faculty of mathematics and natural sciences

## On curves with constant curvatures

### Bachelor Project Mathematics

June 12, 2015

Student: Y.M. Ebbens

First supervisor: Prof.dr. G. Vegter Second supervisor: Prof. dr. H.W. Broer

Abstract

In Euclidean geometry a curve is determined completely by its curvatures, up to a rigid transformation. Affine geometry, which studies geometric invariants under the group of volume preserving linear transformations, gives rise to the so called affine curvatures of a curve. In my thesis I will first derive two necessary and sufficient conditions for a curve to have constant Euclidean curvatures. Firstly, the tangents at any two points make the same angle with the line segment connecting these points. Secondly, the distance between points on the curve does not depend on the actual position of these points, but only on the arc length of the curve segment between the points. Then I will derive similar conditions for a curve to have constant affine curvatures. Firstly, the volume of a full-dimensional simplex formed by points on the curve does not depend on the actual positions of these points, but only on the affine arc length of the curve segments between the points. Secondly, the affine arc length of an off-set curve in the tangential direction is proportional to the affine arc length of the curve itself.

Keywords: Euclidean geometry, affine geometry, differential geometry, curvature, affine cur- vature, constant curvature, curve

### Contents

1 Introduction 4

2 Euclidean differential geometry 6

2.1 Arc length . . . 6
2.2 Curvature and torsion in R^{3} . . . 6
2.3 Euclidean curvatures in R^{m} . . . 7

3 Affine differential geometry 10

3.1 Affine arc length and curvature in R^{2} . . . 10
3.2 Affine arc length and curvature in R^{m} . . . 11

4 Curves with constant Euclidean curvatures 14

5 Characterizations of curves with constant affine curvatures 18
5.1 Characterization in R^{2} . . . 18
5.2 Characterization in R^{m} in terms of volume . . . 21
5.3 Characterization in R^{m} in terms of offset curves . . . 27

6 Conclusion 30

7 Acknowledgements 30

### 1 Introduction

Many people have an intuitive idea of the curvedness of a curve. A straight line is not curved
at all, while a constantly curved curve in the plane is a piece of a circle. In three-dimensional
space we add helices as curves with constant curvatures. This thesis will be about generalizing
the notion of constantly curved curve to R^{m}. Furthermore, we will not restrict ourselves to
Euclidean geometry, but we will also look at affine geometry.

Since Klein’s Erlanger Programm [9] in 1872, geometry is regarded as the study of invariants under a certain transitive transformation group. In Euclidean geometry this is the group of rigid transformations, consisting of translations, rotations and reflections. The natural 2-point invariant of this group is the distance between pairs of points. Affine geometry studies geomet- ric invariants under the larger group of volume preserving linear transformations, also known as equi-affine transformations [7, 139-147]. Every equi-affine transformation can be represented as T (x) = Ax + b, where the fact that T is volume preserving means that det A = 1. The natural invariant of this group is (signed) volume. As Euclidean geometry speaks of Euclidean curvatures, affine geometry speaks of affine curvatures.

Euclidean geometry goes back to the beginning of civilization, but differential geometry could
only be introduced after the introduction of differential calculus and multilinear algebra. At first
Euclidean differential geometry focused on curves and surfaces in two- and three-dimensional
space, as do many introductions today, e.g., do Carmo [4]. For three-dimensional curves Frenet
[5] and Serret [11] independently found the so called Frenet-Serret frame, an orthonormal moving
frame that completely determines a curve up to a rigid transformation. Jordan [8] generalized
the idea of curvature for curves in R^{m} by constructing a similar moving frame using the Gram-
Schmidt Orthogonalization Process twenty years later.

In the first half of the 20^{th} century affine differential geometry was studied following the same
methods of Euclidean differential geometry. At first only two- and threedimensional space curves
were regarded. In his standard work Blaschke [2] systematically describes affine invariants, such
as affine arc length and affine curvature. Later these notions were generalized to R^{m}.

Parametrizations for curves with constant curvatures, Euclidean or affine, are usually included
in books on differential geometry, but only for curves in R^{m} for m ≤ 3. For example, Blaschke
[2] proves that conics are the only plane curves with constant affine curvature. While Guggen-
heimer [7, 170-173] describes affine curvatures in R^{m}, he only describes curves with constant
affine curvature in R^{3}. Admittedly, parametrizations for curves with constant affine curvatures
soon become quite complicated, which is why we focus on deriving necessary and sufficient
geometric conditions.

Even though our primary focus is mathematical, applications of curves with constant affine cur- vature can be found in Computer Aided Geometric Design. Coined by Barnhill and Reisenfeld in 1974 [1], this field of research studies the use of geometric algorithms. As one of the simplest curves, curves with constant (affine) curvature are useful tools in approximating more complex curves and surfaces.

In Section 2 and 3 of this thesis we will first present all necessary Euclidean and affine differen- tial geometry. In Section 4 we will look at curves with constant Euclidean curvatures. We will derive two necessary and sufficient conditions for a curve to have constant Euclidean curvatures.

Firstly, the tangents at any two points make the same angle with the line segment connecting these points. Secondly, the (Euclidean) distance between points on the curve does not depend

on the actual position of these points, but only on the arc length of the curve segment between the points.

Section 5 will be about curves with constant affine curvatures. First we will give a proof of
the already known fact that conics are the only curves in R^{2} with constant affine curvature to
illustrate the approach we will use later on in the more general case. Furthermore, we will prove
that every arc of such curves is equi-areal, that is, at any two points the parallellograms formed
by the tangent vectors and the line segment connecting these points have equal area. However,
these conditions cannot be generalized to R^{m}. Instead we will derive two other necessary and
sufficient conditions for a curve to have constant affine curvatures. Firstly, the volume of a
full-dimensional simplex formed by points on the curve does not depend on the actual positions
of these points, but only on the affine arc length of the curve segments between the points.

Secondly, the affine arc length of an off-set curve in the tangential direction is proportional to the affine arc length of the curve itself. Section 6 will conclude with topics for future work.

### 2 Euclidean differential geometry

In this section we will first recall the notions of arc length, curvature and torsion from elementary
differential geometry. Subsection 2.3 describes a generalization of Euclidean curvature to R^{m}.
2.1 Arc length

Let the differentiable curve γ : I → R^{m} be regular. To find an approximation of the arc length
of γ([a, b]), subdivide the interval [a, b] into

a = t0 < . . . < tn= b.

The approximation is then given by the length of the polygonal curve formed by the sequence
of points γ(t0), . . . , γ(tn). This approximation can be made as accurate as wanted by letting
the distance between the t_{i} go to zero [4, 10]. The arc length is then given by:

Z _{b}

a

|γ^{0}(t)|dt, (1)

where |·| denotes the Euclidean norm. We say that γ is parametrized by arc length if its tangent vector has unit length at every point. Then the length of γ([a, b]) is given by b − a. From now on we will use s as the parameter for a curve if it is parametrized by arc length. In that case, any derivatives are with respect to s, unless otherwise stated.

Figure 1: Geometric definition of arc length

Remark 2.1. Critical to the definition of arc length is that it is invariant under rigid transforma- tions. Indeed, this holds, since the derivative of γ is invariant under translations and rotations and reflections preserve the length of a vector. See also [4, 20].

2.2 Curvature and torsion in R^{3}

If γ : I → R^{3} is parametrized by arc length we can differentiate the relation

|γ^{0}(s)|^{2} = 1,

to get

hγ^{0}(s), γ^{00}(s)i = 0,

where h · , · i denotes the standard inner product. Thus, γ^{00}is perpendicular to the tangent t(s).

The curvature of γ is defined as

κ(s) = |γ^{00}(s)|, (2)

which gives a measure of the rate of change of the tangent. In points where k(s) 6= 0 we define a unit normal as

n(s) = γ^{00}(s)

|γ^{00}(s)|. (3)

In points where n(s) is defined, the vectors t(s) and n(s) determine a plane, the so called osculating plane. The unit vector

b(s) = t(s) ∧ n(s), (4)

where ∧ denotes the outer product in R^{3}, is normal to this osculating plane and is called the
binormal vector. We can use the same method as before to show that b^{0}(s) is perpendicular to
b(s) and t(s). Hence,

b^{0}(s) = τ (s)n(s), (5)

for some function τ (s), called the torsion of γ. The torsion measures how rapidly a curve pulls
away from the osculating plane. A simple calculation show that n^{0}(s) = −κ(s)t(s) − τ (s)b(s),
so that we get the Frenet-Serret formulas [4, 16-19]:

t n b

0

=

0 κ 0

−κ 0 −τ

0 τ 0

t n b

. (6)

2.3 Euclidean curvatures in R^{m}

To generalize the method of the previous subsection to R^{m}, note that in fact we constructed
an orthonormal moving frame {t, n, b}. The curvature and torsion were then computed from
the derivatives {t^{0}, n^{0}, b^{0}}. Furthermore, torsion could only be defined if γ^{0} and γ^{00} were linearly
independent.

Now, let γ : I → R^{m} be parametrized by arc length and suppose that {γ^{0}(s), . . . , γ^{(m)}(s)}

is a linearly independent set of vectors, where γ^{(k)} denotes the k-th derivative of γ. Then by
applying the Gram-Schmidt Orthogonalization Process we obtain an orthonormal moving frame
{F_{1}(s), . . . , Fm(s)}, see [6]. The next theorem describes how the Euclidean curvatures of γ can
be obtained from this moving frame.

Theorem 2.2. Let γ : I → R^{m} be parametrized by arc length with associated orthonormal
moving frame {F_{1}(s), . . . , F_{m}(s)}. Then there exist functions κ_{1}, . . . , κ_{m−1}, called the Euclidean
curvatures of γ, such that

F_{1}
F_{2}
F3

... Fm−2

F_{m−1}
Fm

0

=

0 κ1 0 · · · 0 0 0

−κ_{1} 0 κ_{2} · · · 0 0 0

0 −κ_{2} 0 . .. 0 0 0

... ... ... . .. ... ... ...

0 0 0 . .. 0 κ_{m−2} 0

0 0 0 · · · −κ_{m−2} 0 κm−1

0 0 0 · · · 0 −κ_{m−1} 0

F_{1}
F_{2}
F3

... Fm−2

F_{m−1}
Fm

.

These functions are invariant under rigid transformations and determine the curve γ completely, up to a rigid transformation.

Proof. By differentiating

hF_{i}(s), Fj(s)i = δij,
we get

hF_{i}^{0}(s), Fj(s)i + hFi(s), F_{j}^{0}(s)i = 0. (7)
Note that by construction we have for 1 ≤ i ≤ m

Span(γ^{0}(s), . . . , γ^{(i)}(s)) = Span(F1(s), . . . , Fi(s)), (8)
so F_{i}(s) is a linear combination of γ^{0}(s), . . . , γ^{(i)}(s). It follows that F_{i}^{0}(s) is a linear combination
of γ^{0}(s), . . . , γ^{(i+1)}(s), hence also of the orthonormalized basis F1(s), . . . , Fi+1(s). Hence, (7)
reduces to

hF_{i}^{0}(s), F_{j}(s)i = 0,

except possibly for j = i − 1 or j = i + 1. Thus, there exist functions κ_{1}, . . . , κ_{m−1} such that
for 2 ≤ i ≤ m − 1

F_{1}^{0}(s) = κ1(s)F2(s), (9)

F_{i}^{0}(s) = −κ_{i−1}(s)F_{i−1}(s) + κ_{i}(s)F_{i+1}(s). (10)
Invariance of the curvatures under rigid transformations follows from the same reasoning we
used for invariance of arc length. Complete determination of a curve by its curvatures is proved
in [4, 20-21] for R^{3}. This proof can be easily generalized to R^{m}.

It may happen that we cannot make an orthonormal frame from the derivatives of γ. If
we have for some n < m that γ^{0}(s), . . . , γ^{(n)}(s) are linearly independent for all s ∈ I, but
γ^{0}(s), . . . , γ^{(n+1)}(s) are not, then we can only define κ_{1}, . . . , κ_{n−1}. Such a curve lies in an n-
dimensional subspace of R^{m}. This means that by applying an appropriate rigid transformation
we can find a curve in R^{n}corresponding to γ. In other words, if for γ we only have κ1, . . . , κn−1,
then we can reduce the dimension of the ambient space without changing the curve. Therefore,
we will henceforth assume that κ_{1}, . . . , κ_{m−1} can be defined for a curve in R^{m}. In the following
lemma we will see that this implies that most of them are nonzero.

Lemma 2.3. The Euclidean curvatures of a curve γ : I → R^{m} are all nonzero on their domain,
except possibly κm−1.

Proof. Suppose we have for some 1 ≤ i < m and s0 ∈ I that κ_{i}(s0) = 0. Then
F_{i}^{0}(s_{0}) = −κ_{i−1}(s_{0})F_{i−1}(s_{0}).

It follows that γ^{(i+1)}(s_{0}) can be written as a linear combination of F_{1}(s_{0}), . . . , F_{i−1}(s_{0}), which
contradicts the linear independency of γ^{0}, . . . , γ^{(m)}. For i = m we get

F_{m}^{0} (s_{0}) = −κ_{m−1}(s_{0})F_{m−1}(s_{0}),

which is no problem, since there is nothing preventing γ^{(m+1)} from being linearly dependent on
γ^{0}, . . . , γ^{(m)}.

We can start writing the derivatives of γ in terms of the moving frame by using (9) and (10):

γ^{0} = F1,
γ^{00} = κ1F2,

γ^{000} = κ_{1}κ_{2}F_{3}+ κ^{0}_{1}F_{2}− κ^{2}_{1}F_{1}.

A general expression is given by the following lemma, which will prove useful when discussing curves with constant curvatures.

Lemma 2.4. Let γ : I → R^{m} be parametrized by arc length. Then for 1 ≤ i ≤ m we can write:

γ^{(i)}=

i

X

j=1

c_{ij}F_{j}, (11)

where c_{ii}= κ_{1}κ_{2}. . . κ_{i−1} and the other c_{ij} depend on κ_{1}, . . . , κ_{i−2} and their derivatives.

Proof. We will prove this with induction. Clearly, this holds for i = 1. Now assume it holds for i = n. Then

γ^{(n+1)}=

n

X

j=1

c_{nj}F_{j}^{0}+ c^{0}_{nj}F_{j},

= c_{n1}κ_{1}F_{2}+

n

X

j=2

− c_{nj}κ_{j−1}F_{j−1}+ c^{0}_{nj}F_{j}+ c_{nj}κ_{j}F_{j+1},

=

n+1

X

j=1

˜

c_{n+1,j}F_{j},

where ˜c_{n+1,n+1} = c_{nn}κ_{n} = κ_{1}κ_{2}. . . κ_{n} and the other ˜c_{ij} depend on κ_{1}, . . . , κ_{n−1} and their
derivatives.

### 3 Affine differential geometry

Following the same approach as the previous section, we will now define affine arc length and
affine curvatures. We will first look at R^{2} and then generalize this to R^{m}.

3.1 Affine arc length and curvature in R^{2}

To define arc length in affine geometry we cannot use the approach as in Euclidean geometry, since the length of line segments is not necessarily preserved under equi-affine transformations.

However, instead of adding the lengths of the edges of an inscribed polygon, we can add the areas of the triangles formed by the edges of the polygon and the corresponding tangents to the curve in the vertices, as in [2, 9-10], or of the triangles formed by triples of points on the curve, as in [10, 37-38].

(a) [2, 9-10] (b) [10, 37-38]

Figure 2: Geometric definition of affine arc length Both methods yield the same formula for the affine arc length of γ([a, b]):

Z b a

||γ^{0}(t), γ^{00}(t)||^{1/3}dt, (12)

where ||v, w|| denotes the determinant of the matrix formed by the vectors v and w and the derivatives are with respect to t. We then say that γ is parametrized by affine arc length if for all t ∈ I

||γ^{0}(t), γ^{00}(t)|| = 1. (13)

Then the affine arc length of γ([a, b]) is given by b − a. Note that regularity implies that a curve
can be reparametrized by arc length, while reparametrization by affine arc length needs the
extra condition ||γ^{0}(t), γ^{00}(t)|| 6= 0, that is, the curve must have non-zero Euclidean curvature.

From now on we will use r as the parameter for a curve if it is parametrized by affine arc length.

In that case, any derivatives are with respect to r, unless otherwise stated. Now, if a curve is parametrized by affine arc length, it follows that

||γ^{0}(r), γ^{000}(r)|| = 0. (14)

This means that there is a differentiable function k such that

γ^{000}(r) + k(r)γ^{0}(r) = 0. (15)

This function k is called the affine curvature of γ. By combining (13) and (15), we get

k(r) = ||γ^{00}(r), γ^{000}(r)||. (16)

By differentiating (14) we see that

||γ^{00}(r), γ^{000}(r)|| + ||γ^{0}(r), γ^{(4)}(r)|| = 0,
so

k(r) = −||γ^{0}(r), γ^{(4)}(r)||. (17)

Differentiating (16) and (17) we get

k^{0}(r) = ||γ^{00}(r), γ^{(4)}(r)|| = −||γ^{00}(r), γ^{(4)}(r)|| − ||γ^{0}(r), γ^{(5)}(r)||. (18)
Affine arc length can also be defined the other way around. This way of defining affine arc
length will prove especially useful in R^{m}.

Proposition 3.1. Let γ : I → R^{2} be a regular curve with nonzero Euclidean curvature. Then
defining affine arc length as

r(t) = Z t

t0

||γ^{0}(u), γ^{00}(u)||^{1/3}du (19)

is equivalent with choosing r = r(t) such that

dγ
dr,d^{2}γ

dr^{2}

= 1. (20)

Proof. Since

1 =

dγ
dr,d^{2}γ

dr^{2}

=

dγ
dt,d^{2}γ

dt^{2}

dt dr

3

, we see that

r(t) = Z t

t0

||γ^{0}(u), γ^{00}(u)||^{1/3}du,

where γ^{0}(u) denotes the derivative of γ with respect to t evaluated at u. Reading backwards
proves the equivalence.

3.2 Affine arc length and curvature in R^{m}

The definition of Euclidean arc length is the same regardless of the dimension of the ambient
space, but in affine geometry we cannot use the affine arc length in R^{2} as affine arc length in
R^{m}, since only full-dimensional volumes are invariant. Therefore we will base our definition of
affine arc length for curves in R^{m} on Proposition 3.1.

Given a curve γ = γ(t) in R^{m} with nonzero Euclidean curvatures, that is,

dγ
dt,d^{2}γ

dt^{2}, . . . ,d^{m}γ
dt^{m}

6= 0,

the definition of the affine arc length parameter r = r(t) follows from

dγ
dr,d^{2}γ

dr^{2}, . . . ,d^{m}γ
dr^{m}

= ±1.

Here ±1 is used instead of 1, because only for m ≡ 0 mod 4 or m ≡ 3 mod 4 the sign is
invariant under equi-affine transformations [7, 170]. As in R^{2} we calculate

dγ
dr,d^{2}γ

dr^{2}, . . . ,d^{m}γ
dr^{m}

=

dγ
dt,d^{2}γ

dt^{2}, . . . ,d^{m}γ
dt^{m}

dt dr

p

, where the power p is given by

p =

m

X

i=1

i = ^{1}_{2}m(m + 1).

Thus,

r(t) = Z t

t0

||γ^{0}(u), γ^{00}(u), . . . , γ^{(m)}(u)||

2

m(m+1)du. (21)

Similarly to in R^{2}, we say that γ is parametrized by affine arc length if

||γ^{0}(t), γ^{00}(t), . . . , γ^{(m)}(t)|| = 1

for all t ∈ I. We will write γ as function of r if it is parametrized by affine arc length. In that case any derivatives are with respect to r. The affine arc length of γ([a, b]) is then given by b − a.

We will now turn to affine curvatures with the following theorem:

Theorem 3.2. Let γ : I → R^{m} be parametrized by affine arc length and have nonzero Euclidean
curvatures. Then there exist functions k1, . . . , km−1, called the affine curvatures of γ, such that

γ^{0}
γ^{00}
...
γ^{(m−1)}

γ^{(m)}

0

=

0 1 0 · · · 0 0

0 0 1 · · · 0 0

... ... . .. ... ... ...

0 0 0 · · · 0 1

k_{1} k_{2} k_{3} · · · k_{m−1} 0

γ^{0}
γ^{00}
...
γ^{(m−1)}

γ^{(m)}

.

These functions are invariant under equi-affine transformations and determine the curve γ completely, up to an equi-affine transformation.

Proof. If γ = γ(r) is parametrized by affine arc length, we have that

||γ^{0}, . . . , γ^{(m)}|| = 1, (22)

where γ^{(k)} denotes the k-th derivative of γ and the derivatives are with respect to affine arc
length. By differentiating we get

||γ^{0}, . . . , γ^{(m−1)}, γ^{(m+1)}|| = 0, (23)
so there exist differentiable functions k1, . . . , km−1 such that

γ^{(m+1)}= k1γ^{0}+ . . . + km−1γ^{(m−1)}. (24)
The ki are called the affine curvatures of γ. Note that γ^{0}, . . . , γ^{(m−1)} are linearly independent,
so the affine curvatures are uniquely defined. Explicit formulas for the affine curvatures follow
from

||γ^{0}, . . . , γ^{(i−1)}, γ^{(m+1)}, γ^{(i+1)}, . . . , γ^{(m)}|| = ||γ^{0}, . . . , γ^{(i−1)},

m−1

X

j=1

k_{j}γ^{(j)}, γ^{(i+1)}, . . . , γ^{(m)}|| = k_{i}.
(25)

The first formula should be interpreted as follows: start with the determinant of the matrix with the first till m-th derivatives, then replace the i-th derivative by the (m + 1)-th derivative.

The result is ki, because in the sum all terms vanish except for the one with kiγ^{(i)}.

Since equi-affine transformations preserve determinants, the affine curvatures are invariant. We leave it to the reader to adapt the proof of determination of a curve by Euclidean curvatures to the affine analogue.

Using the expressions above we can also give an expression for the derivatives of the affine curvatures:

k_{1}^{0} = ||γ^{(m+2)}, γ^{00}, . . . , γ^{(m)}||, (26)

k_{i}^{0} = ||γ^{0}, . . . , γ^{(i−2)}, γ^{(i)}, γ^{(m+1)}, γ^{(i+1)}, . . . , γ^{(m)}|| + ||γ^{0}, . . . , γ^{(i−1)}, γ^{(m+2)}, γ^{(i+1)}, . . . , γ^{(m)}||,
(27)
for 2 ≤ i ≤ m − 1. Note that the notation may seem confusing. For example, for i = 2 we do
not get γ^{0} nor γ in the first term, because γ^{0} and γ^{(i−1)} coincide in (25). The usage of γ^{(i−2)}
signifies that γ^{(i−1)} is missing, but that all terms before it (if any) remain.

Remark 3.3. In R^{2} we saw that the affine curvature k of a curve γ is given by
k(r) = ||γ^{00}(r), γ^{000}(r)||,

while the generalization to R^{m} gives us

k_{1}(r) = ||γ^{000}(r), γ^{00}(r)|| = −k(r).

This can also directly be seen by comparing (15) and (24). Authors typically use k instead of
k_{1}when they only regard planar curves, because this yields nice geometric properties for curves
with constant affine curvature, see also section 5.1.

### 4 Curves with constant Euclidean curvatures

Now that we have presented all necessary differential geometry, we will prove two necessary and sufficient conditions for a curve to have constant Euclidean curvatures (see Theorem 4.3). One of these conditions is that the curve is equi-angular:

Definition 4.1. A regular curve γ : I → R^{m} is called equi-angular if the tangents at any two
of its points make the same angle with the line segment connecting these points.

Figure 3: Equi-angularity means that the designated angles are equal.

Assume γ is arc length parametrized. If we take two points on the curve γ(t_{1}), γ(t_{2}) an expression
of the angle between γ^{0}(t_{1}) and γ(t_{1}) − γ(t_{2}) is given by

cos θ = hγ^{0}(t_{1}), γ(t_{1}) − γ(t_{2})i

|γ(t_{1}) − γ(t_{2})| . (28)

Hence, equi-angularity of γ means that for all t1, t2 ∈ I

hγ(t_{1}) − γ(t_{2}), γ^{0}(t_{1}) − γ^{0}(t_{2})i = 0.

In the proof of the main theorem of this section (Theorem 4.3) we use an induction argument to prove that an equi-angular curve has constant curvatures. The following lemma, based on personal notes of my supervisor, G. Vegter, shows that if the first i − 1 curvatures are constant, then there is a simple expression for the derivative of the i-th curvature.

Lemma 4.2. Let γ : I → R^{m}be parametrized by arc length with constant curvatures κ_{1}, . . . , κ_{i−1}
for i < m. Then

κ^{2}_{1}κ^{2}_{2}. . . κ^{2}_{i−1}κiκ^{0}_{i} = hγ^{(i+1)}, γ^{(i+2)}i.

Proof. As shown before, we can write

γ^{(l)}=

l

X

j=1

c_{lj}F_{j}.

We claim that (under the assumptions of the lemma) γ^{(l)}, 1 ≤ l ≤ i contains only F_{j} with odd
j if l is odd, and only Fj with even j if l is even. For l = 1 this is clearly true. Suppose it holds
for l = n, where n is odd. Then we know

γ^{(n+1)}= X

1≤j≤n j odd

c_{nj}F_{j}^{0},

= c_{n1}κ_{1}F_{2}+ X

3≤j≤n j odd

c_{nj} − κ_{j−1}F_{j−1}+ κ_{j}F_{j+1},

= cn1κ1F2+ X

2≤j≤n−1 j even

cn,j+1 − κ_{j}Fj + κj+1Fj+2

Similarly, it can be shown that if we start with n even, we get that γ^{(n+1)} only contains Fj with
odd j, which proves the claim. Note that we used that the c_{nj} depend on κ_{1}, . . . , κ_{n−2}, which
are constant.

Now, if i is odd, then
γ^{(i)}= X

1≤j≤i j odd

c_{ij}F_{j},

γ^{(i+1)} = c_{i1}κ_{1}F_{2}+ X

2≤j≤i−1 j even

c_{i,j+1} − κ_{j}F_{j}+ κ_{j+1}F_{j+2},

γ^{(i+2)} = ci1κ1F_{2}^{0} + ciiκ^{0}_{i}Fi+1+ X

2≤j≤i−1 j even

ci,j+1 − κ_{j}F_{j}^{0}+ κj+1F_{j+2}^{0} ,

= ciiκ^{0}_{i}Fi+1+ X

1≤j≤i+2 j odd

˜ cijFj.

We see that γ^{(i+1)} contains only terms F_{j} for even j, while γ^{(i+2)} contains only F_{j} for odd j,
except for Fi+1. Hence,

hγ^{(i+1)}, γ^{(i+2)}i = c^{2}_{ii}κ_{i}κ^{0}_{i}= κ^{2}_{1}κ^{2}_{2}. . . κ^{2}_{i−1}κ_{i}κ^{0}_{i}.
If i is even, the reasoning is very similar.

We are now able to state the main result of this section. Part of this theorem, namely the equivalence 1 ⇔ 3, is based on the before mentioned personal notes.

Theorem 4.3. For a connected regular curve in R^{m} the following statements are equivalent:

1. The curve has constant Euclidean curvatures.

2. The distance between two points on the curve does not depend on the actual positions of these points, but only on the arc length of the curve segment between the points.

3. Every arc of the curve is equi-angular.

Figure 4: Statement 2 means that the line segments AC, BD must have equal length if the arc lengths AC, BD are equal.

Remark 4.4. It is important to remark that both conditions are invariant under rigid transfor- mations, since they are defined in terms of angles, lengths and arc length.

Proof. (1 ⇒ 2): In even dimensions, say m = 2n, a curve γ : [a, b] → R^{m} with constant
Euclidean curvatures has normal form

γ(s) = (a1cos b1s, a1sin b1s, a2cos b2s, a2sin b2s, . . . , ancos bns, ansin bns), (29) wherePn

i=1a^{2}_{i}b^{2}_{i} = 1 due to arc length parametrization. Let t_{1}, t_{2} ∈ [a, b]. Then

|γ(t_{2}) − γ(t_{1})|^{2} =

n

X

i=1

a^{2}_{i}(cos^{2}b_{i}t_{2}− 2 cos b_{i}t_{2}cos b_{i}t_{1}+ cos^{2}b_{i}t_{1}+ sin^{2}b_{i}t_{2}− 2 sin b_{i}t_{2}sin b_{i}t_{1}+ sin^{2}b_{i}t_{1}),

=

n

X

i=1

a^{2}_{i}(2 − 2 cos(bi(t2− t_{1}))),

where the last equality follows from the cosine difference formula. In odd dimensions, m = 2n+1, γ has normal form

γ(s) = (a1cos b1s, a1sin b1s, a2cos b2s, a2sin b2s, . . . , ancos bns, ansin bns, an+1s), (30)

where a^{2}_{n+1}+Pn

i=1a^{2}_{i}b^{2}_{i} = 1 due to arc length parametrization. Let t_{1}, t_{2}∈ [a, b]. Then

|γ(t_{2}) − γ(t1)|^{2} = a^{2}_{m+1}(t2− t_{1})^{2}+

n

X

i=1

a^{2}_{i}(2 − 2 cos(bi(t2− t_{1}))).

We conclude that in both cases the distance between γ(t_{2}) and γ(t_{1}) does not depend on t_{1}, t_{2}
individually, but only on t2− t_{1}, the arc length of the curve segment between γ(t2) and γ(t1).

(2 ⇒ 3): Suppose that

F (t_{1}, t_{2}) = |γ(t_{2}) − γ(t_{1})|^{2} = f (t_{2}− t_{1}),
for some function f . Since

∂F

∂t_{1} +∂F

∂t_{2} = −f^{0}(t2− t_{1}) + f^{0}(t2− t_{1}) = 0,
we know that

hγ^{0}(t_{2}), γ(t_{2}) − γ(t_{1})i + h−γ^{0}(t_{1}), γ(t_{2}) − γ(t_{1})i = 0.

This means that γ is equi-angular.

(3 ⇒ 1): Equi-angularity of γ means that

F (t_{1}, t_{2}) = hγ(t_{1}) − γ(t_{2}), γ^{0}(t_{1}) − γ^{0}(t_{2})i
is identically zero as are its partial derivatives. We also have

∂F

∂t1

= hγ^{0}(t1), γ^{0}(t1) − γ^{0}(t2)i + hγ(t1) − γ(t2), γ^{00}(t1)i,

= 1 − hγ^{0}(t1), γ^{0}(t2)i + hγ(t1) − γ(t2), γ^{00}(t1)i,

∂^{2}F

∂t1∂t2

= −hγ^{0}(t_{1}), γ^{00}(t_{2})i − hγ^{0}(t_{2}), γ^{00}(t_{1})i.

Repeated differentiation yields

∂^{2i+2}F

∂t^{i+1}_{1} ∂t^{i+1}_{2} = −hγ^{(i+1)}(t_{1}), γ^{(i+2)}(t_{2})i − hγ^{(i+1)}(t_{2}), γ^{(i+2)}(t_{1})i. (31)
Now we set t_{1}= t_{2} and use that F is identically zero to obtain

hγ^{(i+1)}, γ^{(i+2)}i = 0 (32)

By using Lemma 4.2 with i = 1 we see that κ1 is constant. Now assume that κ1, κ2, . . . , κi−1

are constant, then by the same lemma, κ_{i} is constant or one of κ_{1}, κ_{2}, . . . , κ_{i−1} is zero. The
latter case is not possible due to Lemma 2.3. Hence, by induction, all Euclidean curvatures are
constant.

### 5 Characterizations of curves with constant affine curvatures

In this section we will discuss curves with constant affine curvatures. We will first look at
R^{2}, because there the expressions are simple enough that we can find all curves with constant
affine curvature analytically. Furthermore, in R^{2} we can find an analogue of equi-angularity. In
Subsection 5.2 and 5.3 we will look at R^{m}.

5.1 Characterization in R^{2}

The fact that all plane curves with constant affine curvature are given by conics was already described by Blaschke [2, 18]. We will first state a proof of this and then look at equi-areality, the affine analogue for equi-angularity.

Theorem 5.1. A planar curve with constant affine curvature is a conic arc.

Proof. Let γ be parametrized by affine arc length. To find curves with constant affine curvature
in R^{2} we have to solve the following differential equation for constant k:

γ^{000}(r) + kγ^{0}(r) = 0.

Define

Mγ(r) = [γ^{0}(r), γ^{00}(r)],

where [v, w] denotes the matrix with the vectors v, w as columns. Then the differential equation can be rewritten as

M_{γ}^{0}(r) = Mγ(r)0 −k

1 0

, so the solution is given by

M_{γ}(r) = M_{γ}(0) exp 0 −k

1 0

r

.

With a suitable equi-affine transformation we can choose Mγ(0) = I. Let K :=0 −k

1 0

.
First assume k = 0. Since in this case K^{2}= 0, we get

M_{γ}(r) = exp 0 −k

1 0

r

= I + Kr =1 0 r 1

. (33)

Hence, γ(r) = (r,^{1}_{2}r^{2}). Secondly, assume that k > 0. Then the eigenvalues of K are given by
λ = ±i√

k, with corresponding eigenvectors (±i√

k, 1). Using the diagonalization of K we get the following solution:

Mγ(r) = exp 0 −k

1 0

r

=

"

cos(√

kr) −√

k sin(√ kr)

√1

ksin(√

kr) cos(√ kr)

# .

Hence,

γ(r) = (^{√}^{1}

ksin(

√

kr), −_{k}^{1}cos(

√

kr)). (34)

Lastly, assume k < 0. Now the eigenvalues are λ = ±p|k|, with corresponding eigenvectors (±p|k|, 1). The solution is given by

Mγ(r) = exp 0 −k

1 0

r

=

"

cosh(p|k|r) p|k| sinh(p|k|r)

√1

|k|sinh(p|k|r) cosh(p|k|r)

# .

Hence,

γ(r) = (√^{1}

|k|sinh(p|k|r),_{|k|}^{1} cosh(p|k|r)). (35)
We can conclude that curves with constant negative, vanishing or positive affine curvature are
hyperbolic, parabolic or elliptic arcs, respectively.

Equi-angularity is defined in terms of angles, which are invariant in Euclidean geometry. Since areas are variant in affine geometry, it makes sense to use this for an analogue condition, called equi-areality:

Definition 5.2. A planar curve γ is called equi-areal if for any two points γ(s), γ(t) the area
of the parallellogram formed by γ^{0}(s) and γ(t) − γ(s) is equal to the area of the parallellogram
formed by γ^{0}(t) and γ(s) − γ(t).

Figure 5: Equi-areality

Denote the areas by A1, A2, respectively. If we regard v = (v1, v2) ∈ R^{2}as vector in R^{3}by setting
v = (v1, v2, 0), we know that the area of the parallellogram formed by γ^{0}(s) and γ(t) − γ(s) is
given by the norm of the cross product of these two vectors. Hence,

A_{1}= |γ^{0}(s) × (γ(t) − γ(s))| = ||γ^{0}(s), γ(t) − γ(s)||. (36)
Then A1 = A2 is equivalent to

D(s, t) := [γ^{0}(s) + γ^{0}(t), γ(s) − γ(t)] = 0. (37)

Theorem 5.3. A connected planar curve has constant affine curvature if and only if every arc of the curve is equi-areal.

Proof. Suppose γ is parametrized by affine arc length and has constant affine curvature. As shown above, γ has one of the following three forms (up to equi-affine transformations):

1. γ(r) = (^{√}^{1}

ksin(√

kr), −^{1}_{k}cos(√

kr)) if k > 0,
2. γ(r) = (r,^{1}_{2}r^{2}) if k = 0,

3. γ(r) = (√^{1}

|k|sinh(p|k|r),_{|k|}^{1} cosh(p|k|r)) if k < 0.

In case 1:

γ^{0}(r) = (cos(√
kr),^{√}^{1}

ksin(√ kr)), D(s, t) =

cos(√

ks) + cos(√

kt) ^{√}^{1}

ksin(√

ks) − ^{√}^{1}

ksin(√ kt)

√1

ksin(√

ks) + ^{√}^{1}

ksin(√

kt) −^{1}_{k}cos(√

ks) + ^{1}_{k}cos(√
kt)

,

= −^{1}_{k}cos^{2}(

√

ks) + _{k}^{1}cos^{2}(

√

kt) −^{1}_{k}sin^{2}(

√

ks) + _{k}^{1}sin^{2}(

√ kt),

= −^{1}_{k}+_{k}^{1} = 0.

In case 2:

γ^{0}(r) = (1, r),
D(s, t) =

2 s − t

s + t ^{1}_{2}s^{2}−^{1}_{2}t^{2}
,

= s^{2}− t^{2}− (s + t)(s − t) = 0.

In case 3:

γ^{0}(r) = (cosh(p|k|r),√^{1}

|k|sinh(

√ kr)),

D(s, t) =

cosh(p|k|s) + cosh(p|k|t) √^{1}

|k|sinh(p|k|s) −√^{1}

|k|sinh(p|k|t)

√1

|k|sinh(√

ks) + √^{1}

|k|sinh(√

kt) _{|k|}^{1} cosh(p|k|s) −_{|k|}^{1} cosh(p|k|t)
,

= _{|k|}^{1} cosh^{2}(

√

ks) −_{|k|}^{1} cosh(p|k|t) − _{|k|}^{1} sinh^{2}(

√

ks) + _{|k|}^{1} sinh^{2}(

√ kt),

= _{|k|}^{1} − _{|k|}^{1} = 0.

In all three cases γ is equi-areal.

Now suppose that γ is equi-areal. We will use that D(s, t) is identically zero, as are its partial derivatives. First we calculate

∂D

∂s(s, t) = ||γ^{00}(s), γ(s) − γ(t)|| + ||γ^{0}(s) + γ^{0}(t), γ^{0}(s)||,

= ||γ^{00}(s), γ(s) − γ(t)|| + ||γ^{0}(t), γ^{0}(s)||,

∂^{2}D

∂s∂t(s, t) = −||γ^{00}(s), γ^{0}(t)|| + ||γ^{00}(t), γ^{0}(s)||,

∂^{5}D

∂s∂t^{4}(s, t) = −||γ^{00}(s), γ^{(4)}(t)|| + ||γ^{(5)}(t), γ^{0}(s)||,
Then, taking s = t and using (18) we get

0 = −||γ^{00}(s), γ^{(4)}(s)] + [γ^{(5)}(s), γ^{0}(s)|| = k^{0}(s).

Since this holds for any point s ∈ I, γ has constant affine curvature.

5.2 Characterization in R^{m} in terms of volume

A natural way to generalize Theorem 5.3 to R^{m} is to look at the volume of the simplex formed
by vectors γ^{0}(t_{1}), γ(t_{2}) − γ(t_{1}), . . . , γ(t_{m}) − γ(t_{1}) for points γ(t_{1}), . . . , γ(t_{m}). However, already
in R^{3} it can be seen that this does not hold. Instead we can look at the analogue of Theorem
4.3 part 2:

Theorem 5.4. A connected regular curve in R^{m} with nonzero Euclidean curvatures has constant
affine curvatures if and only if the volume of the simplex formed by m + 1 points on the curve
does not depend on the actual positions of these points, but only on the affine arc length of the
curve segments between the points.

Figure 6: Above condition means that the areas are equal if the affine arc lengths AB and BC on the one hand, and CE and BD on the other hand are equal.

Before we will prove this theorem we will first look at the structure of curves with constant affine curvature.

Curves with constant affine curvature are described by the following differential equation with constant ki:

γ^{(m+1)}(r) − km−1γ^{(m−1)}(r) − km−2γ^{(m−2)}(r) − . . . − k1γ^{0}(r) = 0. (38)
The solution of this differential equation depends on its characteristic equation:

λ^{m}− k_{m−1}λ^{m−2}− k_{m−2}λ^{m−3}− . . . − k_{2}λ − k_{1}= 0 (39)
We will only look at particular solutions, since the affine curvatures of a curve determine the
curve completely, up to an equi-affine transformation. First note that if λ_{1}, . . . , λ_{m} are the zeros
of (39), then we can write

λ^{m}− k_{m−1}λ^{m−2}− k_{m−2}λ^{m−3}− . . . − k_{2}λ − k1= (λ − λ1) . . . (λ − λm). (40)

By comparing the coefficients of λ^{m−1} we see that

λ1+ . . . + λm= 0. (41)

If (39) has m distinct, nonzero, real zeros λ1, . . . , λm, then γ is given by

γ(r) = (a_{1}e^{λ}^{1}^{r}, a_{2}e^{λ}^{2}^{r}, . . . , a_{m}e^{λ}^{m}^{r}), (42)
where a_{1}, . . . , a_{m} are constants such that γ is parametrized by affine arc length.

If (39) has d distinct pairs of conjugate complex zeros µ_{1} ± iν_{1}, . . . , µ_{d}± iν_{d} and n distinct,
nonzero, real zeros λ1, . . . , λn, then γ is given by

γ(r) = (a1e^{µ}^{1}^{r}cos ν1r, a2e^{µ}^{1}^{r}sin ν1r, . . . , a2d−1e^{µ}^{d}^{r}cos νdr, a2de^{µ}^{d}^{r}sin νdr, a2d+1e^{λ}^{1}^{r}, . . . , ame^{λ}^{n}^{r}),
(43)
where a_{1}, . . . , a_{m} are constants such that γ is parametrized by affine arc length.

If (39) has a double, nonzero zero, we would get that in ||γ^{0}, . . . , γ^{(m)}|| two rows are identical
up to a constant, which means that γ cannot be parametrized by affine arc length.

If d of the roots of (39) are zero, and the remaining n zeros λ_{1}, . . . , λ_{n} distinct, nonzero and
real, then γ is given by

γ(r) = (a_{1}r, a_{2}r^{2}, . . . , a_{d}r^{d}, a_{d+1}e^{λ}^{1}^{r}, . . . , a_{m}e^{λ}^{n}^{r}). (44)
If there is a combination of zero zeros, nonzero real zeros and complex zero pairs, the different
elements of each type of solution are combined.

Theorem 5.4 mentions being dependent only on differences instead of the variables itself. The following lemma will state an equivalent statement which will be useful when proving the the- orem.

Lemma 5.5. A C^{1} function g = g(t_{1}, . . . , t_{m+1}) with domain D_{1} ⊂ R^{m+1} satisfies the partial
differential equation

m+1

X

i=1

∂g

∂t_{i} = 0 (45)

if and only if there exists a C^{1} function f : D_{2} ⊂ R^{m}→ R for

D2 = {(t1− t_{m+1}, . . . , tm− t_{m+1}) ∈ R^{m}| (t_{1}, . . . , tm+1) ∈ D1}
such that

g(t1, . . . , tm+1) = f (t1− t_{m+1}, . . . , tm− t_{m+1}). (46)
Proof. Suppose g satisfies (45). We will use the following change of coordinates:

τ_{i} = t_{i}− t_{m+1}, 1 ≤ i ≤ m,
τ_{m+1} =

m+1

X

i=1

t_{i}.

Since for 1 ≤ i ≤ m

∂g

∂ti

=

m+1

X

j=1

∂g

∂τj

∂τj

∂ti

= ∂g

∂τi

+ ∂g

∂τm+1

,

∂g

∂t_{m+1} =

m+1

X

j=1

∂g

∂τ_{j}

∂τj

∂t_{m+1},

= ∂g

∂τ_{m+1} −

m

X

j=1

∂g

∂τ_{j},
we have

0 =

m+1

X

i=1

∂g

∂ti

=

m

X

i=1

∂g

∂τi

+ ∂g

∂τm+1

+ ∂g

∂τm+1

−

m

X

j=1

∂g

∂τj

,

= (m + 1) ∂g

∂τm+1

. Hence,

g(t1, . . . , tm+1) = f (τ1, . . . , τm) = f (t1− t_{m+1}, . . . , tm− t_{m+1}),
for a function f .

Reversely, suppose (46) holds. If we let

τi = ti− t_{m+1}, 1 ≤ i ≤ m,
we have that

g(t_{1}, . . . , t_{m+1}) = f (τ_{1}, . . . , τ_{m}).

Then for 1 ≤ i ≤ m

∂g

∂t_{i} = ∂f

∂τ_{i},

∂g

∂t_{m+1} = −

m

X

j=1

∂f

∂τ_{j}.
Hence,

m+1

X

i=1

∂g

∂ti

=

m

X

i=1

∂f

∂τi

−

m

X

j=1

∂f

∂τj

= 0.

The structure of curves with constant affine curvatures together with the previous lemma can be used to determine the structure of the volume of the before mentioned simplex. The following lemma will describe this structure.

Lemma 5.6. If γ : [a, b] → R^{m} is an affine arc length parametrized curve with constant affine
curvatures, where d of the zeros of the characteristic equation are zero, and the remaining n are
given by λ1, . . . , λn∈ C, then

||γ(t_{1}) − γ(t_{m+1}), . . . , γ(t_{m}) − γ(t_{m+1})||

has the following structure:

1. it is a sum of functions of the form cf e^{g} for a constant c ∈ R and polynomials f, g,
2. (a) each g is the sum of terms of the form λitj,

(b) each g contains each λ_{i} exactly once, and each t_{j} at most once,

3. (a) if d 6= 0, each f can be written containing only terms of the form ti− t_{j},
(b) if d = 0, then f = ±1,

Proof. 1. From (42),(43) and (44) we immediately see that c = Q_{m}

i=1a_{i}, so henceforth we
will keep this constant out of the determinant. If all zeros are real, there appear no
other functions than polynomials and exponential functions in the determinant. Hence,
by cofactor expansion we can only get products of polynomials and exponential functions.

Since each exponent is linear in some t_{i}, the exponents in such a product add up to a
polynomial. If there is a pair of complex eigenvalues µ ± iν too, so that rows contain a sine
and cosine, as in (43), then we first expand along rows corresponding to the zero zeros,
until the minors do not have such rows anymore. In this minors we use row operations
to get e^{(µ+iν)r}, e^{(µ−iν)r} instead of e^{µr}cos νr, e^{µr}sin νr. Since the determinant does not
change by this row operations, we will henceforth only look at the case with all real zeros.

2. (a) This follows from the reasoning in 1.

(b) Every λ_{i} is in a distinct row, so cofactor expansion can never put one λ_{i} twice in
one g. After expanding along the rows corresponding to the zero zeros, the minors
contain n rows of exponentials, hence each g contains n λi’s. Every tj, j 6= m + 1 is
in a distinct column, so by the same argument one tj can never appear twice in one
g. Regarding j = m + 1, by using column operations we know

||γ(t_{1})−γ(t_{m+1}), . . . , γ(t_{m})−γ(t_{m+1})|| = ||γ(t_{1})−γ(t_{m+1}), γ(t_{2})−γ(t_{1}), . . . , γ(t_{m})−γ(t_{1})||.

Since in the right-hand side tm+1 cannot appear more than once in each g, it neither appears in the left-hand side more than once in each g.

3. (a) First we expand along rows corresponding to nonzero zeros. Then the minors have the following form:

M =

t_{1}− t_{m+1} . . . t_{d}− t_{m+1}
... . .. ...
t^{d}_{1}− t^{d}_{m+1} . . . t^{d}_{d}− t^{d}_{m+1}

.

Note that the t1, . . . , t_{d}can be any collection of d ti’s from t1, . . . , tm, depending on
which minor we are looking at. However, symbolically all these cases have the form

M_{k}=

t1− t_{k+1} . . . tk− t_{k+1}
... . .. ...
t^{k}_{1}− t^{k}_{k+1} . . . t^{k}_{k}− t^{k}_{k+1}

.

We will prove by induction that M_{k} depends only on expressions of the form t_{i}− t_{j}
for 1 ≤ i, j ≤ k + 1. If k = 1, this is trivial. Suppose that it holds for all k × k
determinants with the structure of M_{k}. Then by expanding along the last row we