Intersections & Developments
(Text Chapter 31)
UAA ES A103
Week #11 Lecture
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Introduction
• Intersections and developments are commonly
found in many engineering disciplines.
• Two surfaces that meet to form a line of
intersection.
• A development is the outside surface of a
geometric form laid flat.
• As an engineer you should be able to create the
development of common shapes, such as cones,
prisms, and pyramids.
Visibility
• Visibility is the clear and correct
representation of the relative positions or
two geometric figures, in multiview
drawings.
More Visibility
• In each view, the visibility of the figures is
indicated by drawing the figure that is front
with object lines while drawing the second
figure with both object lines and hidden
lines.
Intersections
• An intersection is a point or line where two
geometric forms, such as lines or surfaces,
meet or cross each other.
Intersecting lines
• Two lines that intersect share a common point.
– If the lines do not have a common point that projects
from view to view, the lines are nonintersecting.
Intersection of a Line
and a Plane
• The intersection of a line and a plane is
referred to as the piercing point.
• A line will intersect a plane if the line is
not parallel to the plane.
• Locations of piercing points can be found
using either the EDGE VIEW or
CUTTING PLANE methods.
Edge View Method
• Create an auxiliary view showing the plane
as an edge.
• Project the intersecting point of the LINE
to all views
Cutting Plane Method
• Create a cutting plane that includes the line and is an edge
view in one of the given views.
• Trace the cutting plane intersection onto the other view.
• The piercing point is where the cutting plane intersection
with the plane intersects the line.
Intersection of Two Planes
• The intersection of two planes is a straight
line all of whose points are common to
both planes.
• The line of intersection between two planes
is determined by locating the piercing
points of lines from one plane with the
other plane and drawing a line between the
points.
Edge
View
Method
Cutting
Plane
Method
Intersection of a Plane
and A Solid
• To find the intersection between a plane
and a solid,
– determine the piercing points using either
cutting planes or auxiliary views,
– then draw the lines of intersection.
A Plane & A Prism
A Plane & A Cylinder
A Plane & A
Cone
Intersection of Two Solids
• Solids are represented by lines, planes and
curved surfaces.
• The basic principles of lines intersecting
planes apply here.
– Piercing points caused by lines one once
surface intersecting another surface can be
found in the various views and be connected
with lines.
Two Prisms
• Prisms are
made up of
planar surfaces
so all
intersections
will be
straight lines.
– This means
that you only
need to find
the ends of
each
intersection
line.
A Prism & A Pyramid
Intersections in Axonometric,
Oblique and Perspective Drawings
• The author stops short of this discussion.
• The same principles apply.
• If one given view shows edge views of one
of the intersecting planes then only two
views are required. Otherwise an auxiliary
view may be required.
Axonometric & Oblique Views
• May need to draw the multi-view
equivalents on the principle faces.
• Use cutting planes to find piercing
points then connect the piercing
point.
Oblique
Example
Developments
• A development is the unfolded or unrolled, flat or
plane figure of a 3-D object.
– Called a pattern, the plane figure may show the true
size of each area of the object. When the pattern is
cut, it can be rolled or folded back into the original
object.
• A true development is one in which no stretching
or distortion of the surface occurs, and every
surface of the development is the same size and
shape as the corresponding surface on the 3-D
object.
– Only polyhedrons and single-curved surfaces can
produce true developments.
Approximate Development
• An approximate development is one in which
stretching or distortion occurs in the process of
creating the development.
• The resulting flat surfaces are not the same size
and shape as the corresponding surfaces on the 3D object.
• Warped surfaces do not produce true
developments, because pairs of consecutive
straight-line elements do not form a plane.
• Double-curved surfaces such as spheres do not
produce true developments.
Types of Developments
•
•
•
•
Parallel-line
Radial-line
Triangulation
Approximation
Examples
Some Observations
• All lines are true length
• All planes are true size
• Each planar surface is either a principle
view or an auxiliary view.
Inside Pattern Development
Development of a Right
Rectangular Prism
Truncated Prism / Cylinder
Truncated Cylinder
REMEMBER!
All lines are
TRUE LENGTH
All planar surfaces are
TRUE SIZE
A Right Circular Cone
Truncated Right Circular Cone
Oblique Cone
Truncated Pyramid
Oblique Prism
Oblique Cylinder