Todays Programme

1
Todays Programme

• Determination of Helical Surface
• Important Curves
• Classification of Helical Surfaces
• Examples of Ruled Helical Surfaces
• Examples of Circular Helical Surfaces
• Ruled Helical Surface in Monge Projection

2
Helical Surfaces
3
Determination of Helical Surface

• Helical Motion (axes, hand, lead).
• Generating curve (planar or space not a helix
  generated by the helical motion not straight a
  line parallel to the axes).
• Helical surface is a union of all helices
  generated by the points of the generating curve.
• gt a helix and the curve (revolved and translated
  generating curve) pass through each point of the
  surface

4
Important Curves
Meridian intersection of the helical surface
with an axial plane.
Cross section intersection of the helical
surface with a plane perpendicular to the axis of
helical motion.
5
Important Curves
Gorge helix helix of a point in generating
curve that is nearest to the axis (if the
generating curve does not intersect the axis) gt
the helix with the steepest slope. Equatorial
helix helix of a point in the generating curve
that is farthest from the axis gt the helix with
the gentlest slope.
6
Classification of Helical Surfaces
(with respect to the generating curve)
ruled (straight line p)
circular (circle k)
skew (p?o)
right (p?o)
closed (pno?Ø)
open (pnoØ)
(according to the relative position of the
generating straight line and the axes of the
helical motion)
(according to the relative position of the
generating straight line and the axes of the
helical motion)
7
Classification of Helical Surfaces
(with respect to the generating curve)
ruled (straight line p)
circular (circle k)
closed (kno?Ø)
open (knoØ)
(according to the relative position of the
generating circle and the axes of the helical
motion)
8
Examples of Ruled Helical Surfaces
Right helical conoid helicoid Conoid directing
straight line axis directing curve
helix directing plane perpendicular to the axis
Helical Stairs Auger
9
Examples of Ruled Helical Surfaces
Skew Closed Ruled Helical Surface Roof of
helical stairs. Gimlet, drill bit, Withwords
thread (cross section of line and circular
segments).
10
Examples of Ruled Helical Surfaces
11
Examples of Circular Helical Surfaces
Circle is Meridian Adornment of Columns
especially in Baroque period. Chute for granular
(sand, grit) and powdery materials. Vault of a
helical stairs (meridian half of
circle). Gimlet, drill bit, Withwords thread
(meridian line and circular segments).
12
Examples of Circular Helical Surfaces
Circle is Cross Section Adornment of Columns
especially in Baroque period. Chute for granular
(sand, grit) and powdery materials.
13
Examples of Circular Helical Surfaces
Circle in Normal Plane of Helix Adornment of
Columns especially in Baroque period. Helical
Canal Surface. Spring. Helical Ball
Bearing.
14
Ruled Helical Surface in Monge Projection
Draw one turn (13 positions of AB well
proportioned) of a surface generated by line
segment AB in helical motion. Monge projection
O 10, 12 Helical motion o ? (x, y) o1
0, 6, 0 right-hand v 12 Line segment A
5, 6, 0 B -2, 6, 2.5
15
Ruled Helical Surface in Monge Projection