Properties of some special surfaces h(z)=f(x)+g(y)

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L. Bal
Institutul de Calcul

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L. Bal, Propriétés de certaines surfaces spéciales h(z)=f(x)+g(y). (Romanian) Acad. R. P. Romîne. Fil. Cluj. Stud. Cerc. Mat. 9 1958 39–44.

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Studii si Cercetari Matematice

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Academy of the Republicof S.R.

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https://ictp.acad.ro/scm/journal/article/view/27

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PROPERTIES OF SPECIAL SURFACES $h (z) = f (x) + g (y)$

OFLET'S GO BALL

Paper presented at the Session of the «V. Babeş» and «Bolyai» Universities in Cluj, on May 27, 1958

In nomography, the aim is to approximately solve equations or systems of equations using nomograms. For this purpose, various types of nomograms are used, among which those containing networks of straight lines or rectilinear scales stand out for their simplicity of use and accuracy. This solution requires that the equations meet certain conditions [1]. Thus, to solve the equation

F (x, y, z) = 0

using a nomogram formed by three bundles of lines or three rectilinear scales, the equation must satisfy the Saint-Robert condition. In this case the equation with three variables is of the form

\begin{matrix} (1) & h (z) = f (x) + g (y) . \end{matrix}

If the functions

h, f

and

g

satisfy certain continuity and differentiability conditions, equation (1) represents an important class of surfaces. This class contains as particular cases the well-known types of surfaces

z = f (x) + g (y),

translation surfaces,

f (z) = x^{2} + y^{2}

surfaces of rotation and quadrics.
To our knowledge, these surfaces have not been sufficiently studied. Recently, Professor Maurice Frechet [2] in a memoir of over 50 pages determines the minimal surfaces of type (1). Radó Francisc in a paper
published in this volume starts from the functional equation that characterizes equation (1) and solves with his own method numerous functional equations that intervene in nomography.

In this paper, we aimed to study ruled and unfoldable surfaces of this type and several types of particular configurations of families of curves located on these surfaces.

We assume that the functions

h (z), f (x)

and

g (y)

are continuously differentiable whenever we need them and admit an inverse. In this case surfaces of type (1) can be represented parametrically by the equations

\begin{matrix} (2) & x = φ (you), y = ψ (V), z = χ (you + V) . \end{matrix}

The coefficients of the two fundamental forms are easily calculated using the derivatives of the three functions

φ, ψ

and

χ

,

\begin{aligned} It is = φ^{' 2} + χ^{' 2}, F = χ^{' 2}, G = ψ^{' 2} + χ^{' 2}, It is G - F^{2} = {(φ^{'} ψ^{'})}^{2} + {(φ^{'} χ^{'})}^{2} + {(φ^{'} ψ^{'})}^{2} \\ D = \frac{ψ^{'} (φ^{'} χ^{''} - χ^{'} φ^{''})}{\sqrt{It is G - F^{2}}}, D^{'} = \frac{φ^{'} ψ^{'} χ^{''}}{\sqrt{It is G - F^{'}}}, D^{''} = \frac{φ^{'} (ψ^{'} χ^{''} - χ^{'} ψ^{''})}{\sqrt{It is G - F^{2}}} \end{aligned}

The condition for surfaces (2) to be deployable is

φ^{' 2} ψ^{'} χ^{'} ψ^{''} χ^{''} + ψ^{' 2} φ^{'} χ^{'} φ^{''} χ^{''} - χ^{' 2} φ^{'} ψ^{'} φ^{''} ψ^{''} = 0

or, assuming that the derivatives

φ^{'}, ψ^{'}, χ^{'}, φ^{''}, ψ^{''}

and

χ^{''}

does not cancel identically,

\begin{matrix} (3) & \frac{χ^{'}}{χ^{''}} = \frac{φ^{'}}{φ^{''}} + \frac{ψ^{'}}{ψ^{''}} \end{matrix}

This is a functional equation between three functions that depend on the variables

u, v

, and

u + v

To find the solution, we first solve the functional equation of known form,

\begin{matrix} (4) & A (u + v) = B (u) + C (v) \end{matrix}

and we find

A (u + v) = a (u + v), B (u) = a u, C (v) = a v (a fiind constantă)

The differential equation (3) leads through simple quadratures to the following parametric equations of real developable surfaces

\begin{aligned} x = M_{1} (u + a)^{m} + N_{1} \\ (5) & y = M_{2} (v + b)^{m} + N_{2} \\ z = M_{3} (u + v + a + b)^{m} + N_{3} \end{aligned}

where

m, M_{ı}, N_{i} a

and

b

are real numbers.
The functional equation that characterizes the ruled surfaces is expressed quite complicatedly using the 3rd order derivatives of the functions

φ

,

ψ

and

χ

and we did not address it in this article.

Before moving on to tissues on surfaces of type (1), we mention that the topological study of tissues was initiated by Prof. W. Blaschke [3] and

Fig. 1

numerous works, especially at the end of the third decade and in the fourth decade of this century on the theme of "Topological Questions of Differential Geometry".
T. Dubourdieu [4], characterizes tissues with the help of an invariant

ρ

, called a topological invariant. If the families of curves of the tissue are written in differential form

L_{i} = φ_{i} (u, v) d u + ψ_{i} (u, v) d v (i = 1, 2, 3)

we can always determine these forms so that we have

L_{1} + L_{2} + L_{3} = 0

or

\begin{aligned} φ_{1} + φ_{2} + φ_{3} = 0 \\ ψ_{1} + ψ_{2} + ψ_{3} = 0 \end{aligned}

If we note

D = | \begin{array}{ll} φ_{1} & ψ_{1} \\ φ_{2} & ψ_{2} \end{array} | = | \begin{array}{ll} φ_{2} & ψ_{2} \\ φ_{3} & ψ_{3} \end{array} | = | \begin{array}{ll} φ_{3} & ψ_{3} \\ φ_{1} & ψ_{1} \end{array} |

then

D \neq 0

represents the condition with the family of curves to form a tissue in a domain in which the functions are defined

φ_{i}

and

ψ_{i}

Noting

cu Λ_{i}

OPERATOR

Λ_{i} = \frac{1}{D} (ψ_{i} \frac{\partial}{\partial u} - φ_{i} \frac{\partial}{\partial v})

and with

p_{i} = \frac{1}{D} (\frac{\partial ψ_{i}}{\partial u} - \frac{\partial φ_{i}}{\partial v})

then

\begin{matrix} (6) & ρ \equiv Λ_{1} ρ_{2} - Λ_{2} ρ_{1} \equiv Λ_{2} ρ_{3} - Λ_{3} ρ_{2} \equiv Λ_{3} ρ_{1} - Λ_{1} ρ_{3}, \end{matrix}

is the topological invariant of the weave.
Among the third-order weaves, the hexagonal ones are remarkable, which have the following closure property.

If

M

is a point on the surface through which a curve from each family passes (fig. 1), then we take a point

P

on a curve in the family (3). We carry through

P

the corresponding curve in family (1) that cuts in

Q

the given curve from family (2). By

Q

we take the corresponding curve from family (3), which intersects the curve given from (1) in

R

and we continue this construction until we reach the initial curve again at the point

P^{'}

If the points

P

and

P^{'}

coincide, then the hexagon

P Q R S T U P^{'}

is closed and the weave is called hexagonal. This property is easily verified for the weave formed by three parallel line bundles.
G. Thomsen [5] demonstrated that hexagonal weaves are topologically mapped onto three parallel line bundles and J. Dubourdieu [4] showed that the necessary and sufficient condition for a third-order weave to be hexagonal is obtained by canceling the topological invariant

ρ

. This condition is easy to
verify when the families of tissues or the differential forms of these families are given. Let us prove a series of theorems related to particular tissues located on surfaces of type (1).

Theorem I. Given a surface

z = f (x, y)

the necessary and sufficient condition for the tissue formed by the families

\begin{aligned} x = const. \\ (7) & y = const. \\ z = const. \end{aligned}

to be hexagonal is for the surface to be of the form (1), that is

h (z) = f (x) + g (y)

We can write the differential forms of the families as

\begin{aligned} - f_{x} d x & = 0 \\ - f_{y} d y & = 0 \\ f_{x} d x + f_{y} d y & = d z = 0 \end{aligned}

The topological invariant has the expression

ρ = \frac{1}{f_{x} f_{y}} \frac{\partial^{2}}{\partial x \partial y} \log \frac{f_{x}}{f_{y}}

provided

\begin{matrix} (8) & ρ = 0 \end{matrix}

if

f_{x} \neq 0, f_{y} \neq 0

, leads to

\begin{matrix} (9) & \frac{\partial^{2} \log \frac{f_{x}}{f_{y}}}{\partial x \partial y} = 0 \end{matrix}

which is precisely the condition of Saint Robert. It follows from this

z = h^{- 1} [f (x) + g (y)]

or

h (z) = f (x) + g (y) .

Theorem II. The tissue formed by the contour lines of the surface (1), the curves of highest slope, and one of the families of coordinate curves is hexagonal.

Considering the surface given by equations (2), we find for the considered families, the differential forms

\begin{aligned} χ^{' 2} d (u + v) + ψ^{' 2} d v & = 0 \\ - ψ^{' 2} d v & = 0 \\ - χ^{' 2} (u + v) d (u + v) & = 0 \end{aligned}

These forms being exact total differentials, it is obvious that they verify condition (8).
Theorem III. The tissue formed by the orthogonal trajectories of the coordinate lines and the lines of greatest slope or the orthogonal trajectories of the family

u - v =

const. is hexagonal.

The differential forms of these families are

\begin{aligned} χ^{' 2} d (u + v) + φ^{' 2} d u & = 0 \\ χ^{' 2} d (u + v) + ψ^{' 2} d v & = 0 \\ 2 χ^{' 2} d (u + v) + φ^{' 2} d u + ψ^{' 2} d v & = 0 \\ φ^{' 2} d u - ψ^{' 2} d v & = 0 \end{aligned}

and verify, however three are taken, condition (8).
Theorem IV. The networks formed by the orthogonal trajectories of the coordinate lines, the lines of greatest slope and the orthogonal trajectories of the family

u - v ==

const. are diagonal.

The families of these networks are

\begin{aligned} A (u + v) + B (u) = C_{1}, 2 A (u + v) + B (u) + C (v) = C_{3}, \\ A (u + v) + C (v) = C_{2}, B (u) - C (v) = C_{4}, \end{aligned}

where

A^{'} (u + v) = χ^{' 2}, B^{'} (u) = φ^{' 2}, C^{'} (v) = ψ^{' 2} .

Noting

cu u_{1}

and

v_{1}

the curves in the first network and

cu u_{2} v_{2}

, the curves in the second network, it is seen that between the curves of the two systems we have the relations

\begin{aligned} u_{2} = u_{1} + v_{1}, \\ v_{2} = u_{1} - v_{1}, \end{aligned}

which characterize, according to W. Blaschke [3], diagonal networks.
These theorems applied to particular surfaces of the type (1) surfaces give interesting interweavings between families of curves. They are also valid for any surface, but in these cases the families do not contain either the level lines or the lines of greatest slope.

Department of Geometry, University «V. Babes»- Chuj

PROPERTIES OF SPECIAL FLOORS $h (z) = f (x) + g (y)$

SHORT CONTENT

The author deals with the class of surfaces of type (1), found in nomographies. They contain, as special cases, rotational surfaces, translational surfaces and second-order surfaces. The author defines then unfolding surfaces of the type (1). Ih parameters-

Czech equations are expressed by formulas (5). Next, some hexagonal fabrics on the considered surfaces are considered, and the families of these fabrics are the intersections of the coordinate planes with the surface and their orthogonal trajectories. For this, the results of Dubourdie [3] are used.

PROPRIÉTÉS DE CERTAINES SURFACES SPÉCIALES

h (z) = f (x) + g (y)

SUMMARY

Dans ce travail, l'auteur s'occupe d'une classe de surfaces du type (1), intervenant en nomographie. Elles comprennée, comme des cas particuliers, les surfaces de rotation, les surfaces de translation et les quadriques. The author then determines the développable surfaces of type (1). Continuing, it examines some hexagonal networks located on the surfaces considered, the families of these networks being the intersections of the coordinate planes with the surface and their orthogonal trajectories. A cette fin, on se sert des résultats de J. Dubourdieu [3].

BIBLIOGRAPHY

L. Bal, F. Rado, Nomography Lessons. Technical Publishing House, Bucharest, 1956.
M. Fréchet, Determination of minima surfaces of type $a (x) + b (y) = c (z)$ . Rend. circus. mat. of Palermo, ser. II, 5, 238-260 (1956); idem 6, 5-12, (1957).
W. Blaschke, G. Bol, Geometrie der Gewebe. J. Springer, Berlin, 1938.
$^{4.}$ MJ Dubourdieu, Topological questions of differential geometry. Mem. often Sci. Mat., Paris, fasc. 78 (1936).
G. Thomsen, A topological theorem on curves. Boll. della Unione Mat. Ital., 6, 80-85 (1927).

1958

Properties of some special surfaces h(z)=f(x)+g(y)

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