Geometry and Constraints
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Transcript 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
•
•
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Block with radius
Variable guide
Offset pipe
Transition