John Olive, Ph.D.

University of Georgia

A shorter version of this paper will appear in a special volume on **Dynamic Geometry**, James King and Doris Schattschneider (Eds.), to be published by **The Mathematical Association of America**

Nikolai Joukowski (1847-1921) was a Russian mathematician who did research in
aerodynamics (James and James, 1992). He used circle inversion in the complex
number plane to study airfoil shapes. By applying Joukowski's transformation
to the mathematical models of airflow over a cylinder (or a circle in
two-space) aeronautical engineers could predict the amount of lift and drag
that certain airfoil shapes would produce in an aircraft wing built using the
airfoil shape as its cross-section. (For an explanation of the aerodynamics of
airplane flight see Foote, 1952 and Ludington, 1943.) With the aid of dynamic
geometry software, such as *The Geometer's Sketchpad,* it is now possible
for students to use Joukowski's transformation to create and explore their own
airfoil shapes. In so doing, the students gain an appreciation for the
important application of complex numbers, their representation as points in
two-space, and the geometry of circle inversion. The recursive scripting
facility in *Sketchpad* also provides students with an opportunity to
investigate results of recursive applications of the Joukowski transformation
that lead to new, aesthetically interesting curves and seemingly chaotic
behavior. These recursive extensions can help students make connections
between art and mathematics, as well as involve them in investigations of the
new mathematics of chaos through dynamic visualization of iterations of
functions in the complex plane.

I begin with a brief review of complex numbers and their representation as
points in the plane. The Joukowski transformation involves the vector addition
of a point and its reciprocal in the complex plane. The reciprocal of a point
in the complex plane is explained as a composite of two geometric
transformations in the plane: inversion in the unit circle and reflection
about the x-axis. I present two different ways of constructing the circle
inversion using *Sketchpad.* The Joukowski transformation is then
achieved using *Sketchpad's* dynamic vector composition.

Having constructed the Joukowski transformation of a free point in the plane,
students can investigate the locus of the Joukowski point as the free point
moves on a circle. The resulting locus can be regarded as the image of the
animation circle under the Joukowski transformation in the complex plane.
Special relations between the animation circle and the unit circle produce
airfoil shapes. Version three of *Sketchpad* provides a facility to
construct the locus of the Joukowski point as a dynamic curve that will change
as the relations between the animation circle and unit circle are changed.

The recursive extensions of this investigation are achieved by simply applying the Joukowski transformation to the constructed Joukowski point, and constructing the locus of the resulting point as the original free point moves on the animation circle.

A complex number can be represented as z=x+iy, where i is and x and y are real numbers. The complex number plane is the set of points (x, y) in the plane such that z=x+iy. The unit circle in the complex plane is the set of points (x, y) such that . The reciprocal of any complex number z in the plane is that number whose complex product with z gives 1; it is represented as . Now thus , which is the same as . Geometrically, is the image of z under the composite of an inversion in the unit circle and a reflection in the x-axis.

**Inversion in a circle. **The *inverse* of a point in a given circle
is a point on the line through the center of the circle and the original point
such that the ratio of the distance of the original point to the center and the
radius of the circle is equal to the ratio of the radius and the distance of
the inverse point to the center. In Figure 1 this identity can be expressed in
the following way:
or, alternatively as: * *AC.AG=radius^{2}. A is the center of the
circle, C the original point, and G the inverse point.

Figure 1. (For a GSP sketch Click here)

The inverse of a free point in a circle can be constructed with the
*Geometer's Sketchpad* in several ways (see King, 1995, pp. 162-164). The
above proportion can be constructed geometrically using similar triangles (see
Figure 2) or by using the dynamic dilation facility built into the
*Sketchpad*.

In Figure 2, point G is the inverse of point C in the circle with center A and radius AB. To construct G an arbitrary point, D is placed on the circle and the segment CD constructed. The line AC and the ray AD are also constructed. In the triangle ACD, AD is the radius of the circle. Point E is the intersection of AC with the circle; thus, AE is also the radius of the circle. By constructing a line through E parallel to CD a triangle AEF, similar to ACD is produced by the intersection (point F) of this new line with the ray AD. This pair of similar triangles provides us with the necessary ratios to produce the inverse point of C in the circle AB: , as AE and AD are both radii of the circle. The inverse point, however needs to be on a line connecting C to the center of the circle. The intersection of the circle (center A) through F with the ray AC gives us the inverse point G, as AG=AF.

Figure 2 (For a GSP sketch Click here)

All of the construction lines, circles and points can be hidden leaving us with the situation in Figure 1. As point C was a free point in the sketch it can be moved around in order to investigate the relation of C to its inverse point G.

A more efficient way to obtain the necessary relation between point C and its
inverse is to use *dynamic dilation.* Point G can be obtained through
the dilation of point E about the center A using the *directed *ratio
as the dilation ratio. To do this in *Sketchpad* students only need to
construct the line through A and C and the intersection point E of this line
with the circle. They then select A as the *center of dilation* and mark
the ratio
by selecting the points A, C, and E **in that order**, and, while holding
the *option* key down (on a Mac) or Shift key (Windows), select **Mark
Ratio
**
from the **Transform** menu. The student then selects the point E and
dilates it by the marked ratio. I leave it to the reader to verify that the
dilated image of point E about the center A of the circle by the ratio
satisfies the inverse relationship with point C.

I now return to the construction in the complex plane of the point
representing the reciprocal
,
from the point (x, y) representing z=x+iy. Figure 3 illustrates the
construction of
from the point z. The point z' is the geometric inverse of point z in the
**unit** circle: thus |z||z'|=radius^{2}=1; the point
is the reflection of z' in the horizontal line through the center of the unit
circle (the x-axis).

Figure 3: Construction of

(For a GSP sketch Click here)

The Joukowski transformation of the complex plane is defined as **w=z+**
.
This transformation has the unique property of mapping both z and
to the same point:

z --> z+ , and -->+z=z+ (as addition of complex numbers is commutative). (I leave it to the reader to verify that the reciprocal of is z.)

Addition of complex numbers can be represented as vector addition in the
complex plane. The complex number z=x+iy is represented by the point (x, y).
The point (x, y) can also be thought off as representing the vector from
the origin (0, 0) to the point (x, y). For two complex numbers
represented by the points (x, y) and (x', y'), the result of their
addition is: (x+iy)+(x'+iy')=(x+x')+i(y+y'). This result is represented by
the point ((x+x'), (y+y')), which is the vector addition of (x, y) and
(x', y'). Geometrically, the point ((x+x'), (y+y')) is constructed by
translating the point (x, y) by the vector (x', y'). Figure 4 shows
the construction of **w**, the Joukowski transformation of the point z. The
construction was accomplished in *Sketchpad* by selecting the points 0 and
as a **Marked Vector **and then translating the point z by this marked
vector.

Figure 4: w=z+

(For a GSP sketch Click here)

The Joukowski airfoil profile is produced by the locus of **w** as z moves
on a circle that passes through the point z=-1 (the left-hand intersection of
the unit circle with the x-axis), and has the point z=+1 in its interior. The
circle also needs to be off-set slightly above the x-axis (see Figure 5)

Figure 5: Joukowski Airfoil Profile

(For a GSP sketch Click here)

With version 3 of *The Geometer's Sketchpad *users can construct the locus
of the airfoil profile as a continuous curve (rather than the trace of
**w**) by selecting the point **w**, the free point z and the circle with
center G (in that order), and while pressing the Option key (Mac) or Shift key
(Windows) select **Locus** from the **Construct** menu. Students can use
this sketch to investigate the unique properties of the Joukowski airfoil by
changing the size and position of the circle (center G) that is transformed
into the airfoil profile. (I shall refer to this circle as circle 2.)

**Investigating airfoils. **James and James (1992) state that the point
z=+1 must be in the *interior* of the transformed circle in order to
produce an airfoil profile. Students can test this property by reducing the
size of circle 2 slowly until the point z=+1 is outside the circle. Figure 6
illustrates what happens to the airfoil profile as z=+1 intersects circle 2,
but still passes through z=-1.

Figure 6: Joukowski profile with z=+1 intersecting Circle 2.

Further reduction of the size of circle 2 (Figure 7) produces a profile very similar to the airfoil shape in Figure 5 with z=+1 outside circle 2!

Figure 7: An airfoil profile with z=+1 exterior to circle 2.

Students can find out how different parts of circle 2 are mapped to the airfoil profile by animating z on circle 2 slowly and following the respective traced paths of z and w. Of particular interest are the parts of the Joukowski profile that correspond to the arcs of circle 2 that are interior to the unit circle, those that are exterior to the unit circle, those that are above the x-axis and those arcs that are below the x-axis.

Students can investigate the critical roles of the points +1 and -1 with respect to circle 2 in determining the shape of the Joukowski profile. Figure 8 illustrates a resulting profile when circle 2 does not pass through either point.

Figure 8: A non-airfoil profile.

In Figure 8, the center of circle 2 (point G) is on the x-axis. Students can
investigate what happens when the center G is *below* the x-axis. Figure
9 shows an upside down airfoil facing the opposite direction.

Figure 9: Flying upside down!

Students should be encouraged to find as many different kinds of profiles as possible and to form ways of categorizing them. In our investigations of the airfoil profile we discussed the relative merits of different airfoil shapes for the design of aircraft wings. With G very close to the x-axis, slim, aerodynamic airfoils can be produced. Such airfoils would give minimal lift to an aircraft but would also generate minimum drag. In contrast, as the center of circle 2 moves further above the x-axis highly curved airfoils are created. Such airfoils would create greater lift for an airplane wing but would also increase the drag on the wing. It became more clear to us why it is that large jet aircraft change the shape of their wings to a profile that is more curved for take-off and landing (when maximum lift is required), while at cruising speeds (when minimum drag and lift are required) the shape of the aircraft wing is more like the slim, aerodynamic airfoil.

What might be the result if the point **w **is transformed by the Joukowski
transformation to create the point **w'= w+ **
?
By investigating this recursive application of the Joukowski transformation
students can create curves suggestive of aquatic creatures, helmets and Greek
letters.

Students can create a script in *Sketchpad *to produce **w** from z by
following the constructions in figures 3 and 4. The script should require
three given points: the center and radius points of the unit circle and the
free point z. Students should apply their script to the same center and
radius points that define the unit circle but they should select **w**
instead of z as the third given.

The locus of **w'** as z moves on a free circle passing through the point
z=-1 can be constructed in the same way that the locus of **w **was
constructed (see Figure 5 above). Figure 10 shows one possible result. I find
this shape aesthetically pleasing and suggestive of some kind of water
creature, dinosaur or bird. I would like to call it a *RAT* profile for
Recursive Airfoil Transformation.

Figure 10: Recursive Airfoil Transformation

(For a GSP sketch Click here)

Students can investigate the*RAT* by repositioning and changing the size
of the transformed circle. By gradually moving the circle so that it is
offset* below* the x-axis, the point **w** creates an upside-down
airfoil (see Figure 9). What happens to the associated *RAT *is
demonstrated in the Figures 11-14.

Figure 11: Joukowski Airfoil and its *RAT*

Figure 12: A symmetric airfoil and its *RAT*

Figure 13: An Upside-Down Airfoil with a Helmet *RAT*

Figure 14: The Omega *RAT*

*The Geometer's Sketchpad* provides the capability to create recursive
scripts that will automatically replay the script on the new objects created by
the script. The user determines the depth of recursion when playing the
script. A *Recursive Joukowski script* can be used to make further
iterations of the Joukowski transformation. (For a recursive GSP script Click here) Figure 15 shows the result of
playing a recursive Joukowski script to a depth of 24 recursions. It shows an
apparent hyperbolic sequence of transformed points for a particular z just
below the center of the unit circle. The first transformed point is D'''. The
pattern appears to show subsequent transformations approaching the x-axis from
above. Further iterations of the transformation verify this trend.

Following are some of the questions that were raised by students during this investigation. Are there points in the z-plane for which the iterations appear to diverge rather than converge? Are there regions of the complex plane that appear to converge to the real line (the x-axis)? Or to the imaginary line (the y-axis)? Why might this be so? Are there points around which the transformed points appear to act chaotically? All of the above questions were answered in the affirmative. Of special interest was the transformation of the unit circle. Under the first Joukowski transformation the unit circle is mapped to the segment -2, 2 on the real line. Subsequent iterations appear to oscillate either side of zero on the real line, but move further and further away from zero. With 31 iterations, the traced paths reached approximately as far as +135 and -45.

Figure 15: 24 iterations of the Joukowski transformation

(For a GSP sketch Click here)

Apparently chaotic behavior was observed as z approached the imaginary axis from points not on the unit circle or the real line. Figure 16 shows the dramatic result of a tiny change in the position of z (from the point (0.123, -0.196) to the point (0.000, -0.196)) so that z ends up directly below the center of the unit circle. All points are now mapped to the y-axis instead of approaching the x-axis! (The iteration was extended to 51 points.)

Figure 16: Trace of movement of the transformed points

(For a GSP sketch Click here)

Figures 17, 18 and 19 illustrate even more dramatic results when the
transformation is iterated to a depth of 100 recursive calls (giving 101
iterations of the **w** transformation).

Figure 17: 101 iterations of **w**

Figure 18: small change in real part of z

Figure 19: real part of z is zero

(For a GSP sketch Click here)

This seemingly chaotic behavior in the iterated function occurs only when z is
very close to the imaginary axis. The dramatic change in the location of the
iterated points can be partially explained by the apparent convergence of the
iteration to the real axis for any z with a non-zero real part. When the real
part of z becomes zero, the iterations must stay on the imaginary axis. What
is not explained by this necessary shift from the real axis to the imaginary
axis is the seemingly unpredictable locations on the imaginary axis for each
iterated point. The relative order in the iterated points as they converge
towards the real axis is completely lost when they shift to the imaginary axis.
What is more, this lack of relative order only becomes apparent when the point
representing z moves towards the imaginary axis very slowly. The above traces
were obtained in *The Geometer's Sketchpad* by creating a point on the
imaginary axis and using a **Movement** button to move the point z slowly to
the point on the imaginary axis. The apparent chaos is lost when animating the
point z around a circle concentric with the unit circle as can be seen in
Figure 20. Even a slow-speed animation advances z around the circle in too
large increments for the chaotic shift to the imaginary axis to show in the
trace of the iterated points as z crosses the imaginary axis on the circle.

Figure 20: Trace of 30 iterated Joukowski points as z moves on a circle

(For a GSP sketch Click here)

Figure 20 suggests that the trace of the iterated points might be bounded by
the upper and lower quarter arcs of the oval that is the trace of **w**, the
first Joukowski point. Each iteration of **w** appears to be pushing this
oval in from the top and bottom while stretching it out along the real axis.
The upper and lower arcs of the trace cross the real axis at some point in the
iteration but eventually appear to converge on the real axis as z moves around
the inner circle in Figure 20. Figures 16-19, however, illustrate that the
apparent bounded symmetry of Figure 20 was just an illusion!

In order to check whether or not the behavior we were seeing was a result of
mathematical inexactitude in the *Sketchpad* program, we investigated the
chaotic behavior using the *Excel* spreadsheet, calculating values out to
ten decimal places. The spreadsheet values supported the wild oscillations
visible in Figure 19. While no immediate significance can be given to the
chaotic behavior of the iterated Joukowski function near x=0, it does open up
the possibilities for rich mathematical investigations. The trace function in
*The Geometer's Sketchpad* provides a unique way of visualizing this
chaotic phenomena.

Foote, D. K. (1952). *Aerodynamics for model airplanes; how and why a model
airplane flies.* New York: A. S. Barnes.

James, R. C. & James, G. (1992). *Mathematics dictionary, fifth
edition.* New York: Van Nostrand Reinhold.

King, J. (1996). *Geometry through the circle with The Geometer's
Sketchpad.* Berkeley, CA: Key Curriculum Press.

Ludington, C. T. (1943). *Smoke streams; visualized air flow.* New York:
Coward-McCann, Inc.