Adv Syst Sci Appl 2025; 1:44–54
Published online at https://ijassa.ipu.ru.

Legendre Transformation and its Applications

Natalia Pavlova1,2 Alexey Remizov1*

1Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Russia
2RUDN University, Moscow, Russia

Abstract: We present the Legendre transformation in a geometric way based on the procedure
of the Legendrian lift. This approach allows us to understand some interesting properties of it,
in particular, the reason for the appearance of singularities of dual curves. Also we consider
application of the Legendre transformation to the Clairaut differential equation. Finally, we
say a few words class of contact transformations and present an infinite group of contact
transformations different from the Legendre transformation.

Keywords: Legendre transformation, duality, contact transformation, contact structure, pedal
curve, singular point, Clairaut equation

INTRODUCTION

The paper has several goals.
First, we present the Legendre transformation in a geometric way based on the procedure

of the Legendrian lift. Therefore, we start with the Legendre transformation of planar
curves and then go to the Legendre transformation of functions. As we shall see below,
this approach has some important advantages in compare with the standard definition of
the Legendre transformation of functions, which is thrown in the reader’s face without
motivations. Moreover, the presented geometric approach allows us to understand the
meaning of restrictions in the standard definition of the Legendre transformation of functions
(for example, the convexity) and the reason for the appearance of singularities of dual curves.

Second, we consider application of the Legendre transformation to ordinary differential
equations at the example of the Clairaut equation. For this, the geometric interpretation of the
Legendre transformation is more suitable.

Finally, we embed the Legendre transformation into a class of contact transformations and
discuss some examples of contact transformations different from it. In particular, we consider
the pedal transformation (apparently the earliest example of contact transformations) and with
its aid we obtain an infinite group of contact transformations, which was first constructed by
Sophus Lie.

It should be remarked that many of results presented in the paper were previously
known, but we presented these results from a more general point of view and explained the
relationship between them.

We start with a small section presented some notions and results about planar curves and
their singularities, which are necessary for understanding the main ideas of the paper.

∗Corresponding author: alexey-remizov@yandex.ru



LEGENDRE TRANSFORMATION AND ITS APPLICATIONS 45

1. CURVES AND SINGULARITIES

Let γ be an arbitrary curve on the plane given in the form

x = ϕ(t), y = ψ(t), (1.1)

where ϕ, ψ are smooth (C∞) functions. Further, we shall assume that all considered functions
and mappings are smooth (C∞) unless otherwise stated.

A point of the curve γ corresponding the parameter t0 is called or critical if

ϕ′(t0) = ψ′(t0) = 0. (1.2)

Otherwise a point is called regular. The curve γ is called regular is all its points are regular.
In a neighborhood of every regular point, the curve is diffeomorphic to its tangent line, while
for singular points it is not true.
Example 1.1:
In various applications, the following types of singular points often appear:

x = ϕ(t) = αtn + o(tn), y = ψ(t) = βtn+1 + o(tn+1), αβ ̸= 0. (1.3)

Fig. 1.1 presents the graphs of such curves for even and for off n. The singular point of the
curve (1.3) with n = 2 is called semicubic casp or simply casp.

It is proved that for n = 2 or 3 the germ of any curve (1.3) at zero can be brought to the
simplest form x = tn, y = tn+1 by means of an appropriate change of variables (x, y) and a
change of the parameter t. However, for n ≥ 4 it is not true [6].

Fig. 1.1. The curves (1.3) with even n (left) and odd n (right). The picture is taken from [1].

2. LEGENDRE TRANSFORMATION

The geometric definition of the Legendre transformation requires the notion of the Legendrian
lift. We start with the latter notion, which is an important mathematical procedure. It can be
applied to curves, functions and many other objects. The Legendrian lift appears not only
in mathematics and physics, but also in various technical applications and even in the living
nature. We shall say a few words about it below.

2.1. Legendrian Lift
The lift or 1-graph of the curve γ given by formula (1.1) is the curve

γ : x = ϕ(t), y = ψ(t), p =
ψ′(t)

ϕ′(t)
(2.4)

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



46 N. PAVLOVA, A. REMIZOV

in 3-dimensional space with the coordinates (x, y, p). This space denoted by J1 is called the
space of 1-jets of functions y(x).

In the latter equality in (2.4) the denominator ϕ′(t) may vanish, and we have to discuss the
correctness of such operation. If ϕ′(t) = 0, but ψ′(t) ̸= 0, we have p = ∞ at the considered
point, that is, the tangent direction of the curve γ is parallel to the y-axis. The dividing
by zero disappears, if we interchange the x-axis with the y-axis. The situation is more
complicated if the curve γ has a singular point where ϕ′(t) = ψ′(t) = 0. However, in the
most of interesting cases the function ψ′(t)/ϕ′(t) has the same one-sided limits, and after an
appropriate definition at the point itself, the function ψ′(t)/ϕ′(t) becomes continuous. For
example, such a situation takes place for the semicubic parabola, see Fig. 2.2.

Fig. 2.2. Legendrian lift of planar curves.

Remark 2.1:
The 1-graph of the curve γ does not depend on the choice of the parametrization the curve γ.

According to modern biological concepts going back to the famous work of Hubel and
Wiesel (1959), the brain of mammals performs the Legendrian lift of apparent contours of
flat images perceived by the retina. There exists an area of the primary cortex of the brain
assiciated with vision (V1) that contains neurons united in groups with complex hierarchy
(columns, hypercolumn). Each column is sensitive to a certain point of the retina and a certain
tangent direction at this point. Flat images perceived by the retina are lifted in the space of
1-jets, and further work of the brain occurs with the lifted objects. See [13] and the references
therein. The only difference between the lift performed by the brain and the lift described
above is that the brain uses the third coordinate θ = arctan p instead of p. It is very natural: θ
is more appropriate for procedure performed biologically, since it is bounded, while p is not.

It is worth observing that this theoretical concept helped to construct some inpainting
algorithms that allow to recover corrupted images. In these algorithms, the Legendrian lift
also plays a crucial role: First, we lift the image, which is considered as a family of contours,
in 3-dimensional space and then we act to the lifted object a hypoelliptic diffusion associated
with the contact structure. For details, see [4, 5].

2.2. Legendre Transformation and Duality
Legendre transformation is an automorphism Λ : J1 → J1 given by the formula (x, y, p) 7→
(X, Y, P ), where the coordinates are connected by the equalities:

x = P, p = X, y + Y = xp = XP. (2.5)

The Legendre transformation can be applied to various objects: curves, functions,
differential equations, etc. The objects obtained in this way are called dual of the original,
they are often denoted by the same symbols with asterisk. Dual objects reflect many important
properties of their preimages.

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



LEGENDRE TRANSFORMATION AND ITS APPLICATIONS 47

Consider the application of the Legendre transformation to the curve γ given by formula
(1.1). It is expressed by the diagram presented in Fig. 2.3.

Fig. 2.3. Definition of the Legendrian lift of a planar curve.

In Fig. 2.3, the vertical upward arrow denotes the Legendrian lift of the curve γ
to the space J1 with coordinates (x, y, p). The horizontal arrow denotes the Legendre
transformation Λ : J1 → J1 given by formula (2.5), the vertical downward arrow denotes
the projection π : (X, Y, P ) 7→ (X, Y ). The composition of these three mappings results in
the Legendre transformation of the curve γ on the (x, y)-plane to a certain curve γ∗ on the
(X, Y )-plane, which is called dual to γ.

The curve γ∗ dual to a curve γ can have singular points even γ is regular. Obviously, they
appear due to the projection π : (X, Y, P ) 7→ (X, Y ).

Example 2.1:
Let us find the dual curves for the line γ1 : y = at+ b, the parabola γ2 : y = x2, and the cubic
parabola γ3 : y = x3.

The line γ1 can be given in the parametric form x = t, y = at+ b. Hence p = a and,
consequently, we get X = a, Y = −b, that is, the dual curve γ∗1 is a single point on the plane.

The parabola γ2 can be presented in the parametric form x = t, y = t2. Hence p = 2t and,
consequently, we get X = 2t and Y = t2. Thus, the dual curve γ∗2 is the parabola Y = X2/4.

The cubic parabola γ3 can be presented in the parametric form x = t, y = t3. This yields
p = 3t2 and X = 3t2, Y = 2t3. This shows that the dual curve γ∗3 is the semicubic parabola
with the casp at the origin. The casp corresponds to the inflection point of the initial curve γ3.

Lemma 2.1:
The Legendre transformation has the following properties.

1. The curve γ∗ dual to the curve γ of the form (1.1) is regular at all points that correspond
to points of the curve γ with non-zero curvature κ(t) ̸= 0. The curve γ∗ has simicubic casp
at points that correspond to points of γ where κ(t) = 0, but κ′(t) ̸= 0.

2. Assume that the curve γ is the graph of a functions y = f(x) such that

df

dx
(0) = · · · = dnf

dxn
(0) = 0,

dn+1f

dxn+1
(0) ̸= 0. (2.6)

Then the curve γ∗ dual to γ has the form (1.3).
3. The Legendre transformation Λ is an involution, that is, Λ2 is the identity

transformation. This is equivalent to the relation Λ = Λ−1, which yields (γ∗)∗ = γ.

The proof is by direct calculation.

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



48 N. PAVLOVA, A. REMIZOV

The lemma shows that regular points of γ∗ correspond to points where f ′′(x) ̸= 0, while
semicubic casps correspond to points where

f ′′(x) = 0, f ′′′(x) ̸= 0,

i.e., simple (cubic) inflection points of the function f .

Example 2.2:
In Fig. 2.4, the curve dual to the graph of sine is presented. (For better visibility, the scale
along the x-axis is increased approximately 20 times.) Casps of the dual curve correspond to
points x = πn with integer n.

Fig. 2.4. The curve dual to y = sinx. From the left to right: x runs from 0 to 2π, 4π, 7π, respectively.

Remark 2.2:
It is worth observing that there exist other ways to define dual curves. For instance, in
algebraic geometry the definition of dual curves is based on the same construction, where
the mapping Λ given by formula (2.5) is replaced with the mapping (x, y, p) 7→ (X̄, Ȳ , P̄ ),
where

X̄ = X/Y, Ȳ = 1/Y, P̄ = dȲ /dX̄, (2.7)

X, Y are defined in (2.5).

Example 2.3:
Consider the curve dual to the ellipse (x/a)2 + (y/b)2 = 1. It is easy to see that this curve is
the hyperbola Y 2 − (aX)2 = b2 in the coordinates (X, Y ) and the ellipse (aX̄)2 + (bȲ )2 = 1
in the coordinates (X̄, Ȳ ). It is not surprising, because all conic sections (which include
ellipses and hyperbolas) are projectively equivalent and the coordinates (X, Y ) and (X̄, Ȳ )
are connected by a projective transformation (2.7).

Now consider an important case that the curve γ is the graph of a function y = f(x). It is
naturally to ask: whether the dual curve γ∗ is the graph of a certain function Y = f ∗(X) as
well. If so, the function f ∗ is called the dual function for f or the Legendre transformation of
f .

Examples considered above show that the dual curve for the graph of a function is not
always the graph of a function. Lemma 2.1 gives the following sufficient condition: if the
function f is strictly convex (f ′′ > 0) or f is strictly concave (f ′′ < 0) on its domain, then
the dual curve is the graph of a function, whence the dual function f ∗ exists. However, the
function f ∗(X) can be defined not for allX . Indeed, sinceX = p, the domain of f ∗ coincides
with the range of f ′(x), and f ∗ is defined for X that belong to the range of f ′ only.

If the function f is strictly convex, its Legendre transformation f ∗ can be defined by the
formula

f ∗(p) = sup
x
(xp− f(x)), p ∈ R, (2.8)

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



LEGENDRE TRANSFORMATION AND ITS APPLICATIONS 49

and if f is strictly concave, its Legendre transformation f ∗ can be defined by the formula

f ∗(p) = sup
x
(f(x)− xp), p ∈ R, (2.9)

where R denoted the range of f ′.
Remark 2.3:
Formulas (2.8) and (2.9) are often used as the definition of the Legendre transformation of
functions. In our opinion, such a definition has a serious drawback compared to the geometric
approach presented above: it is not natural and well motivated.
Remark 2.4:
The Legendre transformation can be obviously adapted for hypersurfaces and, consequently,
for functions depending on any number of variables. In particular, this way brings us to the
same formulas (2.8) and (2.9), where x and p belong to the spaces of the same dimension and
xp means the dot product in orthonormal coordinates.

2.3. Clairaut Differential Equation
The Legendre transformation can be applied to differential equations as well.

Consider the first-order differential equation

F (x, y, p) = 0, p = dy/dx. (2.10)

Passing to the new variables X, Y, P by formula (2.5) and substituting the corresponding
expressions in (2.10), we obtain a new first-order differential equation, which is called dual
to the initial equation (2.10):

F ∗(X, Y, P ) = 0, P = dY/dX, (2.11)

where
F ∗(X, Y, P ) = F (P,XP − Y,X).

In fact, here we have to verify the equality P = dY/dX only, which follows from the equality
p = dy/dx and formula (2.5). It more convenient to write the both equalities via differential
1-forms and check that 1-forms PdX − dY and pdx− dy vanish simultaneously:

PdX − dY = xdp− d(xp− y) = xdp− d(xp) + dy = −(pdx− dy).

From geometric viewpoint, this means that the Legendre transformation (2.5) sends
contact planes pdx− dy = 0 to contact planes PdX − dY = 0, i.e., it preserves the natural
contact structure of the space J1. This implies that the Legendre transformation sends integral
curves of equation (2.10) to integral curves of its dual equation (2.11). In other words, integral
curves of equation (2.11) are dual to integral curves of equation (2.10), and vice versa (since
the Legendre transformation is involution). This fact can be used for solution of differential
equation: if the dual equation can be solved easier that the initial one. The notion of duality
also helps to study singularities of implicit differential equations; see [7].

A nice example of application the Legendre transformation to differential equation is the
Clairaut equation, which has the form

xp− y = f(p), p = dy/dx, (2.12)

where f is an arbitrary function. See, for example, [2, 12].
The Legendre transformation sends differential equation (2.12) to the equation Y =

f(X), which does not contain the derivative. Therefore, the Legendre transformation sends
integral curves of equation (2.12) to points of the (X, Y )-plane filling the graph of the
function Y = f(X). Taking into account Example 2.1, one can see that integral curves of
equation (2.12) are straight lines tangent to a curve dual to the graph Y = f(X).

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



50 N. PAVLOVA, A. REMIZOV

A direct calculation shows that this curve given by formula

x = f ′(p), y = pf ′(p)− f(p) (2.13)

is the discriminant curve of equation (2.12) and the envelope of the family of solutions of
equation (2.12) – its tangent lines. This gives a simple way to get all solutions of equation
(2.12): First, we find the discriminant curve (2.13), which is called singular solution of (2.12).
The remaining solutions are tangent lines to the discriminant curve taken at all its points.
Moreover, it is not hard to see that these tangent lines are given by the formula

y = cx− f(c), c = const. (2.14)

Remark that formula (2.14) can be obtained from equation (2.12) after replacing p with c.
Example 2.4:
In Fig. 2.5, integral curves of equation (2.12) with the function f(p) = p3 are presented. In
this case, the discriminant curve (2.13) is the semicubic parabola

x = 3p2, y = 2p3 (2.15)

with the casp at the origin. The family of tangent lines to the curve (2.15) has concentration
(patch) near this curve and especially near its cusp.

This phenomenon is called a caustic [3]. In physics, a caustic is the envelope of light rays
which have been reflected or refracted by a curve (on the plane) or a surface (in the space).
Fig. 2.5 contains a caustic – the semicubic parabola (2.15).

Fig. 2.5. Integral curves of the Clairaut equation xp− y = p3 are tangent lines to the semicubic parabola (2.15).
The picture is taken from [10]

Finally, let us consider the Clairaut equation of more general type:

F (p, xp− y) = 0, p = dy/dx, (2.16)

where F (u, v) is an arbitrary function of two variables. The Legendre transformation sends
(2.16) to the equation F (X, Y ) = 0 without derivative. Proceeding in the same way as before,
we conclude that integral curves of equation (2.16) are straight lines tangent to the curve
dual to F (X, Y ) = 0. The equation F (X, Y ) = 0 defined the discriminant curves of equation
(2.16), which is its singular solution and the envelope of other solutions – tangent lines given
by the formula

y = ax+ b, F (a,−b) = 0,

which is a generalization of (2.14).

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



LEGENDRE TRANSFORMATION AND ITS APPLICATIONS 51

Example 2.5:
Consider the equation

p3 = (xp− y)2, p = dy/dx, (2.17)

which has the form (2.16) with F (u, v) = u3 − v2.
The curve dual to the curve u3 − v2 = 0 is the cubic parabola y = 4

27
x3, which is the

discriminant curve of equation (2.17) and its singular solutions. The family of non-singular
solutions of (2.17) consists of all tangent lines to the singular solution, which are given by
the formula

y = ax+ b, a3 = b2.

The latter equation can be written in the form

a3 = (ax− y)2, a = const, (2.18)

or in the form y = c2x+ c3, where a = c2. Formula (2.18) can be obtained from equation
(2.17) by replacing the derivative p with the constant c.

3. CONTACT TRANSFORMATIONS

The Legendre transformation is not a unique transformation J1 → J1 preserving the contact
structure of this space. There exist a large group of diffeomorphisms of the space that also
preserve the contact structure, they are called contact transformations. Equivalent definition:
contact transformations are transformations of curves in the plane in which tangent curves
are transformed into tangent curves.

Let us try to determine condition under which a diffeomorphism (x, y, p) → (X, Y, P )
(global or local) preserves the contact structure of the space J1. Namely, consider a
diffeomorphism (x, y, p) → (X, Y, P ) given by the formula

X = F (x, y, p), Y = G(x, y, p), P = H(x, y, p), (3.19)

where the functions F,G satisfy some relation (that we shall find) and the function H is
uniquely defined by F,G due to the condition P = dY/dX . Therefore, the condition of
preservation the contact structure has the form

H(x, y, p) =
dY

dX
=
Gxdx+Gydy +Gpdp

Fxdx+ Fydy + Fpdp
=
Gx + pGy + p′Gp

Fx + pFy + p′Fp

, (3.20)

where p′ = dp/dx.
Since P = H(x, y, p) does not depend on p′, the right hand side of equality (3.20) does

not depend on p′ as well. This yields

Gx + pGy

Fx + pFy

≡ Gp

Fp

or, equivalently,
Fp(Gx + pGy) ≡ Gp(Fx + pFy). (3.21)

The identity (3.21) connecting the functions F and G is a necessary and sufficient condition
for the diffeomorphism (3.19) be a contact transformation. It is not hard to see that the
Legendre transformation satisfies (3.21).

The Legendre transform is the most commonly used and the most useful contact
transformation, but historically it was not the first one. Apparently, the first contact
transformation was the so-called pedal transformation, which is defined as follows. We shall

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



52 N. PAVLOVA, A. REMIZOV

give another example of contact transformations different from the Legendre transform – so-
called pedal transformations, which were investigated by Colin Maclaurin, Arthur Cayley,
and Sophus Lie.

To define pedal transformation, let us fixe an arbitrary pointO on the plane (without loss of
generality,O can be the origin). The pointO is called the pole or center of the transformation.
Let γ be a curve on the plane. Then the pedal curve of γ is the locus of points X so that the
line OX is perpendicular to the tangent line to γ passing through the point X .
Example 3.1:
The pedal curve of the circle is the limacon (Pascal’s snail), the special shape of which is
determined by the distance ρ between the pole O and the circle (Fig. 3.6). Considering ρ as a
parameter, we get a family of the pedal curves, some of which have singular points. Namely,
when the pole O lies on the circle itself (ρ = 0), the pedal curve has the casp at O, this curve
is called cardioid (the third figure from the left).

Fig. 3.6. The pedal curves of the circle (depicted as the dotted line) are limacons, whose shape depends on the
distance between the pole O and the circle

Lemma 3.1:
In the Cartesian coordinate system centered at O, the pedal transformation is given by the
formula

P : (x, y, p) 7→ (X̄, Ȳ , P̄ ),

where
X̄ = XY/(1 +X2), Ȳ = −Y/(1 +X2), P̄ = dȲ /dX̄, (3.22)

the relation between X, Y, P and x, y, p is given in (2.5).
The proof is by direct calculation.
In contrast to the Legendre transformation Λ, the pedal transformation P is not

an involution. Therefore, one can consider the degrees (compositions) of the pedal
transformation. Fixing the pole O, consider the cyclic group generated by the elements Pn

with all integers n:
. . . P−3, P−2, P−1, P0, P1, P2, P3 . . .

A natural question: is thus group finite or infinite? We will get an answer below.
Sophus Lie suggested a nice way how to extend this discrete grout to a continuous group.

For this, consider the polar coordinates (r, ϕ) on the (x, y)-plane and the polar coordinates
(R,Φ) on the (X̄, Ȳ )-plane. As the third coordiante in the space J1 let us take the difference
between the angle of inclination of the radius vector to a given point on the plane and the
angle of inclination of the tangent to the curve at this point such that the orders of subtraction
in the preimage and image are opposite:

x = r cosϕ, y = r sinϕ, ψ = ϕ− arctan p,

X̄ = R cosΦ, Ȳ = R sinΦ, Ψ = arctan P̄ − Φ,
(3.23)

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



LEGENDRE TRANSFORMATION AND ITS APPLICATIONS 53

where x, y, p and X̄, Ȳ , P̄ are connected via (3.22).
It is not hard to check that in the coordinates (3.23), the transformation Pn with every

integer n is given by the formula

R = r| sinψ|n, Φ = ϕ− n(ψ + π
2
), Ψ = ψ. (3.24)

The obtained formulas show shows that Pn ̸= Pm for any n ̸= m. Therefore, the cyclic group
{Pn} is infinite.

If we consider the same formula (3.24) with real n and the operation Pn · Pm = Pn+m,
we obtain a continuous group of contact transformations. This group was constructed by
Sophus Lie.

We will not go deep into the theory of contact transformations, referring the interested
reader to the specialized literature, which is quite extensive. For example, see [8, 9, 11].

4. CONCLUSION

We presented the Legendre transformation in a geometric way, which is based on the notion
of the Legendrian lift. This approach is more natural than the standard definition for the
Legendre transformation of functions. Indeed, it is applicable to various objects (including
functions, curves, differential equations) and it shows that the Legendre transformation of
functions is a partial case of the Legendre transformation of curves.

We also considered the Legendre transformation to ordinary differential equations at the
example of the Clairaut equation and discussed its wonderful properties.

It was shown that the Legendre transformation is a member of a class (group) of contact
transformations. We presented several examples of different contact transformations: the
pedal transformation and an infinite group of contact transformations.

REFERENCES

1. Agakhanova, Ya. S., Nikachev, A. S., & Oblasova, I. N. (2023) On codimension 3
singular points of first-order implicit differential equations, Adv. Syst. Sci. Appl., 23(1),
8–21.

2. Arnol’d, V. I. (1988) Geometrical methods in the theory of ordinary differential
equations, Berlin: Springer.

3. Arnol’d, V. I. (1990) Singularities of caustics and wave fronts, Math Appl. (Soviet
Series), 62. Kluwer Academic Publishers.

4. Boscain, U., Chertovskih, R. A., Gauthier, J. P., & Remizov, A. O. (2014) Hypoelliptic
diffusion and human vision: a semidiscrete new twist, SIAM J. Imaging Sci., 7(2), 669–
695.

5. Boscain, U., Chertovskih, R. A., Gauthier, J. P., Prandi, D., & Remizov, A. O. (2018)
Cortical-inspired image reconstruction via sub-Riemannian geometry and hypoelliptic
diffusion, ESAIM, Proc. Surv., 64, 37–53.

6. Bruce, J. W., Gaffney T. J. (1982) Simple singularities of mappings C, 0 → C2, 0,
J. Lond. Math. Soc., II. Ser., 26, 465–474.

7. Bruce, J. W., Tari, F. (2000) Duality and implicit differential equations, Nonlinearity,
13:3, 791–811.

8. Feld, J. M. (1940) A continuous group of contact transformations containing the
generalized pedal transformation, Tohoku Math. J., 46, 252–260.

9. Gizatullin, M. Kh.(2008) Klein’s conjecture for contact automorphisms of the three-
dimensional affine space, Mich. Math. J., 56(1), 89–98.

10. Izumiya, S., Kurokawa, Y. (1994) On systems of Clairaut type, Kodai Math. J., 17(3),
636–643.

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



54 N. PAVLOVA, A. REMIZOV

11. Lie, S. (1875) Foundations of an invariant theory of contact transformations, Math. Ann.,
8, 215–303.

12. Mitrinović, D. S., Kečkić, J. D. (1981) Variations and generalisations of Clairaut’s
equations, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., 716/734, 11–21.

13. Petitot, J. (2003) The neurogeometry of pinwheels as a sub-Riemannian contact
structure, J. Physiol. Paris, 97, 265–309.

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)


	Curves and singularities
	Legendre transformation
	Legendrian Lift
	Legendre Transformation and Duality
	Clairaut Differential Equation

	Contact transformations
	Conclusion