On curves with constant curvatures

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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

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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 curvature, constant curvature, curve

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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 geometric 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 R³. 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 curvature 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 differential 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

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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 R² 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.

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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

|γ⁰(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 transformations. 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 R³

If γ : I → R³ is parametrized by arc length we can differentiate the relation

|γ⁰(s)|² = 1,

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to get

hγ⁰(s), γ⁰⁰(s)i = 0,

where h · , · i denotes the standard inner product. Thus, γ⁰⁰is perpendicular to the tangent t(s).

The curvature of γ is defined as

κ(s) = |γ⁰⁰(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) = γ⁰⁰(s)

|γ⁰⁰(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 R³, is normal to this osculating plane and is called the binormal vector. We can use the same method as before to show that b⁰(s) is perpendicular to b(s) and t(s). Hence,

b⁰(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 n⁰(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 {t⁰, n⁰, b⁰}. Furthermore, torsion could only be defined if γ⁰ and γ⁰⁰ were linearly independent.

Now, let γ : I → R^m be parametrized by arc length and suppose that {γ⁰(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 {F₁(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 {F₁(s), . . . , F_m(s)}. Then there exist functions κ₁, . . . , κ_m−1, called the Euclidean curvatures of γ, such that





 F₁ F₂ F3

... Fm−2

F_m−1 Fm







0

=







0 κ1 0 · · · 0 0 0

−κ₁ 0 κ₂ · · · 0 0 0

0 −κ₂ 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











 F₁ F₂ F3

... Fm−2

F_m−1 Fm





 .

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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_i⁰(s), Fj(s)i + hFi(s), F_j⁰(s)i = 0. (7) Note that by construction we have for 1 ≤ i ≤ m

Span(γ⁰(s), . . . , γ⁽ⁱ⁾(s)) = Span(F1(s), . . . , Fi(s)), (8) so F_i(s) is a linear combination of γ⁰(s), . . . , γ⁽ⁱ⁾(s). It follows that F_i⁰(s) is a linear combination of γ⁰(s), . . . , γ⁽ⁱ⁺¹⁾(s), hence also of the orthonormalized basis F1(s), . . . , Fi+1(s). Hence, (7) reduces to

hF_i⁰(s), F_j(s)i = 0,

except possibly for j = i − 1 or j = i + 1. Thus, there exist functions κ₁, . . . , κ_m−1 such that for 2 ≤ i ≤ m − 1

F₁⁰(s) = κ1(s)F2(s), (9)

F_i⁰(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 R³. 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 γ⁰(s), . . . , γ⁽ⁿ⁾(s) are linearly independent for all s ∈ I, but γ⁰(s), . . . , γ⁽ⁿ⁺¹⁾(s) are not, then we can only define κ₁, . . . , 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 Rⁿcorresponding 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 κ₁, . . . , 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_i⁰(s₀) = −κ_i−1(s₀)F_i−1(s₀).

It follows that γ⁽ⁱ⁺¹⁾(s₀) can be written as a linear combination of F₁(s₀), . . . , F_i−1(s₀), which contradicts the linear independency of γ⁰, . . . , γ^(m). For i = m we get

F_m⁰ (s₀) = −κ_m−1(s₀)F_m−1(s₀),

which is no problem, since there is nothing preventing γ^(m+1) from being linearly dependent on γ⁰, . . . , γ^(m).

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

γ⁰ = F1, γ⁰⁰ = κ1F2,

γ⁰⁰⁰ = κ₁κ₂F₃+ κ⁰₁F₂− κ²₁F₁.

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_ijF_j, (11)

where c_ii= κ₁κ₂. . . κ_i−1 and the other c_ij depend on κ₁, . . . , 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_njF_j⁰+ c⁰_njF_j,

= c_n1κ₁F₂+

n

X

j=2

− c_njκ_j−1F_j−1+ c⁰_njF_j+ c_njκ_jF_j+1,

=

n+1

X

j=1

˜

c_n+1,jF_j,

where ˜c_n+1,n+1 = c_nnκ_n = κ₁κ₂. . . κ_n and the other ˜c_ij depend on κ₁, . . . , κ_n−1 and their derivatives.

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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 R² and then generalize this to R^m.

3.1 Affine arc length and curvature in R²

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

||γ⁰(t), γ⁰⁰(t)||^1/3dt, (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

||γ⁰(t), γ⁰⁰(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 ||γ⁰(t), γ⁰⁰(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

||γ⁰(r), γ⁰⁰⁰(r)|| = 0. (14)

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This means that there is a differentiable function k such that

γ⁰⁰⁰(r) + k(r)γ⁰(r) = 0. (15)

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

k(r) = ||γ⁰⁰(r), γ⁰⁰⁰(r)||. (16)

By differentiating (14) we see that

||γ⁰⁰(r), γ⁰⁰⁰(r)|| + ||γ⁰(r), γ⁽⁴⁾(r)|| = 0, so

k(r) = −||γ⁰(r), γ⁽⁴⁾(r)||. (17)

Differentiating (16) and (17) we get

k⁰(r) = ||γ⁰⁰(r), γ⁽⁴⁾(r)|| = −||γ⁰⁰(r), γ⁽⁴⁾(r)|| − ||γ⁰(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 → R² be a regular curve with nonzero Euclidean curvature. Then defining affine arc length as

r(t) = Z t

t0

||γ⁰(u), γ⁰⁰(u)||^1/3du (19)

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

dγ dr,d²γ

dr²

= 1. (20)

Proof. Since

1 =

dγ dr,d²γ

dr²

=

dγ dt,d²γ

dt²

dt dr

3

, we see that

r(t) = Z t

t0

||γ⁰(u), γ⁰⁰(u)||^1/3du,

where γ⁰(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 R² 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,d²γ

dt², . . . ,d^mγ dt^m

6= 0,

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

dγ dr,d²γ

dr², . . . ,d^mγ dr^m

= ±1.

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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 R² we calculate

dγ dr,d²γ

dr², . . . ,d^mγ dr^m

=

dγ dt,d²γ

dt², . . . ,d^mγ dt^m

dt dr

p

, where the power p is given by

p =

m

X

i=1

i = ¹₂m(m + 1).

Thus,

r(t) = Z t

t0

||γ⁰(u), γ⁰⁰(u), . . . , γ^(m)(u)||

2

m(m+1)du. (21)

Similarly to in R², we say that γ is parametrized by affine arc length if

||γ⁰(t), γ⁰⁰(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





 γ⁰ γ⁰⁰ ... γ^(m−1)

γ^(m)







0

=







0 1 0 · · · 0 0

0 0 1 · · · 0 0

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

0 0 0 · · · 0 1

k₁ k₂ k₃ · · · k_m−1 0











 γ⁰ γ⁰⁰ ... 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

||γ⁰, . . . , 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

||γ⁰, . . . , γ^(m−1), γ^(m+1)|| = 0, (23) so there exist differentiable functions k1, . . . , km−1 such that

γ^(m+1)= k1γ⁰+ . . . + km−1γ^(m−1). (24) The ki are called the affine curvatures of γ. Note that γ⁰, . . . , γ^(m−1) are linearly independent, so the affine curvatures are uniquely defined. Explicit formulas for the affine curvatures follow from

||γ⁰, . . . , γ⁽ⁱ⁻¹⁾, γ^(m+1), γ⁽ⁱ⁺¹⁾, . . . , γ^(m)|| = ||γ⁰, . . . , γ⁽ⁱ⁻¹⁾,

m−1

X

j=1

k_jγ^(j), γ⁽ⁱ⁺¹⁾, . . . , γ^(m)|| = k_i. (25)

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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:

k₁⁰ = ||γ^(m+2), γ⁰⁰, . . . , γ^(m)||, (26)

k_i⁰ = ||γ⁰, . . . , γ⁽ⁱ⁻²⁾, γ⁽ⁱ⁾, γ^(m+1), γ⁽ⁱ⁺¹⁾, . . . , γ^(m)|| + ||γ⁰, . . . , 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 γ⁰ nor γ in the first term, because γ⁰ 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 R² we saw that the affine curvature k of a curve γ is given by k(r) = ||γ⁰⁰(r), γ⁰⁰⁰(r)||,

while the generalization to R^m gives us

k₁(r) = ||γ⁰⁰⁰(r), γ⁰⁰(r)|| = −k(r).

This can also directly be seen by comparing (15) and (24). Authors typically use k instead of k₁when they only regard planar curves, because this yields nice geometric properties for curves with constant affine curvature, see also section 5.1.

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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(t₁), γ(t₂) an expression of the angle between γ⁰(t₁) and γ(t₁) − γ(t₂) is given by

cos θ = hγ⁰(t₁), γ(t₁) − γ(t₂)i

|γ(t₁) − γ(t₂)| . (28)

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

hγ(t₁) − γ(t₂), γ⁰(t₁) − γ⁰(t₂)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.

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Lemma 4.2. Let γ : I → R^mbe parametrized by arc length with constant curvatures κ₁, . . . , κ_i−1 for i < m. Then

κ²₁κ²₂. . . κ²_i−1κiκ⁰_i = hγ⁽ⁱ⁺¹⁾, γ⁽ⁱ⁺²⁾i.

Proof. As shown before, we can write

γ^(l)=

l

X

j=1

c_ljF_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_njF_j⁰,

= c_n1κ₁F₂+ X

3≤j≤n j odd

c_nj − κ_j−1F_j−1+ κ_jF_j+1,

= cn1κ1F2+ X

2≤j≤n−1 j even

cn,j+1 − κ_jFj + 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 κ₁, . . . , κ_n−2, which are constant.

Now, if i is odd, then γ⁽ⁱ⁾= X

1≤j≤i j odd

c_ijF_j,

γ⁽ⁱ⁺¹⁾ = c_i1κ₁F₂+ X

2≤j≤i−1 j even

c_i,j+1 − κ_jF_j+ κ_j+1F_j+2,

γ⁽ⁱ⁺²⁾ = ci1κ1F₂⁰ + ciiκ⁰_iFi+1+ X

2≤j≤i−1 j even

ci,j+1 − κ_jF_j⁰+ κj+1F_j+2⁰ ,

= ciiκ⁰_iFi+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 = c²_iiκ_iκ⁰_i= κ²₁κ²₂. . . κ²_i−1κ_iκ⁰_i. If i is even, the reasoning is very similar.

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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 transformations, 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=1a²_ib²_i = 1 due to arc length parametrization. Let t₁, t₂ ∈ [a, b]. Then

|γ(t₂) − γ(t₁)|² =

n

X

i=1

a²_i(cos²b_it₂− 2 cos b_it₂cos b_it₁+ cos²b_it₁+ sin²b_it₂− 2 sin b_it₂sin b_it₁+ sin²b_it₁),

=

n

X

i=1

a²_i(2 − 2 cos(bi(t2− t₁))),

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)

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where a²_n+1+Pn

i=1a²_ib²_i = 1 due to arc length parametrization. Let t₁, t₂∈ [a, b]. Then

|γ(t₂) − γ(t1)|² = a²_m+1(t2− t₁)²+

n

X

i=1

a²_i(2 − 2 cos(bi(t2− t₁))).

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

(2 ⇒ 3): Suppose that

F (t₁, t₂) = |γ(t₂) − γ(t₁)|² = f (t₂− t₁), for some function f . Since

∂F

∂t₁ +∂F

∂t₂ = −f⁰(t2− t₁) + f⁰(t2− t₁) = 0, we know that

hγ⁰(t₂), γ(t₂) − γ(t₁)i + h−γ⁰(t₁), γ(t₂) − γ(t₁)i = 0.

This means that γ is equi-angular.

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

F (t₁, t₂) = hγ(t₁) − γ(t₂), γ⁰(t₁) − γ⁰(t₂)i is identically zero as are its partial derivatives. We also have

∂F

∂t1

= hγ⁰(t1), γ⁰(t1) − γ⁰(t2)i + hγ(t1) − γ(t2), γ⁰⁰(t1)i,

= 1 − hγ⁰(t1), γ⁰(t2)i + hγ(t1) − γ(t2), γ⁰⁰(t1)i,

∂²F

∂t1∂t2

= −hγ⁰(t₁), γ⁰⁰(t₂)i − hγ⁰(t₂), γ⁰⁰(t₁)i.

Repeated differentiation yields

∂²ⁱ⁺²F

∂tⁱ⁺¹₁ ∂tⁱ⁺¹₂ = −hγ⁽ⁱ⁺¹⁾(t₁), γ⁽ⁱ⁺²⁾(t₂)i − hγ⁽ⁱ⁺¹⁾(t₂), γ⁽ⁱ⁺²⁾(t₁)i. (31) Now we set t₁= t₂ 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 κ₁, κ₂, . . . , κ_i−1 is zero. The latter case is not possible due to Lemma 2.3. Hence, by induction, all Euclidean curvatures are constant.

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5 Characterizations of curves with constant affine curvatures

In this section we will discuss curves with constant affine curvatures. We will first look at R², because there the expressions are simple enough that we can find all curves with constant affine curvature analytically. Furthermore, in R² 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 R²

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 R² we have to solve the following differential equation for constant k:

γ⁰⁰⁰(r) + kγ⁰(r) = 0.

Define

Mγ(r) = [γ⁰(r), γ⁰⁰(r)],

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

M_γ⁰(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 K²= 0, we get

M_γ(r) = exp 0 −k

1 0

r

= I + Kr =1 0 r 1

. (33)

Hence, γ(r) = (r,¹₂r²). 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), −_k¹cos(

√

kr)). (34)

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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 γ⁰(s) and γ(t) − γ(s) is equal to the area of the parallellogram formed by γ⁰(t) and γ(s) − γ(t).

Figure 5: Equi-areality

Denote the areas by A1, A2, respectively. If we regard v = (v1, v2) ∈ R²as vector in R³by setting v = (v1, v2, 0), we know that the area of the parallellogram formed by γ⁰(s) and γ(t) − γ(s) is given by the norm of the cross product of these two vectors. Hence,

A₁= |γ⁰(s) × (γ(t) − γ(s))| = ||γ⁰(s), γ(t) − γ(s)||. (36) Then A1 = A2 is equivalent to

D(s, t) := [γ⁰(s) + γ⁰(t), γ(s) − γ(t)] = 0. (37)

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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_kcos(√

kr)) if k > 0, 2. γ(r) = (r,¹₂r²) if k = 0,

3. γ(r) = (√¹

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

In case 1:

γ⁰(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_kcos(√

ks) + ¹_kcos(√ kt)

,

= −¹_kcos²(

√

ks) + _k¹cos²(

√

kt) −¹_ksin²(

√

ks) + _k¹sin²(

√ kt),

= −¹_k+_k¹ = 0.

In case 2:

γ⁰(r) = (1, r), D(s, t) =

2 s − t

s + t ¹₂s²−¹₂t² ,

= s²− t²− (s + t)(s − t) = 0.

In case 3:

γ⁰(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 cosh²(

√

ks) −_|k|¹ cosh(p|k|t) − _|k|¹ sinh²(

√

ks) + _|k|¹ sinh²(

√ 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) = ||γ⁰⁰(s), γ(s) − γ(t)|| + ||γ⁰(s) + γ⁰(t), γ⁰(s)||,

= ||γ⁰⁰(s), γ(s) − γ(t)|| + ||γ⁰(t), γ⁰(s)||,

∂²D

∂s∂t(s, t) = −||γ⁰⁰(s), γ⁰(t)|| + ||γ⁰⁰(t), γ⁰(s)||,

∂⁵D

∂s∂t⁴(s, t) = −||γ⁰⁰(s), γ⁽⁴⁾(t)|| + ||γ⁽⁵⁾(t), γ⁰(s)||, Then, taking s = t and using (18) we get

0 = −||γ⁰⁰(s), γ⁽⁴⁾(s)] + [γ⁽⁵⁾(s), γ⁰(s)|| = k⁰(s).

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

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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 γ⁰(t₁), γ(t₂) − γ(t₁), . . . , γ(t_m) − γ(t₁) for points γ(t₁), . . . , γ(t_m). However, already in R³ 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) − . . . − k1γ⁰(r) = 0. (38) The solution of this differential equation depends on its characteristic equation:

λ^m− k_m−1λ^m−2− k_m−2λ^m−3− . . . − k₂λ − k₁= 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 λ₁, . . . , λ_m are the zeros of (39), then we can write

λ^m− k_m−1λ^m−2− k_m−2λ^m−3− . . . − k₂λ − k1= (λ − λ1) . . . (λ − λm). (40)

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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) = (a₁e^λ¹^r, a₂e^λ²^r, . . . , a_me^λ^m^r), (42) where a₁, . . . , a_m are constants such that γ is parametrized by affine arc length.

If (39) has d distinct pairs of conjugate complex zeros µ₁ ± iν₁, . . . , µ_d± iν_d and n distinct, nonzero, real zeros λ1, . . . , λn, then γ is given by

γ(r) = (a1e^µ¹^rcos ν1r, a2e^µ¹^rsin ν1r, . . . , a2d−1e^µ^d^rcos νdr, a2de^µ^d^rsin νdr, a2d+1e^λ¹^r, . . . , ame^λⁿ^r), (43) where a₁, . . . , 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 ||γ⁰, . . . , 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 λ₁, . . . , λ_n distinct, nonzero and real, then γ is given by

γ(r) = (a₁r, a₂r², . . . , a_dr^d, a_d+1e^λ¹^r, . . . , a_me^λⁿ^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 theorem.

Lemma 5.5. A C¹ function g = g(t₁, . . . , t_m+1) with domain D₁ ⊂ 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 C¹ function f : D₂ ⊂ R^m→ R for

D2 = {(t1− t_m+1, . . . , tm− t_m+1) ∈ R^m| (t₁, . . . , 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.

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Since for 1 ≤ i ≤ m

∂g

∂ti

=

m+1

X

j=1

∂g

∂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(t₁, . . . , t_m+1) = f (τ₁, . . . , 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(t₁) − γ(t_m+1), . . . , γ(t_m) − γ(t_m+1)||

has the following structure:

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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^µrcos νr, e^µrsin 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(t₁)−γ(t_m+1), . . . , γ(t_m)−γ(t_m+1)|| = ||γ(t₁)−γ(t_m+1), γ(t₂)−γ(t₁), . . . , γ(t_m)−γ(t₁)||.

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 =

t₁− t_m+1 . . . t_d− t_m+1 ... . .. ... t^d₁− t^d_m+1 . . . t^d_d− t^d_m+1

.

Note that the t1, . . . , t_dcan 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^k₁− 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

On curves with constant curvatures