Download Geometry and Constraints

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
 Coordinate Systems
 Geometric Elements
 Mechanical Drawing Tools
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
 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 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
 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
 The distance from the origin
to the point in the xy plane
is specified as the radius (r)
 The angle measured form the
positive 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
 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
 The distance from the origin
is specified as the radius (r)
 The angle between the x-axis and
the projection of line r on the xy
plane is specified as q
 The angle between line r and the
z-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
 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







A point
A line
A curve
Planes
Closed 2-D elements
Surfaces
Solids
A Point
 Specifies an exact location in space
 Dimensionless
• No height
• No width
• No depth
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
 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
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
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

C
q
B
A
Closed 2-D Elements (planar)
R
 Circles
•
•
•
•
•
•
Radius (R)
Diameter (D)
Angle (1 rev = 360o 0’ 0”)
Circumference (2*3.14159*R)
Tangent
Chord

D
A line perpendicular to the midpoint of a chord
passes through the center of the circle
• Concentric circles
q
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
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
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
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