94

ISSN 1064–5624, Doklady Mathematics, 2009, Vol. 79, No. 1, pp. 94–96. © Pleiades Publishing, Ltd., 2009.Original Russian Text © V.I. Berdyshev, 2009, published in Doklady Akademii Nauk, 2009, Vol. 424, No. 5, pp. 588–590.

1. An observed object moves in the space

R

3

with anobstacle in the form of a closed solid set

G

⊂

R

3

thatcan impede the visibility of the object. An example isthe motion of an object over the ground surface, inwhich case

G

is the ordinate set of the function definingthe relief.

Let the moving object be at a point

t

∈

R

3

\

G

and anobserver be at a point

f

∈

R

3

. The points

t

and

f

are vis-ible to one another; i.e., the interval (

t

,

f

) with its end-points at

t

and

f

does not intersect

G

. Let

0

≤

r

≤

ρ

, where

ρ

=

|

t

–

f

|

,

|

·

|

is the Euclidean norm,

K

r

(

t

,

f

) =

conv

(

V

r

(

t

)

∪

f

)

is the convex hull of the point

f

and the ball

V

r

(

t

) =

{

v

:

|

t

–

v

|

≤

r

}

, and

α

=

. In what follows, we

assume that

r

=

r

(

t

) =

r

(

t

,

f

,

G

)

and

α

=

α

(

t

) =

α

(

t

,

f

,

G

)

are the largest radius

r

and the largest angle

α

for which

where

is the interior of the set

K

r

. The functions

r

(

t

)

and

α

(

t

)

are viewed as visibility characteristics of thepoint

t

. The introduction of these characteristics ismotivated as follows. Assume that an object moving ata constant velocity

v

t

passes through the point

t

, and anobserver moves at the velocity

v

f

. If

v

t

≤

v

f

si

n

α

, thenthe observed object does not escape the ball

V

r

(

t

)

over

the time

τ

=

required for the observer to travel from

the point

f

to

t

, since

v

t

·

τ

=

v

t

·

≤

ρ

sin

α

=

r. More-

over, since (t) ∩ G = , the observed object isalways visible to the observer.

The function r(t) is continuous and satisfies the Lip-schitz condition

(1)

rρ---arcsin

Kr° t f,( ) G∩ , Kr t f,( )\ f( ) G∩ ,≠=

r t( ) t f– α t( ),sin=

Kr°

ρv f

------

ρv f

------

Vr°

r t( ) r T( )– t T– , T Vr t( ).∈≤

Along with r(t) and α(t), we introduce an averagedvisibility characteristic of a point. Let ∆(t) = {l} be the setof rays issuing from the origin such that (t + l) ∩ G ≠ .For a ray l ∈ ∆(t), the point of the set (t + l) ∩ G that isthe nearest to t is denoted by ft, l. The averaged visibilityof a point t is defined as

2. One-sided differentiation of r(t). When a route ofmotion of an object is planned, it is important to knowthe directional derivative of vis(t). This derivative iscalculated in terms of the derivatives of r(t) and α(t).Consider the differentiation of r(t) and α(t) with respectto any direction , | | = 1. We use the notation

The boundary bdKr(t, f ) of the set Kr(t, f ) is the unionof two surfaces: spherical sr(t) and conical κr(t). Specif-ically, sr(t) is the closure of the set of points of thesphere Sr = Sr(t) = {v: |t – v| = r} that are invisible fromf, i.e., the closure of the set {v ∈ Sr: (v, f ) ∩ Sr ≠ }and κr(t) = κr(t, f ) = (bdKr(t, f))\sr(t). The points of(Kr(t, f )\ f ) ∩ G can lie on sr(t) or κr(t).

Given a point f and g ∈ Vρ(t), define the function

Then

Lemma. The function R(t, g) is continuous and hasa directional derivative in any direction that is alsocontinuous in both variables.

The proof of the lemma is based on the followingfact: if g ∈ sr(t)\ (t, f ), where is the closure of κr,then

(2)

for points T from the neighborhood of t; if g ∈ κr(t, f ),then, for points T from the neighborhood of t, we have

vis t( ) α t f t l, G, ,( )sin l.d

∆ t( )∫=

t̃ t̃

r' t( ) ∂r t( )∂ t̃

------------r t λ t̃+( ) r t( )–

λ------------------------------------.

λ +0→lim= =

R t g,( ) = sup r: g Kr t f,( )∉{ } = min r: g Kr t f,( )∈{ }.

r t( ) min R t g,( ): g G∈{ }.=

κr κr

R T g,( ) T g–=

MATHEMATICS

Visibility Characteristic of a Moving PointCorresponding Member of the RAS V. I. Berdyshev

Received September 22, 2008

DOI: 10.1134/S1064562409010281

Institute of Mathematics and Mechanics, Ural Division, Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620219 Russia; e-mail: bvi@imm.uran.ru

DOKLADY MATHEMATICS Vol. 79 No. 1 2009

VISIBILITY CHARACTERISTIC OF A MOVING POINT 95

(3)

where

Lg = {f + ϑ(g – f ): ϑ > 0} and

d(T, Lg) distance from T to a ray Lg;

if g ∈ sr(t) ∩ (t, f), then (2) and (3) hold for T fromthe neighborhood of t if (T – g, f – g) ≥ 0 and (T – g, f –g) < 0, respectively. Here, (·, ·) denotes the scalar prod-uct of vectors.

Theorem 1. The function r = r(t) is differentiable inany direction , | | = 1. Moreover,

(4)

where G(t) = {g ∈ G: R(t, g) = r(t)}.

The proof is based on a theorem of Dem’yanov’s [1,Ch. 4, Section 2] about the differentiation of the maxi-mum function. Note that V.S. Balaganskii proved thedirectional differentiability of the function withoutusing (4)–(6) in the case of a Hilbert space (the proof isnow in press).

Clearing theorems. By clearing, we mean a formalreduction of an optimization problem to an equivalentone that is free of the excessive conditions. Foe exam-ple, by Chebyshev’s classical theorem (see, e.g., [2]),the best approximation of a continuous function on aninterval by nth-degree polynomials is reduced to anapproximation problem on an (n + 2)-point subset ofthe interval. The complexity of problem (4) depends onthe cardinality of the set G(t). In the simplest case,when g – t and are parallel vectors, the value of (5)with g ∈ sr(t) is equal to cosγg, where γg is the anglebetween the vectors t – g and , and the value of (6)with g ∈ κr(t) is equal to sinα(t), where α(t) is the anglebetween the vectors g – f and t – f. Therefore,

In an arbitrary case, we extract from G(t) a subset ofpoints g for which

R T g,( ) d T Lg,( ),=

κr

t̃ t̃

∂r t( )∂ t̃

------------ min∂R t g,( )

∂ t̃-------------------: g G t( )∈

⎩ ⎭⎨ ⎬⎧ ⎫

,=

∂R t g,( )∂ t̃

------------------- =

ddλ------ g t– λ t̃– λ 0= , if g sr t( )∈

ddλ------d t λ t̃+ Lg,( )λ 0= , if g κr t( ),∈⎩

⎪⎨⎪⎧ 5( )

6( )

t̃

t̃

∂r t( )∂ t̃

------------ minγ g: gcos sr t( ) G∩∈α t( )sin : g κr t( ) G.∩∈⎩

⎨⎧

=

∂r t( )∂ t̃

------------R t λ t̃+ g,( ) r–

λ------------------------------------.

λ +0→lim=

Define

Since r' = , there exists a converging

sequence of points gλ ∈ G for which

Let g = . Clearly, g ∈ G(t). The following

cases are possible:

(i) gλ ∈ (tλ), i.e., |gλ – tλ| = rλ.

(ii) gλ ∈ (tλ); i.e., d(tλ, ) = rλ.

The following result holds in case (i).Theorem 2. The point g lies on the circle C in the

intersection of the sphere Sr(t) with the plane throughthe point t + (r · r') and orthogonal to the vector .Moreover,

By virtue of (1), we have |r'| ≤ 1. Therefore, thisintersection is nonempty.

Consider case (ii). Denote by γ the angle betweenthe vectors t – f and , < γ < π. In the plane Z throughthe point t and orthogonal to the vector f – t, we intro-duce a Cartesian coordinate system with the origin atthe point t and with the vertical axis directed along theprojection of onto Z. On the circle x2 + y2 = R2 in thisplane, we choose two points v+ and v– with the sameordinates equal to

where

and choose two rays = { f + ϑ(v± – f ): ϑ > 0},

which contain these points. Let g± = ∩ Sr(t).

Theorem 3. The point g lies on either the ray

or and

r r t( ), tλ t λ t̃ , rλ+ r tλ( ), r'∂r t( )

∂ t̃------------.= = = =

rλ r–λ

------------λ +0→lim

r'R tλ gλ,( ) r–

λ-----------------------------.

λ +0→lim=

gλλ +0→lim

srλ

κrλLgλ

t̃ t̃

∂r t( )∂ t̃

------------tλ g– r–

λ------------------------.

λ +0→lim=

t̃

t̃

rρ r γcos ρr'–( )ρ2 r2–( ) γsin

--------------------------------------,

Rrρ

ρ2 r2–--------------------, ρ t f– ,= =

Lv

±

Lv

±

Lv

+

Lv

–

∂r t( )∂ t̃

------------d tλ Lg,( ) r–

λ-----------------------------.

λ +0→lim=

96

DOKLADY MATHEMATICS Vol. 79 No. 1 2009

BERDYSHEV

Corollary. Theorems 2 and 3 imply that

To conclude, we present a formula for the differen-tiation of sinα(t, f, G) that involves r'.

Theorem 4. Let γ be the angle between the vectorst – f and . Then

The directional derivative of vis(t) is calculatedusing Theorems 1–4.

ACKNOWLEDGMENTSThis work was supported by the Russian Foundation

for Basic Research (project no. 08-01-00325) and theprogram “Leading Scientific Schools” (projectno. 1071.2008.1).

REFERENCES1. V. F. Dem’yanov and V. N. Malozemov, Introduction to

Minimax (Nauka, Moscow, 1972) [in Russian].2. N. I. Akhiezer, Lectures in the Theory of Approximation

(Ungar, New York, 1956; Nauka, Moscow, 1965).

∂r t( )∂ t̃

------------∂R t g,( )

∂ t̃------------------- g∀ C g+ g–.∪ ∪∈=

t̃

α t λ t̃+ f G, ,( )sin α t f G, ,( )sin–λ

----------------------------------------------------------------------------------λ +0→lim

= 1ρ--- ∂r t( )

∂ t̃------------ α t f G, ,( ) γcossin+⎝ ⎠

⎛ ⎞ .