MODELLING THE PROPERTIES OF ELLIPTICITY: LINEAR VARIATIONS

(1)

ɪɚɡɨɜɚɧɢɣɫɞɜɢɝɚɢɝɨɦɨɬɟɬɢɢ. ȼɨɤɪɟɫɬɧɨɫɬɢɧɟɤɨɬɨɪɨɣɬɨɱɤɢɷɥɥɢɩɫɚɩɨɤɚɡɚɧɵɩɪɨɯɨɞɹɳɢɟ ɱɟɪɟɡɧɟɟɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɢɟɥɢɧɢɢ, ɤɨɬɨɪɵɟɦɨɝɭɬɛɵɬɶɢɫɩɨɥɶɡɨɜɚɧɵɜɬɟɯɢɥɢɢɧɵɯɤɨɦ -ɩɨɡɢɰɢɨɧɧɵɯɪɟɲɟɧɢɹɯɞɥɹɮɪɚɝɦɟɧɬɨɜɩɪɨɟɤɬɢɪɭɟɦɵɯɨɛɴɟɤɬɨɜ.

Ʉɥɸɱɟɜɵɟɫɥɨɜɚ:ɦɨɞɟɥɢɪɨɜɚɧɢɟ, ɷɥɥɢɩɬɢɱɧɨɫɬɶ, ɤɜɚɞɪɢɤɚ, ɮɨɤɚɥɶɧɵɟɩɚɪɚɦɟɬɪɵ, ɤɨɧɮɢɝɭɪɚɰɢɢ, ɤɨɧɫɬɚɧɬɚɩɟɪɢɦɟɬɪɢɢ, ɝɟɨɦɟɬɪɨɝɪɚɮɢɹ, ɡɨɥɨɬɚɹɩɪɨɩɨɪɰɢɹ.

.

:

.

-

: «

;

—

,

?» [1].

.

«

»

Ⱥ

ȼ

.

:

,

?

-:

.

,

.

-,

,

-,

.

,

-,

-.

,

«

»,

.



b

R

;



b

R



(

. 1)

(0

0

⋅

R

)

"(0; )

R

(0,618

R

)

y

,

«

»

,

-.

«

»

e

i

,

«

--

» (1; 2; 3; 4),

i

R

.

i

P

_«

_»,

—

e

i

.

,

e

i

,

«

ȿ

—

».

-; 2

F

a



(2)

. 1.

i

e

-y

x

.

«

»

R

i

,

e

i

—

.

(0;

R

i

)

e

i

—

-i

R

_[3].

_«

-»

(0)

(

R



R

i

)

x

P

R

.

( ;

b P

R

)

«

»

R

i

e

i

.

i

R

«

»

,

.

i

e

( ; )

t n

i i

.

R

i

,

i

P

(

−

F P

;

i

)

4. (4

 

e

i

P

i

)

( ),

t

i

( ),

n

i

-( )

t

i

(3

→ −

(

F

)

→

4),

e

i

.

[2].

-.

.

,

.

-.

,

(

₋

F F

; )

₍

_{. 2),}

_e

*

(3)

. 2.

,

-,



.

(

.

. 1)

,

.

—

( ;

R y

i i

),

.

a

-x

,

y

_i

R

.

∆

y

( ; ).

b R

,

a

,

y

7. (7;

x

R

i

)

,

(0)

i

R

e

i

.

,

(7;

x

R

i

)

(0; ).

e

i

-,

(0; )

e

i

{

R

Η ⋅ τ

x

}

.

«

»,

«

»

x

.

-

[4].

,

e

i

,

-,

«

-»

«

».

(

. 3).

-,

,

(4)

. 3.

Ȼɢɛɥɢɨɝɪɚɮɢɱɟɫɤɢɣɫɩɢɫɨɤ

1. ȽɢɥɶɛɟɪɬȾ., Ʉɨɧ-Ɏɨɫɫɟɧɋ. . . : , 1981.

2.ɉɨɥɟɠɚɟɜɘ.Ɉ. - : . . : - , 2010.

3. ȽɢɥɶɛɟɪɬȾ. . . : , 1948. 4. ɄɨɪɧȽ. . . : , 1974.

5. ɋɚɩɪɵɤɢɧɚ ɇ.Ⱥ. . . : - , 2005.

ɉɨɫɬɭɩɢɥɚɜɪɟɞɚɤɰɢɸɜɦɚɟ 2012 ɝ.

: ɉɨɥɟɠɚɟɜɘɪɢɣɈɥɟɝɨɜɢɱ —

-, ɎȽȻɈɍȼɉɈ «Ɇɨɫɤɨɜɫɤɢɣɝɨɫɭɞɚɪɫɬɜɟɧɧɵɣɫɬɪɨɢɬɟɥɶɧɵɣɭɧɢɜɟɪɫɢɬɟɬ» (ɎȽȻɈɍ

ȼɉɈ «ɆȽɋɍ»), 129337, . , , . 26, (499) 183-24-83, graﬁ ka@mgsu.ru;

Ȼɨɪɢɫɨɜɚ Ⱥɧɠɟɥɢɤɚɘɪɶɟɜɧɚ — , ,

, ɎȽȻɈɍȼɉɈ «Ɇɨɫɤɨɜɫɤɢɣɝɨɫɭɞɚɪɫɬɜɟɧɧɵɣɫɬɪɨɢ

-ɬɟɥɶɧɵɣɭɧɢɜɟɪɫɢɬɟɬ» (ɎȽȻɈɍȼɉɈ «ɆȽɋɍ»), 129337, . , , . 26, (499) 183-24-83, graﬁ ka@mgsu.ru.

: ɉɨɥɟɠɚɟɜɘ.Ɉ., ȻɨɪɢɫɨɜɚȺ.ɘ. // . 2012. № 8. . 34—38.

Yu.O. Polezhaev, A.Yu. Borisova

MODELLING THE PROPERTIES OF ELLIPTICITY: LINEAR VARIATIONS

The authors discuss some of the properties of linear variations of ellipticity within the frame-work of planimetry. Six elliptic models were constructed through the employment of geometrogra-phy-related methods: an ellipse interrelated with the (i) “golden proportion” and a (ii) focal plane rectangle; (iii) a constant of the perimetry of the focal diamond; (iv) compression of the base circle in the axial direction (y); (v) differential straight lines of the moving point of an ellipse; (vi) compass incidence, a composition of transformations of the shift and homothety.

Characteristic lines that run in the neighborhood of some point of the ellipse are demonstrated. The characteristic lines in question include those that can be employed as part of various composite solutions related to the fragments of structures being constructed.

A set of closed polygons and curves with selected lines passing through the characteristic points of the circle squaring — these are the geometrographic structures that can form the basis of composite solutions to the problem of design. The authors also believe that the properties employed by the “golden mean” increase the aesthetic constituent of the solution.

(5)

ment of Descriptive Geometry and Graphics, Moscow State University of Civil Engineering (MGSU),

26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; graﬁ ka@mgsu.ru; +7 (499) 183-24-83.