Rotation minimizing frames and spherical curves in simply isotropic and semi-isotropic 3-spaces

Rotation minimizing frames and spherical curves in simply isotropic and semi-isotropic 3-spaces


In this work we are interested in the differential geometry of curves in simply isotropic and semi-isotropic 3-spaces. These are examples of Cayley-Klein geometries whose absolute figure is given by a plane at infinity and a degenerate quadric. Motivated by the success of rotation minimizing (RM) frames in Euclidean and Lorentz-Minkowski geometries, here we show how to build RM frames in isotropic geometries and apply them in the study of spherical curves. Indeed, through a convenient manipulation of osculating spheres described in terms of RM frames, we show that it is possible to characterize spherical curves via a linear equation involving the curvatures that dictate the RM frame motion. For the case of semi-isotropic space, we also discuss on the distinct approaches for the absolute figure in the framework of a Cayley-Klein geometry and prove that they are all equivalent approaches through the use of hyperbolic (or double) numbers, a complex-like system where the square of the imaginary unit is . Finally, we also show how to relate isotropic RM and Frenet frames through the use of Galilean trigonometric functions and dual numbers, a complex-like system where the square of the imaginary unit is zero.

Non-Euclidean geometry Cayley-Klein geometry isotropic space semi-isotropic space spherical curve plane curve
51N25 53A20 53A35 53A55 53B30

1 Introduction

The three dimensional () simply isotropic and semi-isotropic 3-spaces are examples of Cayley-Klein (CK) geometries (12); (19); (23); (30). The basic idea behind a CK geometry is the study of those properties in projective space that preserves a certain configuration, the so called absolute figure. In the spirit of Klein “Erlanger Program” (4); (14), a CK geometry is the study of those properties invariant by the action of the subgroup of projectivities that fix the absolute figure. For example, the Euclidean (Minkowski) space () is modeled through an absolute figure given in homogeneous coordinates by a plane at infinity identified with and a non-degenerate quadric of index zero (index one) identified with (, respectively) (12). In our case of interest, i.e., isotropic space geometries, the absolute figure is given by a plane at infinity, identified with , and a degenerate quadric of index zero or one, identified with : for the simply isotropic figure and for the semi-isotropic one.

Besides its mathematical interest (1); (2); (13); (26); (33), see also (24) and references therein, isotropic geometry also finds applications in economics (3); (8), elasticity (21), and in image processing and shape interrogation (15); (22), just to name a few. Another stimulus to study isotropic geometries comes from the problem of characterizing curves on level set surfaces . Indeed, an idea to approach such a problem is the introduction of a metric induced by (10) and, since it may fail to be non-degenerate, one may be led to the study of an isotropic geometry: e.g., for cylindrical quadrics, i.e., a translation surface with a conic as generating curve, the natural geometric framework is that of an isotropic geometry. In fact, here one basically has , which leads to the geometry of isotropic spaces (12); (27); (31) because such a Hessian may induce in a degenerate metric: simply isotropic space if (24); (29); semi-isotropic space if (2); and doubly isotropic space if (7).

Motived by the success of Rotation Minimizing (RM) frames in the study of spherical curves in both and (6); (10); (11); (20), in this work we develop the fundamentals of RM frames in simply isotropic and semi-isotropic spaces, which in combination with an adequate manipulation of osculating spheres allow us to prove that spherical curves can be characterized through a linear equation involving the coefficients that dictate the frame motion2 (in analogy with what happens in both and (6); (10)). In addition, for the case of semi-isotropic space we discuss in detail the construction of spheres, moving frames along curves, and the distinct approaches to the study of semi-isotropic space as a CK geometry (with the help of hyperbolic numbers (5); (32), we are able to prove that the available approaches are all equivalent). Finally, we also show how to relate RM and Frenet frames, in both and , by using the Galilean trigonometric functions (32) and dual numbers (23); (32).

The remaining of this work is divided as follows. In section 2 we review the concept of RM frames in Euclidean space and its spherical curves. In section 3 we introduce some terminology related to simply isotropic space and, in section 4, we discuss how to introduce moving frames along simply isotropic curves. In section 5 we then study simply isotropic spheres and the characterization of spherical curves. In section 6 we turn our attention to the semi-isotropic space . In section 7 and 8 we study semi-isotropic spheres and moving frames along semi-isotropic curves, respectively. Finally, in section 9 we present a characterization of semi-isotropic spherical curves.

2 Preliminaries: rotation minimizing frames and spherical curves in Euclidean space

Let us denote by the Euclidean space, i.e., equipped with the standard Euclidean metric . The usual way to introduce a moving frame along it is by means of the Frenet frame (16); (17). However, we can also consider any other adapted orthonormal moving frame. For the Frenet frame the equations of motion are given in terms of the curvature function and the torsion : ; ; and . On the other hand, by introducing the notion of a rotation minimizing vector field3, Bishop considered an orthonormal adapted moving frame , where , whose equation of motion is (6)


The basic idea here is that rotates only the necessary amount to remain normal to the tangent (then justifying the terminology). In addition, the coefficients and relate with the curvature and torsion according to (6)


The above relations show that RM frames are not uniquely defined. Indeed, any rotation of and still gives two relatively parallel fields, i.e., there is an ambiguity associated with the group acting on the normal plane which specifies an RM frame up to an additive constant4. Finally, of great interest to us, is that RM frames allow for a simple characterization of spherical curves5:

Theorem 2.1 (Bishop (6))

A regular curve in lies on a sphere if and only if its normal development, i.e., the curve , lies on a line not passing through the origin: the distance of this line from the origin, , and the radius of the sphere, , are reciprocals, i.e., . Finally, straight lines passing through the origin characterize plane curves which are not spherical.

In the following we shall extend this formalism in order to present a way of building RM frames along curves in both simply isotropic and semi-isotropic 3-spaces and then apply them to furnish a unified approach to the characterization of isotropic spherical curves. And, in addition, by employing dual numbers and Galilean trigonometric functions, we will show how to relate (i) a Frenet frame to an RM frame and (ii) an RM frame with another RM frame.

3 Differential geometry in simply isotropic space

In this section we introduce some basic terminology in simply isotropic space. We refer the reader to Sachs’ monograph (24) for more details.

In the spirit of Klein’s Erlangen Program, simply isotropic geometry is the study of those properties in invariant by the action of the 6-parameter group


where . So, is the group of rigid motions in .

Observe that on the plane this geometry looks exactly like the plane Euclidean geometry . The projection of a vector on the plane is called the top view of and we shall denote it by . The top view concept plays a fundamental role in the simply isotropic space , since the -direction is preserved under the action of (6). A line with this direction is called an isotropic line and a plane that contains an isotropic line is said to be an isotropic plane.

One may introduce a simply isotropic inner product between two vectors and as


from which we define a simply isotropic distance as usual7:


Note that the inner product and distance above are just the plane Euclidean counterparts of the top views. In addition, since the isotropic metric is degenerate, the distance from to is zero. In such cases, one may define a codistance by (the codistance is preserved by and then is an isotropic invariant: it may be used to define angles involving isotropic lines and planes (22); (24)).

4 Moving frames along curves in simply isotropic space

Now we introduce some terminology related to curves. A regular curve , i.e., , is parametrized by an arc-length if . In the following we assume that all curves are parametrized by an arc-length (in particular, this excludes the possibility of an isotropic velocity vector). In addition, a point in which is linearly dependent is an inflection point and a regular unit speed curve with no inflection point is called an admissible curve if .

Remark 1

The admissible condition implies that the osculating planes, i.e., the planes that have a contact of order 2 with the reference curve8, can not be isotropic. Moreover, the only curves with are precisely the isotropic lines (24).

4.1 Simply isotropic Frenet frame

The (isotropic) unit tangent, principal normal, and curvature function are defined as usual


respectively. As usually happens in isotropic geometry, the curvature is just the curvature function of its top view and then we may write . To complete the moving trihedron, we define the binormal vector as the (co)unit vector in the isotropic direction. The Frenet frame is linearly independent for each :


The Frenet equations corresponding to the isotropic Frenet frame can be written as


where is the (isotropic) torsion (24), p. 110:


The above expressions for the torsion and curvature are also valid for any generic regular parameter for and, in addition, they are invariant by rigid motions in .

Contrary to the Euclidean space , we can not define the torsion through the derivative of the binormal vector. However, remembering that the idea behind such a definition in is that one can measure the variation of the osculating plane by measuring , we may ask if still characterizes plane curves in . It can be shown that the isotropic torsion is directly associated with the velocity of variation of the osculating plane, see (24), pp. 112-113, and that an admissible curve lies on a non-isotropic plane if and only if its torsion vanishes identically. Observe in addition that, contrary to the isotropic curvature, the torsion is not defined as the torsion of the top view (this would result in ). The isotropic torsion is an intermediate concept depending on its top view behavior and on how much the curve leaves the plane spanned by and .

4.2 Rotation minimizing frames in simply isotropic space

Let be an admissible curve parametrized by arc-length . A normal vector field is a simply isotropic RM vector field if , for some function . We easily see that the binormal is an RM vector field: . Except for plane curves, the principal normal fails to be RM: .

In order to introduce an RM frame in , we just need to look for an RM vector field in substitution to the principal normal. If is a normal vector, we may write


where we suppose (otherwise is just a multiple of ). Now, imposing implies that


The derivative of is


So, if we assume to be an RM vector field, it follows that


Finally, if we impose that has the same orientation as , we conclude that

Remark 2

Using the definition of the Galilean trigonometric functions, i.e., and (32), we can write an RM vector field in terms of the Frenet frame as


This is analogous to RM frames in both Euclidean and Lorentz-Minkowski spaces (6); (20).

Theorem 4.1

Let be a unit normal vector field along . If is RM and has the same orientation as the Frenet frame, then


where is a constant and we shall define . In addition, a rotation minimizing frame in isotropic space satisfies


where the natural curvatures are and .


The expression for follows from the discussion above. On the other hand, we have for the derivative of and


Finally, taking into account that , we find


From the equalities above we find the desired equations of motion for the RM trihedron . ∎

Using the definition for the Galilean trigonometric functions (32), we can relate the RM frame curvatures to the Frenet ones according to


This also shows that two RM frames and differ only by an additive constant, . This issue can be further clarified with the help of the ring dual numbers (23); (32) in isotropic plane (23) (note that the normal plane is always isotropic, since is in it).

Figure 1: Unit length and zero divisors hyperbolic (or double) and dual numbers: (a) Dual numbers and their zero divisors, i.e., nonzero numbers such that (dotted black line), and their unit length numbers (solid blue line). Note that a unit length dual number may be represented as ; (b) Hyperbolic numbers and their zero divisors (dotted black line), and their unit length numbers (solid blue and dashed red lines). Note that there exist four types of unit length hyperbolic numbers, the dashed red lines correspond to and the solid blue lines correspond to . The right dashed red line in particular correspond to , the group of vector space and time orientations preserving hyperbolic rotations (see subsection 6.1).

The standard way to write a dual number is , where the dual imaginary unit satisfies . The real and imaginary parts are and , respectively . The arithmetic operations are defined as


The modulus in isotropic plane may be expressed as , which may be seen as being induced by the degenerate metric in : .

In analogy with , where we can use unit complex number to describe rotations9, we can use a unit dual number in order to describe (Galilean) rotations in through a right multiplication, i.e., (see Fig. 1). Indeed, if we identify a point with the dual number , it then follows that a rigid motion in may be written as


where we used the following linear (matrix) representation for the dual numbers


In short, with the help of the ring of dual numbers, we can interpret an isotropic RM frame as a frame that minimizes isotropic (or Galilean) rotations.

4.3 Moving bivectors in simply isotropic space

In it is not possible to define a vector product with the same invariance significance as in Euclidean space. However, one can still do some interesting investigations by employing in the usual vector product from Euclidean space . Associated with the isotropic Frenet frame, one introduces a (moving) bivector frame as (24)


which satisfies the equation


and (24), Eqs. (7.43a-c), p. 130,


Analogously, we shall introduce the following (moving) RM bivector frame associated with an RM frame

Proposition 1

The moving frame forms a basis for . In addition, a moving RM bivector frame satisfies the equation


where and .


We have


In addition




Then, we find , which shows that the RM bivectors form a basis.

For the equations of motion, we have




5 Simply isotropic spherical curves

5.1 Isotropic osculating spheres

Due to the degeneracy of the isotropic metric, some geometric concepts can not be defined just using . This is the case for spheres.

Definition 1

We define simply isotropic spheres as connected and irreducible surfaces of degree 2 given by the 4-parameter family10


where . In addition, up to a rigid motion (in ), we can express a sphere in one of the two normal forms below

  1. (sphere of parabolic type)

  2. (sphere of cylindrical type)


It can be shown that the quantities and are isotropic invariants. Moreover, spheres of cylindrical type are precisely the set of points equidistant from a given center


Note however, that the center of a cylindrical sphere is not defined. More precisely, any other point with the same top view as , i.e., , would do the same job. We can remedy this by assuming the center located on the plane.

An osculating sphere of an admissible curve at a point is the (isotropic) sphere that has a contact of order 3 with . The position vector of an osculating sphere satisfies, Eq. (7.18) of (24),


where , is the usual inner product in Euclidean space , and and are constants to be determined.

5.2 Characterization of spherical curves in simply isotropic space

Our approach to spherical curves is based on order of contact. More precisely, we investigate osculating spheres in by using RM frames and their associated bivector frames. Then, we use that a curve is spherical when its osculating spheres are all equal to the sphere that contains the curve. We refer to (9) for a similar approach in the simpler setting of Euclidean spherical curves.

Defining a function , where and are constants to be determined, we have for the derivatives of


Imposing the condition (contact of order 3) gives


From the first and third equations above, we find that


for some constant . On the other hand, from the second equation we find that


The reader can easily verify that , and then we can rewrite the expression above as


where in the last equality one should use the expressions form in terms of , see Eq. (20).

In short, the equation for the isotropic osculating sphere (39), with respect to an RM frame and its associated bivector frame, can be written as


where .

Theorem 5.1

An admissible regular curve lies on the surface of a sphere if and only if its normal development, i.e., the curve , lies on a line not passing through the origin. In addition, is a spherical curve of cylindrical type with radius if and only if is constant and equal to .


The condition of being spherical implies that the isotropic osculating spheres are all the same and equal to the sphere that contains the curve. This condition demands




The first condition gives


which, by taking into account the linear independence of , implies


On the other hand, condition (47) implies


where we used that to obtain the second equality. If the curve is not of cylindrical type, we can not have , which is equivalent to , and then we conclude that for a parabolic spherical curve the normal development lies on a line not passing through the origin.

On the other hand, if the curve is of cylindrical type , taking the derivative gives


Then . We have that and, therefore, and is a constant.

Taking the derivative of Eq. (51) gives


Hence, the curvature is a constant and, in addition, .

Reciprocally, if is a (non-zero) constant, define . Taking the derivative gives and then is a constant. Clearly we have . ∎

Remark 3

In the discussion above, we can also use the Frenet frame instead of an RM one. In this case, spherical curves may be characterized by .

Proposition 2

An admissible regular curve lies on a plane if and only if its normal development lies on a line passing through the origin.


It is known that is a plane curve if and only if all its osculating planes are equal to the plane that contains the curve. Define a function , where and (the idea here is that a plane can be represented through a unit vector in the Euclidean sense). Taking the derivatives of twice and demanding a contact of order 2, we have


From these equations we deduce that


where, by applying the definition of the Frenet and RM bivectors, we can write .

The condition of being a plane curve is equivalent to . This leads to


where we used that , , , and .

Finally, it is easy to see that the planarity condition, i.e., , is equivalent to . We then deduce that it is equivalent to and then lies on a line passing through the origin. ∎

6 Differential geometry in semi-isotropic space

Following the Cayley-Klein paradigm, we must specify an absolute figure in order to build the semi-isotropic space. Here, the semi-isotropic absolute is composed by a plane, identified with , and a degenerate quadric of index one, identified with (11). Equivalently, we may say that in homogeneous coordinates the semi-isotropic absolute figure is composed by a plane , , and a pair of lines , , and , . Observe in addition that the point lies in the intersection and, therefore, should be preserved. Hence, the semi-isotropic absolute figure is alternatively given by .

Let us denote a projectivity in by


Imposing that and should be preserved leads to and , respectively.

A projectivity that preserves the absolute figure is said to be a direct projectivity if it takes to , i.e., goes in , and an indirect projectivity if it takes to (), i.e., goes in . The coefficients of a direct projectivity should satisfy the following relations


Adding and subtracting the equations above leads to and .

Going to affine coordinates and denoting , , , , , and (), defines the group of semi-isotropic direct similarities


Let us introduce a metric in according to


If we apply a transformation from to , then the norm induced by the metric above satisfies