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Geometric Construction. Engineering Graphics Stephen W. Crown Ph.D. 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. Overview.

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geometric construction

Geometric Construction

Engineering Graphics

Stephen W. Crown Ph.D.

  • 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
  • Coordinate Systems
  • Geometric Elements
  • Mechanical Drawing Tools
coordinate systems
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)
cartesian coordinate system
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 rightof the origin as shown
    • y-axis
      • vertical axis
      • positive above the origin as shown
    • z-axis (added for a 3-D coordinate system)
      • normal to the xy plane
      • positive in front of the origin as shown
review right hand rule
Review: Right Hand Rule
  • Your thumb, index finger, and middle finger represent the X, Y, and Z axis respectively.
  • Point your thumb in the positive axis direction and your fingers wrap in the direction of positive rotation
polar coordinate system
Polar Coordinate System
  • The distance from the origin to the point in the xy planeis specified as the radius (r)
  • The angle measured form thepositive x axis is specified as q
  • Positive angles are defined according to the right hand rule
  • Conversion between Cartesian and polar
    • x=r*cos q , y=r*sin q
    • x^2+y^2=r^2 , q=tan-1(y/x)
cylindrical coordinate system
Cylindrical Coordinate System
  • Same as polar except a z-axis is added which is normal to the xy plane in which angle q is measured
  • The direction of the positive z-axis is defined by the right hand rule
  • Useful for describing cylindrical features
spherical coordinate system
Spherical Coordinate System
  • The distance from the origin is specified as the radius (r)
  • The angle between the x-axis andthe projection of line r on the xy plane is specified as q
  • The angle between line r and thez-axis is specified as f
  • Positive angles of q are defined according to the right hand rule and the sign of f does not affect the results
  • Conversion between Cartesian and spherical
    • x=r*sinf*cosq , y=r *sinf*sin q , z= r*cosf
redefining coordinates
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) - @(3,2,6)
  • World Coordinate System
    • a stationary reference
  • User Coordinate System (ucs)
    • change the location of the origin
    • change the orientation of axes
geometric elements
Geometric Elements
  • A point
  • A line
  • A curve
  • Planes
  • Closed 2-D elements
  • Surfaces
  • Solids
a point
A Point
  • Specifies an exact location in space
  • Dimensionless
    • No height
    • No width
    • No depth
a line
A Line
  • Has length and direction but no width
  • All points are collinear
  • May be infinite
    • At least one point must be specified
    • Direction may be specified with a second point or with an angle
  • May be finite
    • Defined by two end points
    • Defined by one end point, a length, and direction
a curve
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
  • 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
closed 2 d elements planar
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)
      • A^2+B^2=C^2 (Pythagorean theorem)
      • Sinq=A/C
      • Cosq=B/C





closed 2 d elements planar17
Closed 2-D Elements (planar)


  • Circles
    • Radius (R)
    • Diameter (D)
    • Angle (1 rev = 360o 0’ 0”)
    • Circumference (2*3.14159*R)
    • Tangent
    • Chord
      • A line perpendicular to the midpoint of a chord passes through the center of the circle
    • Concentric circles



closed 2 d elements planar18
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
  • 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
Three dimensional

They have a volume

Regular polyhedra

Have regular polygons for faces

All faces are the same


Two equal parallel faces

Sides are parallelograms


Common intersection point (vertex)




useful tools from mechanical drawing techniques
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