Geometric ConstructionGeometric Construction

Engineering GraphicsEngineering Graphics

Stephen W. Crown Ph.D.Stephen W. Crown Ph.D.

Objective

To review basic terminology and concepts To review basic terminology and concepts related to geometric formsrelated to geometric forms

To present the use of several geometric To present the use of several geometric tools/methods which help in the tools/methods which help in the understanding and creation of engineering understanding and creation of engineering drawingsdrawings

Overview

Coordinate SystemsCoordinate Systems Geometric ElementsGeometric Elements Mechanical Drawing ToolsMechanical Drawing Tools

Coordinate Systems

Origin (reference point)Origin (reference point) 2-Dimensional Coordinate System2-Dimensional Coordinate System

• Cartesian (x,y) Cartesian (x,y) • Polar (r,Polar (r,))

3-Dimensional Coordinate System3-Dimensional Coordinate System• Cartesian (x,y,z)Cartesian (x,y,z)• Cylindrical (z,r,Cylindrical (z,r,))• Spherical (r,Spherical (r,))

Cartesian Coordinate System

Defined by two/three mutually perpendicular axes which Defined by two/three mutually perpendicular axes which intersect at a common point called the originintersect at a common point called the origin• x-axisx-axis

horizontal axishorizontal axis positive to the rightpositive to the right

of the origin as shownof the origin as shown

• y-axisy-axis vertical axisvertical axis positive above positive above

the origin as shownthe origin as shown

• z-axis (added for a 3-D coordinate system)z-axis (added for a 3-D coordinate system) normal to the xy planenormal to the xy plane positive in front of the origin as shownpositive in front of the origin as shown

Review: Right Hand Rule

Your thumb, index finger, and middle finger Your thumb, index finger, and middle finger represent the X, Y, and Z axis respectively.represent the X, Y, and Z axis respectively.

Point your thumb in the positive axis direction and Point your thumb in the positive axis direction and your fingers wrap in the direction of positive your fingers wrap in the direction of positive rotationrotation

Polar Coordinate System

The distance from the originThe distance from the origin to the point in the xy plane to the point in the xy planeis specified as the radius (r)is specified as the radius (r)

The angle measured form theThe angle measured form thepositive x axis is specified as positive x axis is specified as

Positive angles are defined Positive angles are defined according to the right hand ruleaccording to the right hand rule

Conversion between Cartesian and polarConversion between Cartesian and polar• x=r*cos x=r*cos y=r*sin y=r*sin • x^2+y^2=r^2 , x^2+y^2=r^2 , tantan-1-1(y/x)(y/x)

Cylindrical Coordinate System

Same as polar except a Same as polar except a z-axis is added which is z-axis is added which is normal to the xy plane in normal to the xy plane in which angle which angle is measured is measured

The direction of the The direction of the positive z-axis is defined positive z-axis is defined by the right hand ruleby the right hand rule

Useful for describing Useful for describing cylindrical featurescylindrical features

Spherical Coordinate System

The distance from the origin The distance from the origin is specified as the radius (r)is specified as the radius (r)

The angle between the x-axis andThe angle between the x-axis andthe projection of line r on the xy the projection of line r on the xy plane is specified as plane is specified as

The angle between line r and theThe angle between line r and thez-axis is specified as z-axis is specified as

Positive angles of Positive angles of are defined according to the right are defined according to the right hand rule and the sign of hand rule and the sign of does not affect the resultsdoes not affect the results

Conversion between Cartesian and sphericalConversion between Cartesian and spherical• x=r*sinx=r*sin*cos*cosy=r *siny=r *sin*sin *sin z= r*cosz= r*cos

Redefining Coordinates

Absolute coordinatesAbsolute coordinates• measured relative to the originmeasured relative to the origin• LINE (1,2,1) - (4,4,7)LINE (1,2,1) - (4,4,7)

Relative coordinatesRelative coordinates• measured relative to a previously specified pointmeasured relative to a previously specified point• LINE (1,2,1) - @(3,2,6)LINE (1,2,1) - @(3,2,6)

World Coordinate System World Coordinate System • a stationary referencea stationary reference

User Coordinate System (ucs)User Coordinate System (ucs)• change the location of the origin change the location of the origin • change the orientation of axeschange the orientation of axes

Geometric Elements

A pointA point A lineA line A curveA curve PlanesPlanes Closed 2-D elementsClosed 2-D elements SurfacesSurfaces SolidsSolids

A Point

Specifies an exact location in spaceSpecifies an exact location in space DimensionlessDimensionless

• No heightNo height• No widthNo width• No depthNo depth

A Line

Has length and direction but no width Has length and direction but no width All points are collinearAll points are collinear May be infinite May be infinite

• At least one point must be specifiedAt least one point must be specified• Direction may be specified with a second point or with an Direction may be specified with a second point or with an

angle angle May be finiteMay be finite

• Defined by two end pointsDefined by two end points• Defined by one end point, a length, and directionDefined by one end point, a length, and direction

A Curve

The locus of points along a curve are not The locus of points along a curve are not collinearcollinear

The direction is constantly changingThe direction is constantly changing Single curved linesSingle curved lines

• all points on the curve lie on a single planeall points on the curve lie on a single plane A regular curveA regular curve

• The distance from a fixed point to any point on the The distance from a fixed point to any point on the curve is a constantcurve is a constant

• Examples: arc and circleExamples: arc and circle

PlanesPlanes

A two dimensional slice of space A two dimensional slice of space No thickness (2-D)No thickness (2-D) Any orientation defined by:Any orientation defined by:

• 3 points3 points• 2 parallel lines2 parallel lines• a line and a pointa line and a point• 2 intersecting lines2 intersecting lines

Appears as a line when the direction of view is Appears as a line when the direction of view is parallel to the planeparallel to the plane

Closed 2-D Elements (planar)Closed 2-D Elements (planar)

BB

AA

TrianglesTriangles• Three sidesThree sides• Equilateral triangle (all sides equal, 60 deg. Equilateral triangle (all sides equal, 60 deg.

angles)angles)• Isosceles triangle (two sides equal)Isosceles triangle (two sides equal)• Right triangle (one angle is 90 degrees)Right triangle (one angle is 90 degrees)

A^2+B^2=C^2 (Pythagorean theorem)A^2+B^2=C^2 (Pythagorean theorem) SinSin=A/C=A/C CosCosB/CB/C

CC

Closed 2-D Elements (planar)Closed 2-D Elements (planar)

CirclesCircles• Radius (R)Radius (R)• Diameter (D)Diameter (D)• Angle (1 rev = 360Angle (1 rev = 360oo 0’ 0”) 0’ 0”)• Circumference (2*3.14159*R)Circumference (2*3.14159*R)• TangentTangent• ChordChord

A line perpendicular to the midpoint of a chord passes through A line perpendicular to the midpoint of a chord passes through the center of the circlethe center of the circle

• Concentric circlesConcentric circles

DD

RR

Closed 2-D Elements (planar)Closed 2-D Elements (planar)

ParallelogramsParallelograms• 4 sides4 sides• Opposite sides are parallelOpposite sides are parallel• Ex. square, rectangle, and rhombusEx. square, rectangle, and rhombus

Regular polygonsRegular polygons• All sides have equal lengthAll sides have equal length

3 sides: equilateral triangle3 sides: equilateral triangle 4 sides: square4 sides: square 5 sides: pentagon5 sides: pentagon

• Circumscribed or inscribedCircumscribed or inscribed

SurfacesSurfaces

Does not have thicknessDoes not have thickness Two dimensional at every pointTwo dimensional at every point

• No massNo mass• No volumeNo volume

May be planarMay be planar May be used to define the boundary of a May be used to define the boundary of a

3-D object 3-D object

SolidsSolids

• Three dimensionalThree dimensional• They have a volumeThey have a volume• Regular polyhedraRegular polyhedra

Have regular polygons Have regular polygons for facesfor faces

All faces are the sameAll faces are the same

Prisms Prisms • Two equal parallel Two equal parallel

facesfaces• Sides are Sides are

parallelogramsparallelograms PyramidsPyramids

• Common intersection Common intersection point (vertex)point (vertex)

ConesCones CylindersCylinders SpheresSpheres

Useful Tools From Mechanical Drawing TechniquesUseful Tools From Mechanical Drawing Techniques

Drawing perpendicular lines (per_)Drawing perpendicular lines (per_) Drawing parallel lines (offset)Drawing parallel lines (offset) Finding the center of a circle (cen_)Finding the center of a circle (cen_) Some difficult problems for someone who Some difficult problems for someone who

completely relies on AutoCAD toolscompletely relies on AutoCAD tools• Block with radiusBlock with radius• Variable guideVariable guide• Offset pipeOffset pipe• Transition Transition