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## Über dieses Buch

This encyclopedia presents an all-embracing collection of analytical surface classes. It provides concise definitions and description for more than 500 surfaces and categorizes them in 38 classes of analytical surfaces. All classes are cross references to the original literature in an excellent bibliography.

The encyclopedia is of particular interest to structural and civil engineers and serves as valuable reference for mathematicians.

## Inhaltsverzeichnis

### Chapter 1. Ruled Surfaces

A surface formed by the continuous movement of a straight line is called

a ruled surface

or a ruled surface, also known as

a scroll surface S

, is the result of movement of a straight line along a curve.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 2. Surfaces of Revolution

A

surface of revolution

Surface of revolution

is generated by rotation of a plane curve

z

=

f

(

x

Oz

called

the axis of the surface of revolution

. The resulting surface therefore always has

azimuthal symmetry

.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 3. Translation Surfaces

A surface of translation

(

a translation surface

) is a

surface formed

Translation surfaces

by parallel translation of a curve of some direction that is a generatrix curve

L

1

along another curve that is a directrix curve

L

2

(Fig. 1). So, a point

M

0

of the curve

L

1

slides along the curve

L

2

. The same surface can be obtained if we shall take a curve

L

2

as a generatrix but a curve

L

1

as a directrix.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 4. Carved Surfaces

Carved

surfaces

Carved

surface

are called surfaces with one family of plane lines of principal curvatures lying on planes that are orthogonal to the surface. The family of the plane lines of principal curvatures of a carved surface is

geodesic lines

. So the normals of these lines coincide with the normals of the surface.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 5. Surfaces of Congruent Sections

A surface of congruent sections

is called a surface carrying on itself the continuous single-parametric family of plane lines. Such surface is formed by any

moving fixed plane line

(

generatrix

). A single-parametric family of the planes

α

that are carriers of these lines corresponds to single-parametric family of the cross sections of the surface.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 6. Continuous Topographic and Topographic Surfaces

Topographic surfaces

Topographic surfaces

are called surfaces given by the discrete set of their horizontals. Such definition of a topographical surface is used mainly in mining art, building and in topography.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 7. Helical Surfaces

It is well known that apart from the trivial uniform motion, where nothing moves at all and all velocities are zero, there are the following three cases: (1)

uniform translations

, (2)

uniform rotations

with nonzero angular velocity about a fixed axis, and (3)

uniform helical motion

that are the superposition of a uniform rotation and a uniform translation parallel

to the

Helical surface

of variable pitch

rotation’s axis.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 8. Spiral Surfaces

S. N. Krivoshapko, V. N. Ivanov

### Chapter 9. Spiral-Shaped Surfaces

Spiral-shaped surfaces

Spiral-shaped surfaces

bear a resemblance to

spiral surfaces

but these surfaces cannot be related to the same class because the spiral surface has the directrix curve only in the form of a spiral on a right circular cone and the generatrix curve doesn’t change its form in the process of the motion along the conical spiral directrix line. But for a directrix curve of any spiral-shaped surface, one may take arbitrary spiral curve laying on any surface.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 10. Helix-Shaped Surface

A

helical surface

Helix-Shaped Surface

is formed by a rigid curve which rotates uniformly about the helical axis lying in the same plane with the generatrix curve and, at the same time, executes a translational motion along the same axis.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 11. Blutel Surfaces

Blutel surface

[aut] Blutel

Double Blutel conic surface

Double Blutel conic surface

is formed by a single-parametric family of the conics and simultaneously envelopes a two-parametric family of second-order cones.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 12. Veronese Surfaces

Let

M

be a

two-dimensional

Veronese surfaces

manifold;

σ

:

$$M \to S^{4} (1)$$

M

S

4

(

1

)

an immersion into the four-dimensional unit sphere of the real Euclidean space

R

5

.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 13. Tzitzéica Surfaces

The

centroaffine invariant

Tzitzéica surfaces

S. N. Krivoshapko, V. N. Ivanov

### Chapter 14. Peterson Surfaces

Peterson surface

[aut] Peterson K.M.

is a surface having a conjugate net of conical or cylindrical lines which are the main base of the bending. For example,

Monge surfaces with a circular cylindrical directrix surface

, the corresponding

translation surfaces

Translation surfaces

and

surfaces of revolution

are

Peterson surfaces

Peterson surfaces

. The indicatrix of rotations of Peterson surfaces is

right conoid

. In particular,

right helicoid

is the indicatrix for

carved surface

;

equilateral hyperbolic paraboloid

is the indicatrix for translation surface. First, this class of the surfaces was studied by K.M. Peterson as an example of surfaces assuming bending at the main base. Peterson (1866) has pointed at a class of surfaces capable to bend so that two appointed families of lines remain conjugated during all process of bending. Using his terminology, one may say that these lines are main base of bending for the considered surfaces.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 15. Surfaces of Bézier

An elementary

surface

of

Surfaces

of Bézier

Bézier

Elementary

surface of Bezier

is defined with the help of a vector equation:

$$\varvec{R}(u,v) = \sum\limits_{i = 0}^{m} {\sum\limits_{j = 0}^{n} {B_{i}^{m} (u)B_{j}^{n} (v)P_{ij} } } ,\;0 \,\le \,u \,\le \,1;\;0 \,\le \,v \,\le\, 1,$$

R

(

u

,

v

)

=

i

=

0

m

j

=

0

n

B

i

m

(

u

)

B

j

n

(

v

)

P

i

j

,

0

u

1

;

0

v

1

,

where

$$\varvec{P} = \left\{ {\varvec{P}_{ij} } \right\},\;i = 0,1, \ldots ,m;\;j = 0,1, \ldots ,n$$

P

=

P

i

j

,

i

=

0

,

1

,

,

m

;

j

=

0

,

1

,

,

n

is a given data;

$$B_{i}^{m} (u) = \left( {\begin{array}{*{20}c} m \\ i \\ \end{array} } \right)u^{i} \left( {1 - u} \right)^{m - i} ,\;B_{j}^{n} (v) = \left( {\begin{array}{*{20}c} n \\ j \\ \end{array} } \right)v^{j} \left( {1 - v} \right)^{n - j}$$

B

i

m

(

u

)

=

m

i

u

i

1

-

u

m

-

i

,

B

j

n

(

v

)

=

n

j

v

j

1

-

v

n

-

j

are polynomials of S.N. Bernstein.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 16. Quasi-ellipsoidal Surfaces

The forming of

quasi-ellipsoidal surfaces

Quasi-ellipsoidal surface with

Quasi-ellipsoidal surfaces

is based on mathematical transformations applied to a canonic equation of

ellipsoid.

V.A.

Nikityuk picked up

[aut] Nikityuk V.A.

three groups of quasi-ellipsoidal surfaces.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 17. Cyclic Surfaces

A cyclic surface

is formed by motion of a circle of variable or constant radius according to some law in the space (Fig.

1

).

S. N. Krivoshapko, V. N. Ivanov

### Chapter 18. One-Sided Surfaces

One-sided

and

two-sided

surfaces are two types of surfaces differing in the way of their disposition in the space. To be more correct, one-sided and two-sided surfaces are two types of varieties differing in the method of putting of them into complete space. For example, a cylinder is a two-sided surface and a Möbius strip is a one-sided surface, in spite of this, their physical models may be made from the same long rectangular strip. The main difference of these surfaces is the following: the boundary of a cylinder consists of two curves but the boundary of a Möbius strip consists of only one curve.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 19. Minimal Surfaces

Minimal surface

is a surface having the mean curvature

H

equal to zero at all points. Hence, minimal surface is a surface of negative Gaussian curvature. The extensive information on the initial stages of the investigations of

minimal surfaces

Minimal surfaces

is given at Mathematical encyclopedias, monographs and at numerous courses of differential geometry.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 20. Affine Minimal Surfaces

Affine minimal surface

is a surface with the affine mean curvature equal to zero. In contrast to ordinary minimal surfaces consisting only of the saddle points, the affine minimal surface may contain elliptic points. Thus,

the elliptical paraboloid

consists only of the elliptic points but it is an affine minimal surface.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 21. Surfaces with Spherical Director Curve

Surfaces with spherical director

curve

Surfaces

with spherical director curve

have a spherical curve

$$\varvec{E}_{0} \left( u \right) = \varvec{e}_{0} \left( u \right) = a\left( {\varvec{i}\cos u + \varvec{j}\sin u} \right)\cos \omega + \varvec{k}a\,\sin \omega ,$$

E

0

u

=

e

0

u

=

a

i

cos

u

+

j

sin

u

cos

ω

+

k

a

sin

ω

,

at the surface of a sphere of a radius

a

as a director curve;

$$\omega = \omega \left( u \right).$$

ω

=

ω

u

.

The unit vector

$$\varvec{e}_{0} \left( u \right)$$

e

0

u

is a normal of the sphere, at which the director curve is disposed.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 22. Weingarten Surfaces

The Weingarten surface

is a surface, the mean curvature

H

of which is connected with its Gaussian curvature

K

by a functional relation:

$$f\left( {H,K} \right) = 0.$$

f

H

,

K

=

0

.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 23. Surfaces of the Constant Gaussian Curvature

Gaussian curvature

of surface

К

is determined by a formula:

$$K = k_{1} k_{2} = \frac{{LN - M^{2} }}{{A^{2} B^{2} - F^{2} }}.$$

K

=

k

1

k

2

=

L

N

-

M

2

A

2

B

2

-

F

2

.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 24. Surfaces of the Constant Mean Curvature

“Soap bubble” may be called a physical system which is modeled by a surface of constant mean curvature in Euclidian three-dimensional space

R

3

.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 25. Wave-Shaped, Waving, and Corrugated Surfaces

Wave-shaped surfaces

are formed by translational-and-oscillatory motion of a rigid generatrix curve vibrating about a basic surface, a plane, or a line taken in advance. Hence, the generatrix curves of the wave-shaped surfaces are congruent to each other.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 26. Surfaces of Umbrella Type

A cyclic symmetrical spatial structure formed from several identical elements is called

an umbrella dome

. Curves obtained as a result of the intersection of their middle surfaces are the generatrix curves of any dome-shaped surface of revolution.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 27. Special Profiles of Cylindrical Products

Cylindrical products with various cross sections, i.e.,

profile

s, are widely used in civil engineering and different branches of machine building.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 28. Bonnet Surfaces

A surface permitting the isometric transformation with the preservation of the mean curvature is called

a

Bonnet surface

. V.

Lalan

[aut] Lalan V.

(1949) was the first who used the term “

Bonnet surface.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 29. Edlinger’s Surfaces

An oblique ruled surface

the osculating hyperboloids of which are

hyperboloids of revolution

(Fig.

1

) is called

an

Edlinger’s surface

Edlinger’s surfaces

. The Edlinger’s surfaces are characterized by the constant parameter of the distribution and have the lines of principal curvature as striction lines.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 30. Coons Surfaces

A

Coons surface

Coons surfaces

on any four given lines of the contour is determined by a sum of two linear surfaces which are constructed by the motion of a straight line above two corresponding opposite contour lines with the deduction of the oblique plane passed through the angular points of the contour (Fig. 1).

S. N. Krivoshapko, V. N. Ivanov

### Chapter 31. Surfaces Given by Harmonic Functions

A group of the surfaces of negative Gaussian curvature given in the explicit form by an equation.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 32. Surfaces of Joachimsthal

If one family of plane lines of the principal curvatures

v

of surface lies at the planes of a pencil, then this surface is called

a

surface of Joachimsthal.

S. N. Krivoshapko, V. N. Ivanov

are the generalization of

surfaces of negative Gaussian curvature

. A part of arbitrary surface of three-dimensional Euclidean space cut off by arbitrary plane with compact form closure of a contour section is called

a crust

. If we cannot cut off a crust by any plane, then this surface is a saddle surface. For a twice continuously differentiable surface to be a saddle surface, it is necessary and sufficient that at each point of the surface its Gaussian curvature is nonpositive. There are no

closed surfaces

E

3

.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 34. Kinematical Surfaces of General Type

A generatrix curve of

a kinematical surface of general type

transferring from one position to another can keep a certain character of motion but parameters of movements, positions of axes and the direction of infinitesimal displacements of the generatrix line simultaneously change.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 35. The Second Order Surfaces

The second order surfaces

are defined by algebraic equations of the second order relatively to the Cartesian coordinates. The second order surfaces are called also

or

.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 36. Algebraic Surfaces of the High Orders

Algebraic surface is a two-measured algebraic variety. A theory of algebraic surfaces is one of the sections of algebraic geometry.

S. N. Krivoshapko, V. N. Ivanov

### Chapter 37. Polyhedrons and Quasi-polyhedrons

In elementary geometry, a polyhedron (

plural polyhedra

or

polyhedrons

) is a body in three dimensions with flat faces, straight edges and sharp corners or vertices. The word “

polyhedron

” comes from the Classical Greek as poly- (“many”) and -hedron (form of “base” or “seat”).

S. N. Krivoshapko, V. N. Ivanov

### Chapter 38. Equidistances of Double Systems

A set of points

$$P_{ 1} ,P_{ 2} ,\; \ldots$$

P

1

,

P

2

,

equidistant from the figures

$$\varPhi_{1} ,\varPhi_{2} ,\; \ldots$$

Φ

1

,

Φ

2

,

” in the space

R

n

(

n

is a number of the measurements) is called

an equidistance

of the system

$$\varPhi_{1} - \varPhi_{2} - \cdots$$

Φ

1

-

Φ

2

-

” in

R

n

. In this definition,

a figure

is any nonempty set of points and the term “equidistance” is not connected with the same name concept in

the plane geometry of Lobachevski

and was introduced as a comfortable abridgement.

S. N. Krivoshapko, V. N. Ivanov

### Backmatter

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