Descriptive Geometry

Introduction What is Descriptive Geometry?

→ It is the study of points, lines, and planes in space to determine their locations and true shapes.

Coordinate Space→ In order to locate points, lines, planes, or

other geometric forms, their positions must first be referenced to some known position, called a reference point or origin of measurement

Cartesian Coordinate System Commonly used in mathematics and

graphics, locates the positions of geometric form in 2-D and 3-D space.→ 2-D Coordinate System: Establishes an origin at

the intersection of two mutually perpendicular axes, labeled X (horizontal) and Y (vertical)

+X-X

+Y

-Y

0,0origin

Cartesian Coordinate System 3-D space.

→ 3-D Coordinate System: The origin is established at the point where three mutually perpendicular axes (X,Y,Z) meet. The origin is assigned the coordinate values of 0,0,0

+X

-X+Y

-Y

0,0,0origin-Z

+Z

Cartesian Coordinate System Using these coordinate systems, you

can locate any point in 2-D or 3-D space by assigning a unique set of numbers to that point.

+X

-X+Y

-Y

0,0,0origin

-Z

+Z. 3,0,0

Cartesian Coordinate System Right Hand Rule

→ Used to determine the positive direction of the axes.

→ Make a fist with right hand, with your thumb pointing outward.

→ The direction of your thumb is pointing indicates the positive X axis.

→ Straighten your index finger – this is pointing in the positive Y axis direction.

→ Straighten your middle finger – this direction is in the positive Z axis direction.

Absolute and Relative Coordinates Absolute Coordinates are always

referenced to the origin 0,0,0.

0,0,0

0,3,0

X

Y

Z

4,3,0

4,0,0

Absolute and Relative Coordinates Relative Coordinates are always referenced

to a previously defined location. For example, here is the same rectangle with points located with reference to the previous point, moving from A – B, B – C, and C – D.

0,0,0

-4,0,0

X

Y

Z

0,3,0

4,0,0AB

CD

World Coordinate System In AutoCAD, world coordinate system

(WCS), is defined based on intersection of the 3D Cartesian coord system (X,Y,Z).

The WCS always exists in any drawing and cannot be deleted.

The WCS is the default coord system in AutoCAD for defining the position of drawing objects in 2D or 3 D space.

User Coordinate System In AutoCAD, the user can create and save

multiple User Coordinate Systems (UCS) to make construction of a particular 3D geometry easier.

Only one coordinate system can be active at any one time (in any one view) – either the WCS or UCS.

By default the UCS is aligned with the WCS.

We will refer to this more when we begin drawing 3-D objects in AutoCAD.

Orientation of the WCS or UCS The icon near the bottom left corner of

the default AutoCAD graphics window shows the positive X-dir and positive Y-dir of the active coordinate system.

X

YW

2D UCS at WCS

X

Y

3D UCS at WCS

Geometric Elements Geometric elements are categorized as

points, lines, surfaces, or solids. We’ll focus on points and lines.→ Point: A theoretical locations that has

neither width, height, nor depth. Points describe an exact location in space.

→ Line: A geometric primitive that has length and direction, but not necessarily thickness. A line may be straight, curved, or a combination of these.

Points A point is found at the intersection of two

lines or at the end of a finite line. In CAD, it is common to use the word node

to mean point.→ For example, the intersection of geometric entities,

and specific locations along arcs, circles, and splines are called nodes.

Nodes are very important when constructing geometric forms with CAD. CAD systems normally allow the user to locate exactly such important elements as endpoints, centers, and intersections.

These nodes can be used to construct geometric forms more accurately.

Points

Point++

End Points +

Point node at the tangency of 2 circles

+Point node at the midpoint of a line

Point node at the center of a circle

+

+Point node at the intersection of 2 lines

Lines Straight Lines: generated by a point

moving in a constant direction.+ + + + +

Straight Lines can be either finite or infinite.

A straight finite line is a line of specific length.

The relationship of one line to another results in a condition.

Line Conditions Parallel lines

→ Occurs when two or more lines on a plane are a constant distance apart.

Nonparallel lines→ Occurs when two or more lines on one or more planes are

spaced unevenly apart. Perpendicular lines

→ Sometimes called normal, occurs when two or more lines on a plane intersect each other at right angles (90o).

Intersecting lines→ Occurs when two or more lines cross each other at a common

point. Tangent

→ A tangent condition exists when a straight line is in contact with a curve at a single point.

Lines

Finite line

Parallel lines

Tangent condition

Intersecting lines

Nonparallel lines

Curved Lines A curved line is the path generated by

a point moving in a constantly changing direction.

Examples of curved lines include circles, parabolas, spirals, and splines.

+ + + + + +

Circles A regular curve is a constant-radius arc

or circle generated around a single point.

All points on the surface of a circle are equidistant from one point, the center.

+ radius

Elements of a Circle Center: midpoint of the circle Circumference: The distance all the way around the circle. Radius: A line joining the center to any point on the circumference. Chord: A straight line joining any two points on the circumference. Diameter: A chord that passes through the center. Arc: A continuous segment of the circle. Tangent: A line that touches the circle at one and only one point.

+

radius

Ctr

tangent

90O

chord+

diameter

arc

Major axis

A

D

C

B

AB + BC = AD + CD

EllipseAn ellipse is a single-curved primitive. Mathematically, an ellipse is the set of all points in a plane for which the

sum of the distances from two fixed points (the foci) in the plane is constant.

Major diameter (major axis) – the longest straight-line distance between the ellipse sides.

Minor diameter (minor axis) – the shortest straight-line distance between the ellipse sides and is through the bisector of the major axis.

Foci – The two points used to construct the perimeter and are on the major axis. Minor axis

EllipseA line of sight other than 90o changes the

appearance of a circle to an ellipse.

.

Line of sight

Edge view of circle

.

.

What you see

Angles Angles are formed by the apex of two intersecting line or

planes.

Angles are categorized by their degree measurement.

Angle

Apex

Angle

180o

Straight

90o

Right

More than 90o

Obtuse

Less than 90o

Acute

Planes A plane is a 2-dimensional surface that

wholly contains every straight line joining any two points lying on that surface.

Although many drawing are created from simple geometric primitives, such as lines and curves, many real world designs are made of planar surfaces.

Theoretically, a plane has width and length but not thickness.

Planes Planes are formed by three points, two parallel lines, a line and

a point, or two intersecting lines.1+

2 +

+ 3

Plane2 parallel lines

Plane

+

PlanePlane

3 points

Line and point 2 intersecting lines