DESCRIPTIVE GEOMETRY

Descriptive Geometry is the science of graphic representation and solution of space problems.

Coordinate Systems:

World Coordinates: world axes move coincident with the created part

Construction Plane Coordinates: relative to the current construction plane

 

The lines of intersection of the perpendicular projection planes are called folding lines (H/F and F/P)

 

POINT

Theoretically a point has location only and no dimensions.

Point in space is represented by a capital letter (eg. A)

A (x,y,z) , A (80,100,120)

In views of point, identified with lowercase letter plus the appropriate superscript aF, aH, aP

 

LINE

In geometry a line theoretically has no width. The position of a straight line is established by locating any two noncoincident points on the line.

PLANE

  • 3 points
  • 2 intersecting lines
  • 2 parallel lines
  • a point and a line

all the above will represent a plane.

 

3D OBJECT

intersecting planes form an object. Surface of objects are limited planes.

 

INTERSECTION

- Intersection of lines

  • intersecting lines (have a common point)
  • nonintersecting lines (do not have common point)
    • parallel lines
    • skew lines (nonintersecting and nonparallel lines never form a plane)

 

- Intersection of planes

  • nonintersecing (parallel) planes
  • intersecting planes (intersection will be line of intersection (LI), can be limited or unlimited)

 

- Intersection of line and plane

  • parallel
  • intersecting

(unless a line is in or parallel to a plane, it must intersect the plane)

The intersection point is called piercing point.

 

- Intersection of object and line

  • intersecting (piercing points)
  • nonintersecting
  • tangent (1 piercing point)

 

- Intersection of object and plane

  • intersecting (intersection will be line (LI))
  • nonintersecting
  • tangent (point or line)

 

VISIBILITY

An essential step in drawing an orthographic view is the correct determination of the visibility of the lines that make up the view. The outline of a view will always be visible, but the lines within the outline may be visible or hidden, depending on the relative positions of those lines with respect to the line of sight.

 

 

VIEW OF LINE

  • True length of a line: A line is shown in true length, TL, when the line of sight is normal (perpendicular) to the line.
  • Any line of sight not normal to a line results in a view that is shorter than true length.

 

 POINTS ON LINES

  • Any two successive views of a point must lie on a projection line perpendicular to the folding line between the two views.
  • A point may be said to be above or below, in front of or behind, or to the right or to the left of another point.

  

VIEW OF PLANE

  • Edge view: The edge view of a plane is seen when all lines of the plane appear to coincide in a single line (EV)
  • Normal view: The normal view (true surface) of a plane is seen when the line of sight is perpendicular to the plane (TS)

 

POINTS AND LINES IN PLANES

  • If a line is known to be in plane then any point on that line is in the plane.
  • A line may be drawn in a plane by keeping it in contact with (intersecting) any two given lines in the plane.

 

LOCUS

  • A locus is the path of all possible positions of a moving point, line, or curve.
  • The locus of points in a plane and at a specified distance from a given point is a circle.
  • The locus of points in space at a specified distance from a given point is a sphere.

 

PIERCING POINTS

  • Intersection of line-plane is called a piercing point.

 

INTERSECTION OF PLANES

  • The intersection of two planes is a straight line common to the planes and its position is therefore determined by any two points common to both planes.
  • If a given plane is bounded by a closed figure then the line of intersection will be limited.

 

ANGLE BETWEEN PLANES

  • The angle formed by two intersecting planes is called a dihedral angle.
  • The true size of the dihedral angle is observed in a view in which each of the given planes appears in edge view.
  • A view showing a point view of a line common to two planes, the line of intersection (LI), produces an edge view (EV) of each of the planes.

 

ANGLE BETWEEN LINE AND PLANE

  • In order to show true magnitude of the angle between a line and a plane it is necessary to show the line in true length (TL) and plane in edge view.

 

 COMMON PERPENDICULAR

  • Connecting two skew (non-intersecting and non-parallel) lines there is only one line which is perpendicular to both lines.
  • Common perpendicular gives the shortest distance between the lines.

 

 PARALLELISM

Parallel Lines

  • Any two lines in a plane must either intersect or be parallel.
  • Lines parallel in space project as parallel lines in any view except in those views in which they coincide or appear as points.
  • Oblique lines that appear parallel in two or more principle views are parallel in space.
  • Two horizontal, two frontal, or two profile lines that appear to be parallel in two principal views may or may not be actually parallel in space.
  • The true distance between two parallel lines may be obtained either by constructing a view showing the lines as points or by obtaining a normal view of the plane of the two lines.

Parallel Planes

  • If two intersecting lines in one plane are parallel respectively to two intersecting lines in a second plane, the planes are parallel in space.
  • If two planes are parallel, any line in one plane is parallel to the other plane, since it cannot intersect the other plane.
  • If two planes are parallel, any view showing one of the planes in edge view must also show the other plane as a parallel edge view.

Lines Parallel to Planes

  • If two lines are parallel, any plane containing one of the lines is parallel to the other line.
  • If two planes are parallel, any line in one plane is parallel to the other plane, since it cannot intersect the other plane.

 

PERPENDICULARITY

  • If a line is perpendicular to a plane it is perpendicular to any line in the plane

Plane perpendicular to line

  • A plane is perpendicular to a line if the plane contains two intersecting lines each of which is perpendicular to the given lines.

 

 


Copyright 2006 , Middle East Technical University