Geometric Construction

1
Geometric Construction

• Engineering Graphics
• Stephen W. Crown Ph.D.

2
Objective

• To review basic terminology and concepts related
  to geometric forms
• To present the use of several geometric
  tools/methods which help in the understanding and
  creation of engineering drawings

3
Overview

• Coordinate Systems
• Geometric Elements
• Mechanical Drawing Tools

4
Coordinate Systems

• Origin (reference point)
• 2-Dimensional Coordinate System
• Cartesian (x,y)
• Polar (r,q)
• 3-Dimensional Coordinate System
• Cartesian (x,y,z)
• Cylindrical (z,r,q)
• Spherical (r,q,f)

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Cartesian Coordinate System

• Defined by two/three mutually perpendicular axes
  which intersect at a common point called the
  origin
• x-axis
• horizontal axis
• positive to the right of the origin
• y-axis
• vertical axis
• positive above the origin
• z-axis (added for a 3-D coordinate system)
• horizontal axis perpendicular to the x-axis
• positive in front of the origin

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Polar Coordinate System

• The distance from the reference point (origin) is
  specified as the radius (r)
• The angular position is specified (q)
• The angle is measured from a line extending
  horizontally to the right from the origin
• Positive angles are measured in a
  counter-clockwise direction
• Conversion between Cartesian and polar
• xrcos q , yrsin q
• x2y2r2 , qtan-1(y/x)

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Cylindrical Coordinate System

• Same as polar except a z-axis is added which is
  perpendicular to the plane in which angle q is
  measured
• The direction of the positive z-axis is defined
  by the direction of positive rotation in the
  perpendicular plane (right hand rule)
• Useful for describing cylindrical features

8
Spherical Coordinate System

• The distance from the reference point (origin) is
  specified as the radius (r)
• The projected angular position in one plane is
  specified as q
• The projected angular position in a perpendicular
  plane is specified as f
• Positive angles are measured in a
  counter-clockwise direction
• Conversion between Cartesian and spherical
• xrcosfcosq , yr cosfsin q , z rsinf

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Right Hand Rule

• Thumb points in the positive x direction
• Index finger points in the positive y direction
• Middle finger points in the positive z direction
• Used to define positive rotation
• Point thumb in the positive direction along the
  axis which is perpendicular to the plane of
  rotation
• The fingers point in the direction of positive
  rotation

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Redefining Coordinates

• Absolute coordinates
• measured relative to the origin
• LINE (1,2,1) - (4,4,7)
• Relative coordinates
• measured relative to a previously specified point
• LINE (1,2,1) - _at_(3,2,6)
• World Coordinate System
• a stationary reference
• User Coordinate System (ucs)
• change the location of the origin
• change the orientation of axes

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Geometric Elements

• A point
• A line
• A curve
• Planes
• Closed 2-D elements
• Surfaces
• Solids

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A Point

• Specifies an exact location in space
• Dimensionless
• No height
• No width
• No depth

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A Line

• Has length and direction but no width
• All points are collinear (straight lines)
• May be infinite
• At least one point must be specified
• Direction may be specified with a second point or
  with an angle
• Ray (end point and direction)
• May be finite
• Defined by two end points
• Defined by one end point, a length, and direction

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A Curve

• The locus of points along a curve are not
  collinear
• The direction is constantly changing
• Single curved lines
• all points on the curve lie on a single plane
• A regular curve
• The distance from a fixed point to any point on
  the curve is a constant
• Examples arc and circle

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Planes

• A two dimensional slice of space
• No thickness (2-D)
• Any orientation defined by
• 3 points
• 2 parallel lines
• a line and a point
• 2 intersecting lines
• Appears as a line when the direction of view is
  parallel to the plane

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Closed 2-D Elements (planar)

• Triangles
• Three sides
• Equilateral triangle (all sides equal, 60 deg.
  angles)
• Isosceles triangle (two sides equal)
• Right triangle (one angle is 90 degrees)
• A2B2C2 (Pythagorean theorem)
• SinqA/C
• CosqB/C

C
A
q
B
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Closed 2-D Elements (planar)
R

• Circles
• Radius (R)
• Diameter (D)
• Angle (1 rev 360o 0 0)
• Circumference (23.14159R)
• Tangent
• Chord
• A line perpendicular to the midpoint of a chord
  passes through the center of the circle
• Concentric circles

q
D
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Closed 2-D Elements (planar)

• Parallelograms
• 4 sides
• Opposite sides are parallel
• Ex. square, rectangle, and rhombus
• Regular polygons
• All sides have equal length
• 3 sides equilateral triangle
• 4 sides square
• 5 sides pentagon
• Circumscribed or inscribed

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Surfaces

• Does not have thickness
• Two dimensional at every point
• No mass
• No volume
• May be planar
• May be used to define the boundary of a 3-D
  object

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Solids

• Three dimensional
• They have a volume
• Regular polyhedra
• Have regular polygons for faces
• All faces are the same

• Prisms
• Two equal parallel faces
• Sides are parallelograms
• Pyramids
• Common intersection point (vertex)
• Cones
• Cylinders
• Spheres

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Useful Tools From Mechanical Drawing Techniques

• Drawing perpendicular lines (per_)
• Drawing parallel lines (offset)
• Finding the center of a circle (cen_)
• Some difficult problems for someone who
  completely relies on AutoCAD tools
• Block with radius
• Variable guide
• Offset pipe
• Transition