“Scalar Frenet Equations” and Focal
Curvatures for Curves in Rm+1
Ricardo Uribe–Vargas
Université Paris 7, Équipe Géométrie et Dynamique.
UFR de Math. Case 7012. 2, Pl. Jussieu, 75005 Paris.
[email protected] http://www.math.jussieu.fr/∼uribe/
Abstract. The focal curve of an immersed smooth curve γ : R → Rm+1 , θ → γ(θ),
in the Euclidean space Rm+1 , consists of the centres of its osculating hyperspheres. The
connected components of the focal curve may be parametrized as Cγ (θ) = (γ + c1 n1 +
c2 n2 + · · · + cm nm )(θ), where (t, n1 , . . . , nm ) is the Frenet frame of γ and c1 , . . . , cm−1 are
smooth functions that we call focal curvatures of γ. We found a remarkable expression of
the Euclidean curvatures ki , i = 1, . . . , m, in terms of the focal curvatures. We show that
the focal curvatures satisfy some “scalar Frenet equations”. Using the focal curvatures,
we give necessary and sufficient conditions for the radius Rl of the osculating l–sphere to
be critical, for l = 1, . . . , m. We also give necessary and sufficient conditions for a point
of γ to be a vertex.
Introduction
The focal curve of an immersed smooth curve γ : R → Rm+1 , θ → γ(θ),
in the Euclidean space Rm+1 , consists of the centres of its osculating hyperspheres. The centres of the osculating hyperspheres of γ are well defined only
for the points of γ where all curvatures are different from 0. The connected
components of the focal curve corresponding to the arcs of the curve γ (whose
Frenet frame is t, n1 , . . . , nm ) at which all curvatures are nowhere 0 may be
parametrized as Cγ (θ) = (γ +c1n1 +c2 n2 +· · ·+cm nm )(θ), where c1 , . . . , cm−1
are smooth functions that we call focal curvatures of γ. For curves in the
Euclidean space Rm+1 , with the preceding conditions and parametrized by
arc length, we found a remarkable expression of the Euclidean curvatures ki ,
i = 1, . . . , m, in terms of the focal curvatures (Th.2 below). We also show
(Th.1 below) that the focal curvatures satisfy some “scalar Frenet equations”.
Using the focal curvatures, we give necessary and sufficient conditions for
which the radius Rl of the osculating sphere of dimension l be critical, for
l = 1, . . . , m. We also give necessary and sufficient conditions for which a
point of γ be a vertex (a point at which the osculating hypersphere and the
curve have higher order of contact than the usual one).
In §0, we introduce some basic definitions as order of contact, osculating
hypersphere, focal curve and so on. In §1, we state the results of the paper.
1
In §2, we use the techniques of singularity theory to describe the geometric
properties of the focal set of a curve (the envelope of the normal lines to that
curve). In §3, we prove our results.
Acknowledgements. The author is grateful to V.I. Arnold for careful
reading the initial version of the paper and useful remarks and to V.D. Sedykh
who stimulate the author to publish this paper.
§0. Preliminary Definitions and Remarks
In order to give the definition of osculating k−spheres of a curve (at a
point of it) we need to introduce the following definition:
Definition – Let M be a d–dimensional submanifold of Rn , considered
as a complete intersection: M = {x ∈ Rn : g1 (x) = · · · = gn−d (x) = 0}.
We say that a (regularly parametrized) smooth curve γ : t → γ(t) ∈ Rn has
k-point contact with the submanifold M or that their order of contact is k,
at a point of intersection γ(t0 ), if each function g1 ◦ γ, . . . , gn−d ◦ γ has a zero
of multiplicity at least k at t = t0 , and at least one of them has a zero of
multiplicity k at t = t0 .
Remark – If one needs to make this definition more invariant, one could
denote the image of γ by Γ and then write that the order of contact at a
point is the minimum of the multiplicity of zero among the functions of the
form g|Γ : Γ → R at that point, where g belongs to the generating ideal
of M. When M is a hypersurface the order of contact coincides with the
multiplicity of intersection.
Remark – Do not confuse our order of contact with the order of tangency:
two perpendicular lines in the plane have order of contact 1 at the point of
intersection, but the order of tangency is 0.
Remark – Here, M will be a d–dimensional affine subspace or a d–
dimensional sphere.
Example – A smooth curve in Rn has 2−point contact with its tangent
line (at the point of tangency) for the generic points of the curve. The curve
y = x3 has 3−point contact with the line y = 0, at the origin: the equation
x3 = 0 has a root of multiplicity 3.
To simplify the notation, we write n = m + 1. Here, Rm+1 always denotes a Euclidean space. We will always assume that the derivatives of γ of
order 1, . . . , m, are linearly independent at any point (this is true for generic
curves). By convention, the k–dimensional affine subspaces of the Euclidean
space Rm+1 will be considered as k–dimensional spheres of infinite radius.
2
Definition – For k = 1, . . . , m, a k-osculating sphere at a point of a
curve in the Euclidean space Rm+1 is a k–dimensional sphere having at least
(k +2)−point contact with the curve at that point. For k = m we will simply
write osculating hypersphere.
Example – A generic plane curve and its osculating circle have 3−point
contact at an ordinary point of the curve.
Given a smoothly immersed curve γ : R → Rm+1 , write k1 , k2 , . . . , km for
its curvatures and t, n1 , . . . , nm for its Frenet frame. We assume that all the
curvatures of our curves are different from 0 at any point.
We recall that for 1 ≤ l < m the osculating l–sphere of γ at a point is
obtained as the intersection of the osculating hypersphere with the (l + 1)–
dimensional osculating plane at that point. In the sequel θ denotes any
regular parameter of the curve and s denotes the arc length parameter.
Definition – Consider a curve γ : R → Rm+1 having all its curvatures
different from 0 at any point. The curve Cγ : θ → Cγ (θ) ∈ Rm+1 consisting of
the centers of the osculating hyperspheres of the curve γ : θ → γ(θ) ∈ Rm+1
is called the parametrized focal curve of γ.
Remark – At a generic flattening of a curve (i.e. at a point where ki = 0,
= 0) the centre of the osculating hypersphere
i = 1, . . . , m−1, km = 0 and km
is not defined and we will say that “it is at infinity”. If at a point the order of
contact of γ with its osculating sphere of codimension 2, S m−1 , is greater than
the usual one, then the point is a flattening and all hyperspheres containing
S m−1 are osculating, i.e. the centre of the osculating hyperpshere is not
uniquely defined. In particular, the flattenings of a curve lying on a sphere
satisfy these non generic conditions. Such flettenings do not appear in generic
curves. For this reason we consider curves having all its curvatures different
from 0 at any point.
The hyperplane normal to γ at a point consists of the set of centres of all
hyperspheres tangent to γ at that point. Hence the centre of the osculating
hypersphere at that point lies in such normal hyperplane. So (denoting Cγ (θ)
by Cγ , γ(θ) by γ and so on,. . . ) we can write
Cγ = γ + c1 n1 + c2 n2 + · · · + cm nm ,
where c1 , . . . , cm−1 are smooth functions of the parameter of the curve γ.
Definition – We call the function ci the ith focal curvature of γ.
Remark – The function c1 never vanishes: we will see that c1 = 1/k1 .
3
§1. Statement of Results
Consider a smoothly immersed curve γ : R → Rm+1 , write k1 , k2, . . . , km
for its curvatures and t, n1 , . . . , nm for its Frenet frame. We assume that all
the curvatures of our curves are different from 0 at any point.
Theorem 1 – The focal curvatures of γ : R → Rm+1 , parametrized by
arc length s, satisfy the following “scalar Frenet equations” for cm = 0:
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1
c1
c2
c3
..
.
cm−2
cm−1
2 cm − (R2cmm)
⎞
⎛
0 ···
0
0
0
0
k1
⎜ −k1
0
k2 · · ·
0
0
0
⎟ ⎜
⎟ ⎜
..
⎟ ⎜ 0 −k2
.
0
⎟ ⎜
..
⎟ ⎜
.
0 −k3
⎟ ⎜ 0
⎟=⎜ .
⎟ ⎜ ..
0
⎟ ⎜
⎟ ⎜
0
km−1 0
⎟ ⎜
⎠ ⎜ ..
⎝ .
0
km
−km−1
0
0
···
0
−km 0
⎞
⎛
0
⎟
⎟ ⎜ c1
⎟⎜
⎟ ⎜ c2
⎟⎜
⎟ ⎜ c3
⎟⎜
⎟ ⎜ ..
⎟⎜ .
⎟⎜
⎟ ⎜ cm−2
⎟⎜
⎟ ⎝ cm−1
⎠
cm
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
Remark – If the curve is spherical then the last component of the left
hand side vector is just cm .
Theorem 2 – The curvatures of a generic curve γ in Rm+1 , parametrized
by arc length, may be obtained in terms of the focal curvatures of γ by the
formula:
c1 c1 + c2 c2 + · · · + ci−1 ci−1
, for i ≥ 2.
ki =
ci−1 ci
Remark – For a generic curve, the functions ci or ci−1 can vanish at
isolated points. At these points the function c1 c1 + c2 c2 + · · · + ci−1 ci−1 also
vanishes, and the corresponding value of the function ki may be obtained by
l’Hôpital rule.
c21
Denote by Rl the radius of the osculating l–sphere. Obviously Rl2 =
2
+ · · · + c2l . In particular Rm
=
Cγ − γ 2 .
Theorem 3 – The radius Rl of the osculating l–sphere of a generic curve,
parametrized by arc length, in the Euclidean space Rm+1 , m > 1, is critical
if and only if
a) c2 = 0, for l = 1;
b) either cl = 0 or cl+1 = 0, for 1 < l < m;
c) either cm = 0 or cm + cm−1 km = 0 for l = m.
From the formula Rl Rl = cl cl+1 kl+1 , for 1 ≤ l < m, obtained below in the
proof of Theorem 3, one obtains the
4
.
Corollary 1 – If the lth focal curvature cl vanishes at a point, then the
radii Rl and Rl−1 of the osculating l−sphere and (l − 1)−sphere are critical
at that point.
Definition – A vertex of a curve in Rn is a point at which the curve has
at least (n + 2)−point contact with its osculating hypersphere.
Example – A non-circular ellipse in the plane R2 has 4 vertices. They
are the points at which the ellipse intersects its principal axes.
Theorem 4 – A point of a generic curve parametrized by arc length in
, m > 1, is a vertex if and only if
R
m+1
cm + cm−1 km = 0 at that point.
Corollary 2 – A smoothly immersed curve, parametrized by arc length,
in the Euclidean space Rm+1 , m > 1, is spherical if and only if
cm + cm−1 km ≡ 0.
Remark – Item c) of Theorem 3 may be reformulated, using Theorem 4,
as follows:
The radius of the osculating hypersphere of a curve at a point is critical
if and only if either
1) the point is a vertex or
2) at that point, the centre of the osculating hypersphere lies in the osculating hyperplane.
Item c) of Theorem 3 was known to V.D. Sedykh [4] (under this reformulation). We note moreover that item b) of Theorem 3 implies that:
In case 2) the radius of the osculating (m − 1)–sphere is also critical.
The points of a curve for which the centre of the osculating hypersphere
lies in the osculating hyperplane (i.e. cm = 0) are called symmetry points of
that curve [4]. As a corollary of Theorem 4 and of item c) of Theorem 3, one
obtains the following theorem, also known to V.D. Sedykh [4]:
Theorem 5 – Write V , F and S for the number of vertices, flattenings
and symmetry points of a generic closed curve, smoothly immersed in Rm+1 .
Then V + S ≥ F .
§2. Study of the Focal Set of a Curve
Definition – The focal set of a submanifold of positive codimension in
the Euclidean space Rn (for instance, of a curve in R3 ) is defined as the
envelope of the family of normal lines to the submanifold.
5
We will study the focal set of a generic curve in Rn .
We assume that the derivatives of order 1, . . . , n − 1, of our curve are
linearly independent at any point. Let F : Rn × R → R be the n−parameter
family of real functions given by
1
q − γ(s) 2 .
2
Definition – The caustic of a family of functions consists of the parameter
values for which the corresponding function has a non-Morse critical point.
F (q, s) =
The caustic of the family F is the focal set of the curve:
We shall write
Σ(i) = {(q, s)/∂s F (q, s) = 0, ..., ∂si F (q, s) = 0}.
Thus Σ(1) is the set of pairs (q, s) such that q is the centre of some hypersphere of Rn having at least 2−point contact with γ at s (this means that
q is in the normal hyperplane to γ at s). So Σ(2) is the set of pairs (q, s)
such that q is the center of some hypersphere of Rn having at least 3−point
contact with γ at s. It can be seen from the equations that these points
generate a plane of dimension n − 2 contained in the normal hyperplane to γ
at s. If γ(s) is not a flattening of γ then the curve γ has only one osculating
hypersphere at γ(s). So Σ(n) is the set of pairs (q(s), s) such that q(s) is the
centre of an osculating hypersphere at γ(s). Hence, if γ(s) is not a flattening
of γ then the value of F at the point (q(s), s) in Σ(n) is one half of the square
of the radius of the osculating hypersphere at γ(s). The condition for a point
p = γ(s) to be a vertex is equivalent to the fact that the first n+1 derivatives
of F with respect to s vanish at s. Hence Σ(n + 1) is the set of vertices of the
curve. It is a well-known fact of singularity theory [1] that a point belonging
to Σ(n + 1) is a critical point of the restriction of F to Σ(n). So a vertex is
a critical point of the radius of the osculating hypersphere (see also [5]).
Remark – The centers of the osculating hyperspheres at the vertices of
γ are given by the points q ∈ Rn for which there exists a solution s of the
following (n + 1)−system of equations
Fq (s) =
Fq (s) =
..
.
(n+1)
Fq
0
0
(s) = 0.
For a fixed s, the first equation gives the normal hyperplane to the curve
at the point γ(s). The normal hyperplane consists of the set of centres of all
hyperspheres tangent to γ at that point.
6
The first two equations give a codimension 1 subspace of the normal
hyperplane to the curve at the point γ(s). This subspace consists of the set
of centres of all hyperspheres having multiplicity of intersection at least 3
with the curve at that point.
Following this process we obtain (for a generic curve) a complete flag
at each non–flattening point of the curve. The focal curve q(s), formed
by the centers of the osculating hyperspheres, is determined by the n first
equations. The complete flag is the osculating flag of the focal curve. In
particular, the osculating hyperplane of the focal curve at the point q(s) is
the normal hyperplane to the curve γ at the point γ(s). As the point moves
along the curve γ, the corresponding flag (starting with the codimension 2
subspace) generates a hypersurface which is stratified in a natural way by
the components of the flag. This stratified hypersurface is a component of
the focal set of the curve γ. The other component of the focal set is the curve
itself. The stratum of dimension 1 (generated by the 0–dimensional subspace
of the flag, i.e. generated by the center of the osculating hypersphere at the
(n+1)
moving point) is the focal curve of γ. The equation Fq
(s) = 0 gives a
finite number of isolated points on the focal curve. These points correspond
to the vertices.
Remark – The (generating) family F is also useful to calculate the number
of vertices of a curve (in [6] Sturm theory is applied to the generating family
F in order to obtain a formula for the calculation of vertices).
Remark – In terms Lagrangian singularities, the focal set is the caustic of
the Lagrangian map (normal map) defined by the generating family F (q, s)
(For the notions of caustic, Lagrangian map, Lagrangian singularity and
generating family, we refer the reader to [1] and [2]). Thus the vertices of a
curve in Rn correspond to a Lagrangian singularity An+1 of the normal map.
§3. The Proofs
To prove our results we will prove before some lemmas related to the focal
curve. Below, θ denotes any regular parameter of the curve and s denotes
the arc length parameter.
Lemma 1 – Let γ : θ → (ϕ1 (θ), . . . , ϕm+1 (θ)) be a curve in Rm+1 . The
velocity vector q (θ) of the focal curve of γ at θ is proportional to the mth normal vector nm (θ) of γ.
Proof – As in §2, consider the (generating) family of functions F : R ×
R
→ R defined by
m+1
Fq (θ) =
1
q − γ(θ) 2 .
2
7
2
2
2
Write g = γ2 . Using the fact that −F = γ · q − γ2 − q2 , we recall that
the following system of m + 1 equations defines the focal curve q(θ) of γ:
γ · q(θ) − g = 0,
γ · q(θ) − g = 0,
..
.
(∗)
γ (m+1) · q(θ) − g (m+1) = 0.
Deriving each equation with respect to θ, one obtains a second system of
equations:
γ · q (θ) + γ · q(θ) − g = 0,
γ · q (θ) + γ · q(θ) − g = 0,
..
.
(m)
(m+1)
(∗∗)
(m+1)
γ · q (θ) + g
· q(θ) − g
= 0,
γ (m+1) · q (θ) + g (m+2) · q(θ) − g (m+2) = 0.
Combining the ith equation of system (∗∗) with the (i + 1)th equation of
system (∗), for i = 1, . . . , m, one obtains
γ · q (θ) = 0,
γ · q (θ) = 0,
..
.
(∗ ∗ ∗)
γ (m) · q (θ) = 0.
This means that the velocity vector q (θ) is orthogonal to the osculating
hyperplane of γ, i.e. q (θ) is proportional to the mth -normal vector nm . Proposition 1 – A non–flattening point of a curve in Rm+1 is a vertex
if and only if the velocity vector of the focal curve is zero.
Proof – If the point γ(θ) is a vertex of γ, then besides the system of
equations (∗) obtained in the proof of Lemma 1, it also satisfies the equation:
γ (m+2) · q(θ) − g (m+2) = 0,
which combined with the last equation of system (∗∗) gives the equation
γ (m+1) · q (θ) = 0.
The preceding equation together with the system (∗ ∗ ∗) imply that for a
non–flat vertex γ(θ) of the curve γ the velocity vector q (θ) of the focal curve
is zero.
Conversely, if a point γ(θ0 ) is not a vertex then the corresponding point
of the focal curve satisfies the relation
8
γ (m+2) (θ0 ) · q(θ0 ) − g (m+2) (θ0 ) = 0,
which together with the last equation of (∗∗), for θ = θ0 , imply that q (θ0 ) =
0. Proof of Theorem 1 – Let Cγ (s) = (γ + c1 n1 + c2 n2 + · · · + cm nm )(s),
be the focal curve. Applying Frenet equations to the derivative of Cγ with
respect to the arc length of γ (and denoting Cγ (θ), γ(θ) and so on by Cγ ,
γ,. . . ) one obtains:
Cγ = t + c1 (−k1 t + k2 n2 ) + c1 n1 + · · · + cm−1 nm−1 + cm (−km nm−1 ) + cm nm
= (1 − c1 k1 )t + (c1 − k2 c2 )n1 + (c2 + c1 k2 − c3 k3 )n2 + · · ·
+(ci + ci−1 ki − ci+1 ki+1 )ni + · · · + (cm + cm−1 km )nm .
By Lemma 1, the first m − 1 components of Cγ vanish. This implies that
1 = k 1 c1 ,
c1 = k2 c2 ,
c3 = −k2 c1 + k3 c3 ,
.. .. ..
. . .
cm−1 = −cm−2 km−1 + cm km .
(1)
Thus Cγ = (cm + cm−1 km )nm . Using the fact that the radius Rm of the
2
=
Cγ − γ 2 , one obtains
osculating hypersphere satisfies Rm
2 ) = Cγ − γ, Cγ − γ = 2Cγ − γ , Cγ − γ
(Rm
= 2(cm + cm−1 km )nm − t, c1 n1 + · · · + cm nm = 2cm (cm + cm−1 km ).
2
Thus for cm = 0, cm − (R2cmm) = −cm−1 km . This equation together with the
set of equations (1) proves Theorem 1. Remark – Lemma 1, Proposition 1 and part of the calculations of the
preceding proof were also obtained in [3], where the condition to the point
to be a non–flattening is absent (however without this condition their proof
of Proposition 1 does not work).
Proof of Theorem 4 and of its Corollary – In the proof of Theorem 1 we
obtained that Cγ = (cm + cm−1 km )nm . By Proposition 1, a point of the curve
γ is a vertex if and only if cm + cm−1 km = 0. Proof of Theorem 1 – The proof will be by induction. We will use the
“scalar Frenet equations” of Theorem 1. First, we have that
c
c2 + c1 c12
1
c
c1 c1
c + c1 k 2
c2 c2 + c1 c1
and k3 = 2
=
=
.
k1 = , k2 = 1 =
c1
c2
c1 c2
c3
c3
c2 c3
9
Suppose that
ki =
ci−1 ci−1 + · · · + c2 c2 + c1 c1
.
ci−1 ci
(2)
From the “scalar Frenet equations” of Theorem 1, we know that ci+1 ki+1 =
ci + ci−1 ki . Substituting equation (2) one obtains
ci+1 ki+1 = ci +
ci−1 ci−1 + · · · + c2 c2 + c1 c1
ci ci + · · · + c2 c2 + c1 c1
=
. ci
ci
Proof of Theorem 3 – We have Rl2 = c21 + · · · + c2l . Thus Rl Rl = c1 c1 +
· · · + cl cl . Applying the formula of Theorem 2, one obtains
Rl Rl = cl cl+1 kl+1 for 1 ≤ l < m.
For a generic curve in Rm+1 the first m − 1 curvatures are nowhere vanishing
and the mth curvature may vanish at isolated points, which do not coincide
with the points at which Rm−1 is critical. Thus for a generic curve in Rm+1 ,
m > 1, Rl = 0 if and only if either cl = 0 or cl+1 = 0 for 1 ≤ l < m. Moreover,
for a smoothly immersed curve the function R1 = c1 never vanishes. This
proves items a) and b).
2 In the proof of Theorem 1, we obtained that (Rm
) = 2cm (cm + cm−1 km ).
This proves item c). References
[1] Arnol’d V.I., Varchenko A.N., Gussein–Zade S.M., Singularities of Differentiable Maps, Vol. 1, Birkhäser (1986). (French version: Singularités des applications
différentiables, Vol. 1, Ed. Mir, Moscou, 1986. Russian version: Nauka, 1982)
[2] Arnol’d V.I., Singularities of caustics and wave fronts, Kluwer, Maths. and its
Appls., Soviet series, vol.62., 1991.
[3] Romero–Fuster M.C., Sanabria–Codesal E., Generalized evolutes, vertices and
conformal invariants of curves in Rn+1 , Indag. Math. N.S. 10(2), (1999) p.297–305.
[4] Sedykh V.D., Personal comunication, December 2000.
[5] Uribe–Vargas R., On the Higher Dimensional Four–Vertex Theorem, C.R. Acad.
Sci. Paris, t.321, Série I, 1995, p. 1353–1358.
[6] Uribe–Vargas R., 4-Vertex Theorems and Sturm Theory, (To appear)
10