Normat 51:2, 59–62 (2003) 59
A second look at normal curvature
Martin Raussen
Department of Mathematical S ciences
Aalb org University
Fredrik Bajers Vej 7G
DK–9220 Aalborg Øst, Denmark
raussen@math.auc.dk
When lecturing on normal curvature in a course on classical differential geometry
of surfaces in Euclidean space, I was asked by a student for a geometric reason ex-
plaining why the principal directions are perpendicular to each other. I did not have
an easy answer at hand. Moreover, it seemed easy to produce “counterexamples”:
Consider a surface S defined as the graph of a smooth function f : R
2
R.
In polar coordinates, that function is supposed to be of the form f(r, )=r
2
g()
with g : R R a smooth function with g(t + )=g(t) for all t R, (a “radial
parabolic” function).
The surface S has the XY -plane as its tangent plane at the origin O, since all
curves
r(t),(t),r
2
(t)g((t))
have horizontal tangents at O (r =0). The nor-
mal planes are thus all perpendicular to the XY -plane. A normal section, i.e., the
intersection of the normal plane in direction with the surface S, consists there-
fore of the parabola with parameterization
(r)=r
2
g() and normal curvature
k
n
()=2g(). Since we only assumed g to be smooth and to have period ,the
Euler equations (1) relating normal curvatures to the principal curvatures seem
to be violated in general. Nevertheless, the surfaces derived from our construction
lo ok quite “smooth”, cf. Figure 1 and 2 – all figures are produced with the aid of
Maple.
What is wrong? Well, the presentation in Euclidean coordinates of the function
f defining the surface as its graph is f : R
2
R,
f(x, y)=
0 if (x, y) = (0, 0),
(x
2
+ y
2
)g
arctan
y
x
otherwise,
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60 Martin Raussen Normat 2/2003
Figure 1: A smooth surface
f(r, )=r
2
cos(2)
Figure 2: A nonsmooth surface
f(r, )=r
2
cos(4)
and it is scarcely smooth. Smoothness, and in particular the fact that you are
allowed to change the order under double differentiation, is essential in proving
that the differential dN of the Gauss map N from the surface to the 2-sphere is
self-adjoint. This property is crucial in the calculation of the normal cu rvatures
and their relations to the p rincipal curvatures.
In fact, we have the somewhat surprising result about such a function f :
Proposition 1. A function f(x, y)=r
2
g() is differentiable at (0, 0) if and only
if it is a quadratic form
f(x, y)=Ax
2
+2Bxy + Cy
2
, A, B, C R.
Remark 2. A major part of the differential geometry of surfaces proceeds via
the analysis of the best approximating quadratic forms at every point. Proposition
1 shows that the only radial parabolic functions that can be approximated by
quadratic forms are the quadratic forms themselves. These smooth radial parabolic
functions are thus particularly stiff since the coefficients g() have to satisfy the
Euler equations: They attain a minimal value k
1
= g() and a maximal value
k
2
= g( +
2
) and
(1) g( + )=k
1
(cos )
2
+ k
2
(sin )
2
.
To prove the non-trivial part of the statement in Proposition 1 by elementary
means, i.e., without using the Euler e quations, we calculate the partial derivatives
raussen.tex,v 1.7
Normat 2/2003 Martin Raussen 61
of f via polar coordinates:
f
x
=
f
r
x
r
f
y
r
2
=2g()x g
()y
f
y
=
f
r
y
r
+
f
x
r
2
= g
()x +2g()y
for (x, y) = (0, 0) and
f
x
(0, 0) =
f
y
(0, 0) = 0. The second partial derivatives at
(0, 0) are:
2
f
x
2
(0, 0) = lim
x0
1
x
f
x
(x, 0) = 2g(0)
2
f
y x
(0, 0) = lim
y0
1
y
f
x
(0,y)=g
2
2
f
xy
(0, 0) = lim
x0
1
x
f
y
(x, 0) = g
(0)
2
f
y
2
(0, 0) = lim
y0
1
y
f
y
(0,y)=2g
2
The smoothness of f has as a consequence that
(2) g
(0) =
2
f
xy
(0, 0) =
2
f
y x
(0, 0) = g
2
.
Moreover, the differentiability of f /x, resp. f/y at (0, 0) is equivalent to the
existence of the limits
lim
r0
2
g() g(0)
x
g
() g
(/2)
y
r
= lim
r0
2
g() g(0)
cos
g
() g
(/2)
sin
and
lim
r0
g
() g
(0)
x +2
g() g(/2)
y
r
= lim
r0
g
() g
(0)
cos +2
g() g(/2)
sin
.
These limits can only exist if the functions under the limit sign are constants,
i.e., independent of , as well. Since they take the value 0 at =0, resp. at =
2
,
the fu nction g has to satisfy the following two differential equations:
2
g() g(0)
cos
g
() g
(/2)
sin =0(3)
g
() g
(0)
cos +2
g() g(/2)
sin =0.(4)
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62 Martin Raussen Normat 2/2003
Lemma 3. A smooth function g with period satisfies the differential equations
(3) and (4) if and only if it is of the form
(5) g()=A(cos )
2
+2B cos sin + C(sin )
2
, A, B, C R.
It is obvious that Lemma 3 yields Proposition 1.
Proof of Lemma 3: It is routine to check that any function g of the form (5) satisfies
the differential equations (3) and (4).
To see the conve rse, we calculate
1
2
[(3) cos + (4) sin ] and obtain using (2):
g()=g(0)(cos )
2
+ g
(0) cos sin + g(/2)(sin )
2
.
Figure 3: A surface with a flat point
Remark 4. There do exist less stiff functions of type f (r, )=h(r)g() whose
graphs are smooth (twice differentiable) surfaces. But this happens at the expense
of the normal curvatures of that surface at the origin being zero in every direc-
tion, i.e., the origin must be a flat point. An example is given by the function
f(r, )=r
4
cos(4), cf. Fig. 3 above.
Acknowledgment. I would like to thank the referee for pointing out a considerable
shortcut in my original pro of of Lemma 3.
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