Zeszyty Naukowe Politechniki Rzeszowskiej Nr 208 z 2004.pdf

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G. PROKOPSKI, J. HALBINIAK, B. LANGIER: Wpływ czasu dojrze-

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ZESZYTY NAUKOWE POLITECHNIKI RZESZOWSKIEJ

Nr 208

Budownictwo i Inżynieria Środowiska z. 36

2004

Jacek ABRAMCZYK

Rzeszów University of Technology

A NEW PROPOSAL FOR DETERMINATION

OF RULED SURFACES

The paper describes a uniform geometrical way of determination all ruled surfaces

including undevelopable ones. The presented way is based on the properties of

ruled surfaces known from projective and differential geometries but it is different

from the conventional methods. This way has been by the author called the way of

striction line.

1. Introduction

Each ruled surface is made up of infinitely large number of straight lines v i

called rulings [1, 2]. We assume that each ruled surface σ is determined if it is

possible to create every ruling v i of the ruled surface σ [3-5]. A set of geometri-

cal elements including points, lines, planes determining required ruled surface σ

is called the Γ figure determining the σ surface [4, 5]. We can determine each

ruling of σ on the basis of Γ and with the help of well-known operations [3-6].

For example, the Γ figure for a hyperbolical paraboloid can be made up of three

straight lines being skew lines in relation to each other. These lines also have to

be parallel to the same plane and are called directrices of σ. If we want to obtain

any ruling of the paraboloid, we ought to pass a plane through one directrix. This

plane intersects two other directices at two points determining the ruling being

sought. In the same way we can determine the others rulings.

Classical ways of determination of each type ruled surfaces applied in pro-

jective geometry are generally based on separating the ways of creating different

types of developable and undevelopable ruled surfaces [3-5], including the fol-

lowing:

1) various types of the Γ figures and different from each other rules of determi-

nation of the ruled surfaces,

2) different descriptions of creating both types of the ruled surfaces and then

general similarities between various types of these surfaces are shown.

In the paper, a new uniform method is proposed for the creation the σ ruled

surfaces and the Γ figure determining σ. This is a difference – geometric method

J. Abramczyk

the author called the one of striction line. In this method, the determination of all

sorts of the σ surfaces is generally different from the ones known from descrip-

tive [3-6] and differential [1, 2, 7, 8] geometries although it is based on both

classes of geometry. Therefore, it is possible to classify various ruled surfaces in

the way different from those used in teaching geometry at technical schools.

The author proposed to make capital of the presented method of determin-

ing the ruled surfaces for shaping corrugated building shells P b made up of many

deformed profiled sheets in the way describing particular geometric properties

appearing during deformation of the sheets [9-11]. A general ruled surface of

relatively free type can be an ideal geometric model for P b [9, 12]. Each sheet of

P b can be shaped as the central patch W i of a ruled paraboloid ω i , Fig. 1 [10, 13,

14]. The Σ structure having, as a rule, disturbances of the smoothness appearing

themselves along the segments common for each pair of the adjacent W i– 1 and W i

patches, is a result of the composition of these W i , Fig. 2 [12, 16]. Therefore, it is

the imperative to create the Σ structure of which configuration will be approxi-

mated to the σ ruled surface – the ideal model of P b [12, 16].

Fig. 1

Fig. 2

A new proposal for determination ...

2. Assumptions for the method of the striction line

First, the general vector equation of a ruled surface is presented:

uur ur uur

tm c t m p t

()

(1)

uuur – the position vector of the C i point of the c directrix of σ,

()

i ct

()

uur – the direct versor of the rulings of σ,

()

uuur

i ct and – the vectors dependent on the same t parameter as well as σ

[1, 2, 7, 8],

i pt

m – the parameter of the position of each C i point of the v i ruling.

From eq. (1) it follows that the Γ figure is made up of:

1) the c directrix,

2) the { p i } vector field – continuous set of the

i pt

versors starting from the

s point,

3) any interdependence between the C i points of c and the

same O

uuur

i pt

versors called

by the author the K definite homology, Fig. 3.

Fig. 3

The O s point is the center of the Φ sphere of elementary radius. A bundle

of all p i straight lines the author called the X p directing form. Each of the v i rul-

ings of the σ surface is received with the help of the X p directing form in the

following way. The K homology and the C i points of c and the

i pt

()

versors can

be assumed arbitrarily in continuous way. Each v

i ruling is a straight line passing

σ = + ⋅

(, ) ()

where:

i pt

()

uuur

()

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