Evolutes of plane curves and null curves in Minkowski $3$space
Abstract.
We use the isotropic projection of Laguerre geometry in order to establish a correspondence between plane curves and null curves in the Minkowski space. We describe the geometry of null curves (Cartan frame, pseudoarc parameter, pseudotorsion, pairs of associated curves) in terms of the curvature of the corresponding plane curves. This leads to an alternative description of all plane curves which are Laguerre congruent to a given one.
1. Introduction
Two dimensional Laguerre geometry is the geometry of oriented contact between circles in the Euclidean plane (points of are viewed as circles with radius zero). An efficient and intuitive model of Laguerre geometry is the so called Minkowski model [3, 8]. This model establishes a  correspondence, called the isotropic projection [3, 8], between points in the Minkowski space and oriented circles in the Euclidean plane. The group of Laguerre transformations is identified with the subgroup of the group of affine transformations of which is generated by linear (Lorentzian) isometries, translations and dilations of .
A differentiable oneparameter family of circles in the Euclidean plane corresponds, via isotropic projection, to a differentiable curve in the Minkowski space. In particular, the family of osculating circles to a given plane curve corresponds to a certain curve in , which we will call the evolute of . The evolute of a plane curve is a null curve and, conversely, any null curve is the evolute of some plane curve (see Section 5).
The importance of light like (degenerate) submanifolds in relativity theory has been emphasized by many mathematical and physical researchers (see the monographs [5, 6]). Several authors have also investigated the particular case of null curves in the Minkowski space (see the surveys [13, 14]). In the present paper we describe the geometry of null curves (Cartan frame, pseudoarc parameter, pseudotorsion) in terms of the curvature of the corresponding plane curves. This leads us to the notion of potential function (see Definition 5.8). A potential function together with an initial condition completely determines the null curve and the underlying plane curve through the formulae of Theorem 5.10. As a consequence, we obtain an alternative description of all plane curves which are Laguerre congruent to a given curve as follows (see Remark 5.12 for details): starting with the plane curve , compute the pseudotorsion of its evolute; up to scale, the pseudotorsion is invariant under Laguerre transformations of and the potential functions associated to null curves with pseudotorsion are precisely the solutions of a certain second order differential equation; use formulae of Theorem 5.10 to recover from these solutions the corresponding plane curves; in this way, we obtain all the plane curves which are Laguerre congruent with .
In Section 6, we describe, in terms of their potential functions, some classes of associated null curves: Bertrand pairs [1, 13], null curves with common binormal lines [12], and binormaldirectional curves [4]. We shall also prove the following two interesting results on null curves: 1) a null helix parameterized by pseudoarc admits a null curve parameterized by pseudoarc with common binormal lines at corresponding points if, and only if, it has pseudotorsion (see Corollary 6.6); 2) given a null curve parameterized by pseudoarc, there exists a null helix parameterized by pseudoarc with constant pseudotorsion and a  correspondence between points of the two curves and such that, at corresponding points, the tangent lines are parallel (see Theorem 6.10).
2. Curves in the Euclidean plane
We start by fixing some notation and by recalling standard facts concerning curves in the Euclidean plane.
Let be an open interval of and be a regular plane curve, that is, is differentiable sufficiently many times and for all , where and stands for the standard Euclidean inner product on . Consider the unit tangent vector and the unit normal vector of , where is the anticlockwise rotation by . The pair satisfies the Frenet equations and , where is the curvature of . We denote by the arc length at of with respect to the starting point .
Assume that the curvature is a nonvanishing function on . Let be the (signed) radius of curvature at . The evolute of is the curve defined by which is a regular curve if, additionally, is a nonvanishing function on . In this case, the curvature of is given by
(2.1) 
We also have
which means that is an arc length parameter of the evolute.
If is an arc length parameter of , the curve is an involute of . It is well known that the evolute of the involute is precisely and that, for any other choice of arc length parameter, the corresponding involute of is parallel to .
Finally, recall that, given a smooth function , there exists a plane curve parameterized by arc length whose curvature is . Moreover, is unique up to rigid motion and is given by where the turning angle is given by
3. Laguerre geometry and the Minkowski space
Consider on the Lorentzian inner product defined by for and . The Minkowski space is the metric space . If is a spacelike vector, which means that , we denote . The light cone with vertex at is the quadric
We have a  correspondence, called the isotropic projection, between points in and oriented circles in the Euclidean plane, which is defined as follows. Given in , consider the light cone . The intersection of with the Euclidean plane is a circle centered at with radius . The orientation of this circle is anticlockwise if and clockwise if . Points in are regarded as circles of zero radius and correspond to points in with . Making use of this correspondence, Laguerre transformations are precisely those affine transformations of the form , where , and is an orthogonal transformation of . For details see [3].
4. Null curves in Minkowski space
Next we recall some standard facts concerning null curves in the Minkowski space. For details, we refer the reader to [13, 14].
A regular curve with parameter is called a null curve if is lightlike, that is . Differentiating this, we obtain , which means that lies in . We will assume throughout this paper that and are linearly independent for all (in particular, can not be a straight line). Then we have and is spacelike. If is a solution of the differential equation
(4.1) 
then satisfies In this case, we say that is a pseudoarc parameter [7, 13, 14] of .
Suppose now that the null curve is parameterized by pseudoarc parameter , that is . The tangent vector is and the (unit) normal vector is . Define also the binormal vector as the unique lightlike vector orthogonal to satisfying . The Cartan frame of satisfies the following Frenet equations:
(4.2) 
where is called the pseudotorsion of . It follows from (4.2) that
(4.3) 
On the other hand, since is a null curve parameterized by pseudoarc length, we have and . Hence the components of satisfy
(4.4) 
Conversely, given a function , and a fixed basis of satisfying
consider the linear map which is represented, with respect to this basis, by the matrix . This linear map lies in the Lie algebra , which means that we can integrate in order to get a map with , such that the frame satisfies (4.2). Hence defines a regular null curve parameterized by pseudoarc length with pseudotorsion , and is unique up to Lorentz isometry.
The pseudoarc parameter and pseudotorsion are preserved under Lorentz isometries. Regarding dilations, we have the following.
Proposition 4.1.
If is a null curve with pseudoarc parameter and pseudotorsion , and is a real number, then is a pseudoarc parameter of the null curve , which has pseudotorsion
(4.5) 
Proof.
By applying twice the chain rule, we see that the tangent vector and the normal vector of are related with the tangent vector and the normal vector of by
This shows that is a pseudoarc parameter. On the other hand, since the pseudotorsion is the component of along , formula (4.5) can now be easily verified. ∎
5. The evolute of a plane curve
A differentiable oneparameter family of circles in the Euclidean plane corresponds, via isotropic projection, to a differentiable curve in the Minkowski space . In particular, the family of osculating circles to a given plane curve corresponds to a certain curve in , which we will call the evolute of . In the present section, we show that, the evolute of a plane curve is a null curve and that, conversely, any null curve is the evolute of some plane curve if it has nonvanishing third coordinate. We also describe the geometry of null curves in terms of the curvature of the corresponding plane curves.
Definition 5.1.
Consider a regular curve , with curvature and parameter . Assume that and its derivative are nonvanishing functions on . Let be its evolute and its (signed) radius of curvature. The evolute of is the curve in the Minkowski space defined by ; that is, for each , corresponds to the osculating circle of at under the isotropic projection.
Observe that whereas the evolute of a curve is independent of the parameterization, the evolute depends on the orientation: as a matter of fact, if is a orientation reversing reparameterization of , then the trace of the evolute of is that of .
Proposition 5.2.
Given a regular plane curve in the conditions of Definition 5.1, its evolute is a null curve in . Conversely, any null curve is the evolute of some plane curve if is nonvanishing on .
Proof.
Taking into account the Frenet equations for the regular plane curve , we have
Hence , since is a lightlike vector. Moreover, is regular, since on . Hence is a null curve.
Conversely, let be a null curve in with parameter . We must have for all : as a matter of fact, if for some , then since is null; hence and is not regular, which is a contradiction. This means that we can reparameterize with the parameter and assume that is of the form . By hypothesis, . Consider the plane curve defined by where . Observe that is a unit speed curve: since and , we get . Hence is an involute of ; and, consequently, is the evolute of . In particular, the curvature of satisfies . If , then is the evolute of . If , then is the evolute of a orientation reversing reparameterization of .
∎
To make it clear, throughout the rest of this paper, we will assume that

all plane curves are regular, with nonvanishing and ;

all null curves, and the evolutes in particular, are such that and are everywhere linearly independent – in particular, we are excluding null straight lines in and we can always reparameterize by pseudoarc.
Proposition 5.3.
Let be an arc length parameter of and the evolute of . Let be a pseudoarc parameter of , where and is a solution of (4.1). Then
(5.1) 
where is the radius of curvature of .
Proof.
By hypothesis, the null curve satisfies . Taking into account the Frenet equations for plane curves and the chain rule, we have
(5.2) 
and
(5.3) 
Since is a lightlike vector orthogonal to it follows that
(5.4) 
from which we deduce (5.1). ∎
Definition 5.4.
Two plane curves and , with evolutes and , respectively, are said to be Laguerre congruent if the corresponding families of osculating circles are related by a Laguerre transformation, that is, if (up to reparameterization) for some , , and .
The identification induces a natural embedding of the group of all rigid motions of in the group of Laguerre transformations. The subgroup of generated by together with the translation group of will be denoted by . We point out that if corresponds to a translation along the timelike axis , that is, for some real number , then the projections of and into the Euclidean plane coincide, which implies that and are involutes of the same curve and, consequently, they are parallel: where is the unit normal vector of .
We also have the following.
Theorem 5.5.
Two plane curves and are Laguerre congruent if, and only if, the pseudotorsions and of and , respectively, are related by (4.5) for some .
Proof.
5.1. The Tait theorem for osculating circles of a plane curve
The correspondence between null curves and curves in the Euclidean plane allows one to relate an old theorem by P.G. Tait on the osculating circles of a plane curve and the following property for null curves observed by L.K. Graves.
Proposition 5.6 (Graves, [10]).
A null curve starting at lies in the inside of the light cone .
If is a null curve with , then either lies in the inside the upper part of the light cone for all or lies in the inside of the lower part of the light cone for all . Consequently, in both cases, the circle associated to under the isotropic projection does not intersect the circle associated to for all (see Figure 1). This implies the following theorem.
Proposition 5.7 (Tait,[16]).
The osculating circles of a curve with monotonic positive curvature are pairwise disjoint and nested.
For some variations on the P.G. Tait result, see [9].
5.2. The potential function
Let be a regular curve in with arc length parameter and (signed) radius of curvature . Observe that the sign of is changed if the orientation of is reversed.
Definition 5.8.
Take an arc length parameter of such that . In (5.1), choose
and let be the corresponding pseudoarc parameter of the evolute of . The potential function of is the (positive) function
(5.5) 
Proposition 5.9.
If is the potential function of , the pseudotorsion of is given by
(5.6) 
Proof.
Up to congruence, the curve and its evolute can be recovered from its potential function as follows.
Theorem 5.10.
Let be a positive and differentiable function on the open interval . Take and a constant such that is nonvanishing on . Set . Then the null curve given by
(5.9) 
has pseudoarc parameter and is the evolute of some regular plane curve with potential function . Up to a rigid motion in , this plane curve is given by
(5.10) 
where the arc length parameter of satisfies for some strictly monotone function with derivative
(5.11) 
Moreover, if is another regular plane curve with potential function , then coincides with up to rigid motion, for some constant . Consequently, any two plane curves with the same potential function are congruent. Conversely, if and are congruent, then they have the same potential function.
Proof.
Differentiating (5.9) we get
From this we see that , that is, is a null curve. Differentiating again, we obtain
Hence , which means that is a pseudoarc parameter of .
By hypothesis, is nonvanishing on . Hence, by Proposition 5.2, is the evolute of some plane curve. The radius of curvature of this plane curve at is precisely . On the other hand, a simple computation shows that is an arc length parameter for the curve defined by (5.10) and that the radius of curvature of at is . Hence, the fundamental theorem of plane curves assures that, up to rigid motion in , the plane curve whose evolute is coincides with .
Now, take any curve with potential function . Let and be an arc length parameter and the radius of curvature, respectively, of , so that, by definition of potential function,
with . According to our choices in the definition of potential function, we have
where . From the first equation we see that and multiplying both it follows that Hence
(5.12) 
Taking we see from (5.12) that and have the same curvature function and the same arc length parameter , which means that and are related by a rigid motion. In particular, the evolute of is also given by (5.9) up to rigid motion acting on the first two coordinates. We can see this constructively as follows.
Let be the evolute of . By definition of evolute, . On the other hand, we know that the evolute of has arc length and curvature . Since
setting , the curve is given, up to rigid motion, by
by the fundamental theorem of plane curves. Consequently, the evolute is given, up to rigid motion acting on the first two coordinates, by
Finally, if and are congruent, then their evolutes and satisfy , with common pseudoarc parameter , where is a rigid motion acting on the first two coordinates and . Hence, the corresponding curvature radius satisfy . Consequently, , and we are done.
∎
Remark 5.11.
These results provide a scheme to integrate equations (4.4). Given a function , if is a solution to the differential equation (5.6), then the null curve (5.9) has pseudotorsion and pseudoarc parameter . This means that the components of the tangent vector
(5.13) 
of satisfy (4.4). Moreover, all solutions of (5.6) for a given arise in this way.
Remark 5.12.
We have also obtained a description of all plane curves which are Laguerre congruent to a given curve . As a matter of fact, starting with , compute its potential function and the pseudotorsion of its evolute making use of (5.5) and (5.6); in view of Theorem 5.5, find the general solution of the equation
for each ; since this is a second order differential equation, the two initial conditions together with the parameter determine a threeparameter family of potential functions; for any such function , formulas (5.10) and (5.11) define a curve in the plane which is Laguerre equivalent to . Conversely, any curve which is Laguerre congruent to arises in this way, up to rigid motion.
Example 1.
Equation (5.6) is equivalent to . Differentiating this, we obtain the third order linear ordinary differential equation
(5.14) 
For , the general solution of (5.14) is
(5.15) 
and a straightforward computation shows that the solutions of (5.6), with , are precisely those functions (5.15) satisfying . In particular, for and we get the solution . In view of Theorem 5.10, the arc length of the plane curve associated to this potential function is given by and we have
(5.16) 
Up to Euclidean motion, is the logarithmic spiral reparameterized by arc length . The evolute of is given by
This null curve is an example of a Cartan slant helix in . A Cartan slant helix in is a null curve parameterized by pseudoarc whose normal vector makes a constant angle with a fixed direction. Accordingly to the classification established in [4], Cartan slant helices are precisely those null curves whose pseudotorsions are of the form , where and are constants.
Example 2.
Let us consider the Cornu’s Spiral
This is a plane curve with arc length and radius of curvature . From (5.1), (5.5) and (5.6) we can see that the evolute of has pseudoarc , for , the potential function is and the pseudotorsion is . For this pseudotorsion , the general solution of (5.14) is
and the solutions of (5.6) are precisely those functions satisfying
Example 3.
The potential functions associated to the pseudotorsion are of the form
with . For and , formula (5.10), with and , gives , where
In the Figure 2, the curve is represented on the left, for ; on the right, one can see the plane curve associated to the potential function (which corresponds to the choice , and ), obtained by numerical integration of (5.10), with and , also for .
5.3. Pseudotorsion and Schwarzian derivatives
Very recently, Z. Olszak [15] observed that the pseudotorsion of a null curve can be described as follows.
Theorem 5.13.
[15] If is a null curve with pseudoarc parameter , then
(5.17) 
with , for some nonzero function with nonvanishing derivative on . The pseudotorsion of is precisely the Schwarzian derivative of :
5.4. Potential function of the evolute
Theorem 5.14.
Let be a plane curve parameterized by arc length and be its evolute. The potential function and the pseudoarc parameter associated to are related with the potential function and the pseudoarc parameter associated to the evolute by
(5.19) 
and