Where is
the Cheng-Pleijel point of a quadrangle ?
In the above applet, imagine a point O' at a certain height H
above the
flat plane containing the quadrangle ABCD and which orthogonally
projects down directly
to the point O on the plane.
By definition, the Cheng-Pleijel point O
is the point where the sum of the areas of the triangles O'AB, O'BC,
O'CD, O'DA,
are at a minimum for a given value of H. In particular, the
cone formed by the apex O' and the base quadrangle ABCD
is a
minimal surface, or more precisely, a minimal surface with boundary
constraint.
In the case of
a triangle, it is not hard to show that the
Cheng-Pleijel
point is exactly the incenter (center of the inscribed circle) of the triangle
for any value of H.
The situation becomes very different for a quadrangle, and can be quite
complicated. Here, if ABCD can be circumscribed about
a circle, then the Cheng-Pleijel point is again the
center of the inscribed circle for any H.
If not, then the
Cheng-Pleijel point actually changes its position for
different values of H! The interesting case occurs when
the height H
approaches zero, and the limit point O' = O
is now truly a point in the interior of the (convex)
quadrangle, not just a projection.
In this applet, you can experiment with different positions of A,B,
C,D,
and H, and observe how the position of O varies.
To change a point's
position, press the mouse button
on that point and drag it to a new location of your choice. The new location
of the Cheng-Pleijel
point O will automatically be computed on the canvas.
The canvas grid size is
1x1 unit, the height denotes the length of H, and the coordinates
of O are given right below the height value. Coordinates of the vertices
of ABCD are supplied in the side tables, as well as the length of the
sides. Be aware that if ABCD is not convex, then singularities can occur
and if this happens, typing any key will reset the applet to its
original state.
I created this applet by combining the works of many other java programs,
and in particular, the Geometry applet written by D. Joyce has been of
great help. The main innovation in this applet is the ChengPleijel class
which enables the program to find the Cheng-Pleijel
point using the method of steepest descent. Many questions still remain
unanswered regarding the Cheng-Pleijel point.
For example, can you find a formula for this point involving only the points
A,B,C,D, and H ? Is there a simple geometric construction for
locating the point O ? what kind of curve is
traced out by O as H changes its value ?
And if you happen to discover
some unusual property of the Cheng-Pleijel point
while playing with this applet, don't hesitate to let me know and I'll be
happy to write an article about it with you as a co-author.
Happy exploring!