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 3-D 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