Interactive Geometry and Critical Points
Thomas F. Banchoff
Providence, RI 02912
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.
How does an
program make it possible to treat a broad subject like critical points
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
information about the shape of the graph, in particular the numbers of
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
by interacting with images, in particular by deforming them in a
using special sets of parameters. In
this way we can investigate entire families of functions of one and
variables, showing how the behavior of critical points changes when we
the parameters in different ways.
interactive Java demonstrations presented in this article are intended
illustrate the power of software generated by teams of students over a
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
such modifications it is necessary to be running an expanded program on
server. Furthermore it is possible to
incorporate programs like those illustrated in this article into
laboratory settings for use in conjunction with classes in calculus of
more variables. We will not go into
detail about the pedagogical implications of using such demonstrations
conjunction with traditional lecture courses, or in the context of a
programs. A report on such efforts, with
emphasis on assessment, is found in the author’s paper at the
Conference at KAIST, Daejon,
At several places in this article we indicate how the same approaches can
to study more advanced topics in parametric curves and surfaces, in
calculus and in introductory differential geometry and topology. Readers familiar with the geometry of
characteristic classes will recognize ways in which the examples
are fundamental in the study of singularities of mappings of manifolds
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
the segments with negative slope white, we can see immediately where
function has local maxima and minima.Assuming that there are no horizontal segments on the graph, as
from the left endpoint through the domain of the function, we can
the color of the graph changes, from red to white as we pass a local
and from white back to red at a local minimum.
If we start with a red segment and end with a white one, then
be an odd number of color changes, so there are an odd number of
points in the domain. The number of
local maxima is one greater than the number of local minima and in
there is at least one local maximum.
Similarly, if the segments at the endpoints of the interval have
same color, then the number of critical points in the interval is even,
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
approximation of a function, and this can be done in a course in
course in calculus of one variable, the corresponding results are
classical theorems. For the graph of a
differentiable function of one variable f(x), the number of points c
f'(c) = 0 in the interval [a,b] is even if f'(a) and f'(b) have the
and this number is odd if the signs are different, (assuming that there
finite number of critical points in the interval and that the sign of
changes as we go from left to right past a zero of the first
derivative). Beginning students will
difference in two basic examples: the function f(x) = x2 has
critical points in the interval [-2,2] since f’(x) = 2x so f’(-2) and
have different signs; on the other hand, the function f(x) = x3
– 3x has f’(x) = 3x2
– 3 so f’(-2) and f’(2) are
both positive and the function must have an even number of critical
the domain, in this case two.
For the graph of a
differentiable function, we can describe the coloring scheme in a
that will be useful in our generalizations.
The normal line at a point of the graph of a differentiable
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
called the “circular image” of the point.
If we color the left half of this upper semicircle red and the
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
It is easy to illustrate the color
scheme described above by showing a few pictures. What can interactive
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
advanced investigations. We will confine
ourselves to polynomial functions in the introductory parts of the
introduce trigonometric examples when we study parametric curves and
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,
changing the y-intercept b will move the graph up and down without
shape. Also making a change in the
domain gives us a function g(x) = f(x+h) = m(x+h) + b = mx + (mh+b),
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.
2. The Family of Cubic Equations
quadratic function f(x) = ax2 + mx + b with a positive, the graph is a
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) =
a(x+h)2 + m(x+h) + b = ax2
+ (2ah + m)x + (ah2 + mh + b). If a is
non-zero, we can choose h = -m/2a so
that the equation takes the form g(x) = ax2
+ (-m2/4a + b), with no
term and with the graph symmetric with respect to the y-axis.
about cubic equations of the form f(x) = x3
+ ax2 + mx + b? As
before, in order to shift right or
we set g(x) = f(x+h) = (x+h)3 + a(x+h)2 + m(x+h) + b = x3
+ (3h +
3h2 + 2ah + m)x + (h3
+ ah2 + mh + b).
By choosing h = -a/3 we obtain a function of the form g(x) = x3
+ 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.
can now investigate functions of the form f(x) = x3
+ mx by changing
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
derivative f’(x) = 3x2 + m, which will never be zero if m is positive
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
the sign of the derivative does not change at the critical point, the
does for a local maximum or minimum.
coding introduced in the
previous paragraphs makes it easy to see how the critical point
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
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
when the slope of the tangent line is negative, so we have a visual
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
minima, with lighter color).
Family of Quartic Equations
f’’(x) = 6x no matter
what m is, so the concavity of the graph changes from convex downward
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
inflections points of the graph, where the second derivative changes
the concavity of the graph changes.
Figure 1: The family of cubic equations
More interesting is the case of a
quartic, or fourth degree polynomial f(x) = -x4
+ cx3 + ax2 + mx + b. As in
the case of the quadratic and the cubic, changing b only shifts the
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
+ m(x+h) + b = -x4
+ (6h +c)x3
+ lower order terms, so
choosing h = -c/6 we can eliminate the coefficient of x3.
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) = -x4
+ ux2 + vx, for appropriate choices of the parameters u and
Once again, an interactive graphics
programs makes it possible for students to explore the family of graphs
quartic functions by choosing values u and v in the “two-dimensional
space”. Sometimes the number of critical
points is 1, for example when v = 0 and u is negative, and other times
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
and one local maxima. We can show this
algebraically by computing f’(x) = -4x3
+ 2ux = -2x(2x2
-u), which will
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) = -x4
+ vx, then for some values of v near zero, the
points is still 3, but for some values of v, the number of critical
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
+ 2x + v =
0 and f’’(x) = -12x2
+ 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) = -4x3
+ 2ux + v = 0 and f’’(x) = -12x2
+ 2u = 0 so |x| = √u/6 and v = 4x3
style="color: black;"> –
2ux = (4u/6)√u/6 – 2u√u/6 = -(4/3)u√u/6.
“semi-cubical parabola” curve in the “control space” of all possible
u and v will separate the (u,v) choices that give functions with 3
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
configurations of critical points for different choices of u and v.
Figure 2: The family of quartic equations
4. Critical Points for Parametric Curves
Mathematicians familiar with
catastrophe theory will recognize that these two families of functions,
cubic curves and the quartic curves, are two of the most important
examples in that theory.
arising in the sciences of optics and structural design, catastrophe
significant applications in physics and engineering and in differential
geometry and other parts of mathematics.
It is noteworthy that interactive computer graphics makes it
introduce this modern subject to students just beginning the study of
and analytic geometry for functions of one variable.
For a good introduction to elementary
catastrophe theory for functions of one variable,
the Catastrophe Teacher website of Lucien
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
consider graphs of functions of two variables in three-dimensional
eventually moving on to study parametric surfaces.
For a differentiable
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
(-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
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
the vertical and if we color the points in the lower semicircle yellow,
first quadrant will be white, the second will be red, the third will be
and the fourth will be yellow. We can
then color each point on the parametric curve with the same color as
then read off the critical points of the horizontal coordinate function
finding points where the color changes from yellow to white or from red
orange, or conversely.
points of other coordinate function y(t) on the other hand occur where
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
Figure 3: Color-labeled parametric curves in the plane
5. Families of Parametric Curves
point (x(t),y(t)) moves along the curve, the corresponding unit normal
on the unit circle is either moving counterclockwise or clockwise, and
indicate this by making the curve thin in the first case and thick in
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
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
the differential geometry of curves in the plane.
6. Critical Points and Functions of Two Variables
Just as we considered one- and
two-parameter families of function graphs, we can explore families of
parametric plane curves like the cardioid family:
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
less than 1, then a cusp at u = 1 and then a pair of inflection points
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
while for u greater than 1 or less than -1, the unit circles is covered
algebraically. For u between 1 and 2, or
between -1 and -2, there are inflection points where the circular image
back on itself, so that some points of the circle are the circular
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
image of exactly one point on the curve, where the image curve travels
positive directions. We say that
curve has “tangential degree 1”, a concept of great importance the in
differential geometry and topology of
Figure 4: The cardioid family, color-coded
We can generalize the study graphs
of functions of one variable in the plane to the study of graphs of
of two variables in three-space.
restrict ourselves to functions defined on simple domains such as a
As in the case of
functions of one variable, we want to find the range of a function,
smallest rectangular or cylindrical prism (called a “box”) over the
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
the domain or a point on the boundary
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,
an interior global minimum, the bottom plane of the box will contain
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
and in that case the tangent line to the boundary curve will be a
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
-x2 – y2 and the minima represented by the origin for
f(x,y) = x2 + y2,
there are other points where the tangent plane is horizontal, for
origin in the saddle-shaped graph of f(x,y) = x2
- y2, 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) = x2
– y3 or f(x,y) = x3 –
3xy2, 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
that occur are local maxima, local minima, or ordinary saddle points.
Peaks the Geometry of Peaks and Passes
Two particular examples of polynomial
functions give a good idea of the critical point behavior for functions
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].
is the graph of the function f(x,y) = -x4
+ 2x2 – y2 over the domain -1.5 ≤ x ≤
1.5, -1 < y < 1. Even without any
calculus, it is straightforward to show algebraically that f(x,y) ≤ 1
equality occurs at two "peaks" (1,0) and (-1,0). Between
the two peaks there is a saddle point
at (0,0), and there are no other critical points for the function. The level set for z = 1 consists of the two
points (1,0) and (-1,0). For 0 < z
< 1, the level set consists of two closed curves.
The level set for z = 0 is a
self-intersecting "figure eight" curve, and for z < 0, the level
set the portion of a single closed curve inside the rectangular domain. These level set phenomena can be investigated
with a Java applet:
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
on worksheets or by online messages.
They can make conjectures about various configurations that
graphs of certain kinds of equations, and then go on to find algebraic
for these observed phenomena.
8. Crater Lake and the Geometry of Pits and Passes
For Twin Peaks
we can observe that the two local maxima are at the same height, a
that can be removed by a slight perturbation.
In geographic terms, we may consider the effect of a slight
a shearing transformation obtained by adding a linear term mx to the
for f(x,y) = -x4 + 2x2 – y2. In the
study of functions of one variable, we can add a linear term to the
f(x) = -x + 2x2 to get f(x) = -x4
+ 2x2 + mx, and a small non-zero m gives a
function with two local maxima at different heights. The same thing
Twin Peaks. For small m, the graph of
= -x4 + 2x2 + mx - y2 has two peaks at
different heights. When the parameter m
is large enough, for an “extreme earthquake”, the number of peaks goes
to one and the saddle point disappears. Just as in the case of one
students can identify by observation where that crucial changeover
find the values of m for which the function has a degenerate critical
the analogue of a horizontal inflection point.
exhibits a different kind of unstable situation.
graph of g(x,y) = -(x2 + y2)2 + 2(x2 +
y2) is a surface of revolution about the z-axis
with profile curve –x4 +
2x2. The maximum value
of this function
is 1, and there are infinitely many global maxima, lying over the
circle x2 +
y2 = 1 in the domain. There
is one local
minimum at the origin. Once again an
earthquake shear will perturb this situation to give the graph of
g(x,y) = -(x2
+ y2)2 + 2(x2 + y2) + mx, which will still have one local minimum,
circle of maxima will break up into one global maximum and one saddle
least if m is small enough. The level
set at the local minimum will be an isolated point surrounded by a
curve. The level set at the global
maximum will be a single point, and the level set at the saddle point
will be a
"double loop with a single crossing point". Near
the saddle point, the level set looks
like a pair of intersecting lines, as in the case of the figure-eight
Twin Peaks. Between the level of the
and the saddle, the level set is a single curve. Between the saddle and
local minimum, the level set consists of two curves, one inside the
opposed to next to each other as in the case of Twin
Peaks). Below the local
minimum, the level set is
again the part of a single curve lying in the domain.
Figure 6: Graph and slices of Crater Lake
9. The Critical Point Theorem for Graphs of Functions of Two Variables
This description of the critical
points configuration assumes that the earthquake has not been too
If the number m becomes large
enough then we
obtain a function graph with one global maximum and no other critical
The local minimum and the saddle
point have come together and disappeared.
The water in Crater Lake
out. Students can find this point by observation (and compare the value
with m that produces a surface with a degenerate critical point in the
case of Twin Peaks
Both of these topographical
examples suggest ways of designing function graphs with other
critical points, say with n+1 local maxima and n saddle points in
Then we can introduce m
local minima and m
ordinary saddles, of the type found in the tilted Crater
This will produce a
function graph with n+1 maxima, n + m saddles and m minima, so in
the number of maxima plus the number of minima is one greater than the
of saddles, and the total number of critical points will be an odd
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,
will be at least one global maximum.
Furthermore, if all critical points are local maxima, local
ordinary saddles, then the number of critical points is odd and
#maxima - #saddles + #minima = 1. This
conjecture epitomizes fundamental results in Critical Point Theory, of
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
Theorem as well as properties of knotted "strings" in contemporary
biology and theoretical physics.
10. Color-Coding Graphs of Functions and Partial Derivatives
mathematician who popularized critical
point theory was Marston Morse in a series of articles on the subject
He enjoyed giving popular
lectures on the topic for students of all levels, and two of his
unpublished presentations appeared in the November 2007 issue of The
Mathematical Monthly [M].
On a visit to Providence RI
1970’s, he came to the computer graphics laboratory at Brown University
where my computer scientist colleague Charles Strauss and I were
program for analyzing the geometry of surfaces in three and four
Although the graphics
programs were primitive by today’s standards, and very slow, Marston
immediately appreciated the potential of such approaches for teaching,
research, and public exposition of geometric ideas arising in critical
We could not carry out those
projects forty years ago, but now they are possible using interactive
programs and Java applets.
When we analyzed level
functions of two variables, we used a spectrum of colors to indicate
heights of the slices.
For the subject
of partial derivatives, we need only two colors for each variable, four
Just as we colored graphs of
functions f(x) in the plane by red or white depending on the algebraic
f’(x) or the position of the unit normal vector in the left or right
the circular image, we can color the graph of a function f(x,y)
the signs of the partial derivative functions fx
(x,y) and fy(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 fx(x,y)
and fy(x,y) denote the first
partial derivatives of f with respect to x and y respectively, then
(-fx(x,y),-fy(x,y),1) will be a normal vector.
We can color the surface white if fx(x,y)
is positive and red if it is negative
in order to exhibit the points separating those regions, where fx(x,y)
0. Geometrically speaking, a point has
fx(x,y) = 0 if its tangent plane is perpendicular
to the y-z-coordinate
plane. Similarly if we color the surface
white if fy(x,y) is positive and blue if it is negative, we
can identify the
points where fy(x,y) = 0, and where the tangent plane is
perpendicular to the
then use additively to color the surface purple where it is both red
where fx(x,y) and fy(x,y) are both positive.
The surface will be colored red when fx(x,y)
is positive and fy(x,y) is
negative, and blue when when fx(x,y) is negative and fy(x,y)
is positive. If both fx(x,y)
and fy(x,y) are negative, the
surface is colored white. Thus we color
a point white, red, purple, or blue as the point (-fx(x,y),-fy(x,y))
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.
can also express this condition by saying that the point is in the
of the locus fx(x,y) = 0 and the locus fy(x,y)
= 0. Usually those conditions
are expressed and dealt with only algebraically, but interactive
graphics provides a direct visual display of the geometric properties
partial derivative functions.
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
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
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
originally introduced by Gauss.
This method of visualizing critical
point configurations is especially helpful when we are exploring
families of functions, for example the perturbations of Twin Peaks.
8: Color-coded partial derivative graphs
The picture on the left shows the
graph of the first partial derivative of f(x,y) with respect to x,
indicate where the value of that function is positive.
The colored region is separated from the
uncolored region by curves indicating where the fx(x,y)
= 0. The picture on the right shows the
graph for the first partial derivative with respect to y.
The colored region indicates where fy(x,y)
positive, and that region is bounded by the curve where fy(x,y)
9: Color-coded graph of perturbed Twin
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
where the curves corresponding to fx(x,y) = 0 and fy(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
graphs of the two first partial derivatives.
In the domain, we have a pair of curves where fx(x,y)
= 0 and a pair of
intersecting curves where fy(x,y) = 0.
The intersections of these two pairs of curves yield three
points, one global maximum, one ordinary saddle, and one local minimum.
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,
on one side and blue on the other.
is a degenerate situation since the curves where fx(x,y) = 0 and fy(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
of points, with all four colors in a neighborhood of each of these
10: Perturbed Crater Lake, color-coded
11: Color-coded partial derivative graphs
12: Color-coded graph of perturbed Crater
graph of perturbed Crater Lakes function colored according to the
signs of the
two first partial derivatives has four different colors near any
of the function, where the curves corresponding to fx(x,y) =
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
we can generalize from graphs of functions of two variables to
surfaces in space. For such surfaces,
the outer normal vector can point into the upper hemisphere where the
is white, red, purple, and blue, or into the lower hemisphere, for
directions the color is yellow, overlaid as appropriate with red to
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
receives the color of the octant within which its outward normal vector
Figure 13: A torus held vertically
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)
neighborhood of a point is colored with red, white, blue and purple, or
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
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
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
shape of the surface. This result is
basic in the topology and geometry of surfaces, and it is fundamental
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
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
calculus and proceeding to theorems in differential geometry and
topology. These techniques have great
engaging students and general audiences as well as providing fruitful
research, in pedagogy as well as geometry and topology.
Special thanks are due to Michael
Schwarz, who wrote the Java demonstrations used for the illustrations
article. The software for the Java
applets was created at Brown
under the direction of the author.
Banchoff, Thomas “Beyond the Third Dimension”
(1990) Scientific American Library, Freeman Publishing Co.
Banchoff, Thomas, “Interactive Geometry and
Multivariable Calculus on the Internet” CBMS Issues in Mathematical
Education, Vol. 14 (2007), p. 17-31.
Morse, Marston, “Topology and Equilibria”
(November 2007) The American Mathematical Monthly, p. 819-834.