Helical Surfaces

Helical surface can be created by a helical movement of a curve.

Synthetic representation: (* k*,**T**_{S}(*v*) )

Analytic representations:

basic figure - **r**(*u*)=(*x*(*u*), *y*(*u*), *z*(*u*),1), *u*Î<0,1>

generating principle - helical movement with the axis in the coordinate axis *z*

determined by translation vector **a** =( 0, 0, a*z*_{0}, 1 )

corresponding to the revolution angle 1 radian

T_{s}(*v*)=
,* v*Î<0,1>

modelled figure -

**p**(*u*, *v*) = **r**(*u*).**T**_{s}(*v*) = (*x*(*u*)cos a*v*-*y*(*u*)sin a*v*, *x*(*u*)sin a*v*-*y*(*u*)cos a*v*, *z*(*u*)+a*z*_{0}*v*, 1),
(*u*, *v*)Î<0,1>^{2 }

*k*0, *z*_{0}0 is the reduced pitch,
one pitch of the helical surface can be created for a=2p

Parametric *u*-curves on the surface are equal to the basic curve, parametric *v*-curves are helices.

Neck and equator helices have the extremal radii.

in the direction of the helical movement axis

can be graphically represented by Archimedean spiral (Fig. 4. 101).

Polar equation of this curve r=

represents the increment of the

r = * z*_{0} j* v*,
one pitch of the spiral can be created for a=2p.

Axis of the helical movement is usually located, in the Monge method,

to the line parallel to the coordinate axis *z*.

Movement can be determined by the pitch *z _{v}*=2p

Helical surfaces can be distributed with respect to the basic figure to:

**line** (line or its part is subdued to the helical movement),

**cyclical** (cicle or its part is subdued to the helical movement),

**general** (an arbitrary curve or its part is subdued to the helical movement).

Line helical surfaces are:

opened orthogonal …..opened klinogonal7nbsp; ….. closed orthogonal ….. closed klinogonal

If two points on the basic line move on one helix, we speak about surface of bisecants:

opened klinogonal …..closed klinogonal

The only one developable helical surface is the surface of tangents to the helix.

Cyclical helical surface are:

**Vinded column** - basic circle is located in the plane perpendicular to the axis of the helical movement

**Arch surface** - basic circle is located in the plane passing through the axis of the helical movement

**Archimedean serpentine** - basic circle is located in the plane perpendicular to the tangent line to the trajectory of the helical movement - helix of the circle centre

**Tangent plane** in the point on the helical surface is determined by

tangent lines to two curves located on the surface in the tangent point,

trajectory of the movement - helix, and

basic curve in the position moved to the tangent point.

Helix

the point bod

where the helical movement started.

Revolving the point

the related translation

by the development of the point trajectory - helix

using the reduced pitch z

Tangent line

will move in the helical movement

to the tangent line

(its point

Tangent line

by means of the conical surface of tangents to the helix,

where line

can be found.

**Tangent plane to the line helical surface is tangent to the surfacein the basic line.**

Moved position of the line to the tangent point *T* is one line in the tangent plane,

the other one is the tangent line to the trajectory - helix of the tangent point *T*.

In the construction of planar intersections of helical surfaces we usually use only two types of intersection planes:

a) **normal intersection** by plane perpendicular to the surface axis - normal plane,

b) **meridian intersection** by plane passing through the surface axis - meridian plane.

Intersection curves are constructed as sets of separate points,

which are intersections of surface helices with the intersection plane.

Meridian intersection of the anticlockwise helical surface determined by

the basic curve* k* and the pitch *z _{v}*

Points on the basic curve move in the helical movement about the axis

Movement is composed from

the revolution about the axis

and translation by vectors

Coordiantes

related to the revolution angles a

Normal intersection of the clockwise helical surface determined by

the basic curve *k* and pitch *z _{v}* with the plane r is in the Fig. 4.113.

Helical movement can be decomposed to

translation in the direction of the vectors

and revolution in the normal plane r perpendicular to the surface axis.

Length of translation

corresponds to the angle of revolution a