(3u) to get three local
maxima. Once again the critical points of the coordinate
functions will occur when four different colors come together on the
surface, when the color of the surface corresponds to the color of the
normal vector to the surface on the unit sphere.

Figure 14: Warped torus - torus plus multiple of sin(2u)

Figure 15: Warped torus - torus plus multiple of sin(3u)

When a
neighborhood of a point is colored with red, white, blue and purple, or
the
same four colors in the opposite order, the spherical image of the
point is the
north pole, the unit vector along the positive z-axis. When the coloration is orange, yellow, green,
and brown, or these colors in the opposite order, the spherical image
of the
point is the south pole. The Hopf degree
theorem states that the algebraic number of times that the north pole
is hit is
the same as the number of times that the south pole it hit, where the
first
ordering is counted positively and the second negatively. Each of these numbers is half of the “Euler
characteristic of the surface”, a number that expressed the complexity
of the
shape of the surface. This result is
basic in the topology and geometry of surfaces, and it is fundamental
to the
modern theory of extrinsic differential geometry of surfaces. Just as Morse would be pleased to see
interactive demonstrations of critical points, Gauss would appreciate
interactive exploration of surfaces in three-space and higher. Such topics will be dealt with in later
papers on interactive geometry in differential geometry and
combinatorial
topology.

Conclusion

Interactive computer graphics makes
it possible to investigate phenomena connected with critical points of
functions in the plane and in three-dimensional space, starting with
elementary
calculus and proceeding to theorems in differential geometry and
topology. These techniques have great
potential for
engaging students and general audiences as well as providing fruitful
areas for
research, in pedagogy as well as geometry and topology.

Acknowledgements

Special thanks are due to Michael
Schwarz, who wrote the Java demonstrations used for the illustrations
in this
article. The software for the Java
applets was created at Brown
University by
David Eigen
under the direction of the author.

References

[B1] | Banchoff, Thomas “Beyond the Third Dimension” (1990) Scientific American Library, Freeman Publishing Co. | ||

[B2] | Banchoff, Thomas, “Interactive Geometry and Multivariable Calculus on the Internet” CBMS Issues in Mathematical Education, Vol. 14 (2007), p. 17-31. | ||

[M] | Morse, Marston, “Topology and Equilibria” (November 2007) The American Mathematical Monthly, p. 819-834. |