1. Descriptive Geometry

2. 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

3. 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 0,0 origin

4. 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 0,0,0 origin -Z +Z

5. 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 0,0,0 origin -Z +Z. 3,0,0

6. 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.

7. 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

8. 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 4,0,0 A B C D

9. 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.

10. 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.

11. 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 Y W 2D UCS at WCS X Y 3D UCS at WCS

12. 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.

13. 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.

14. 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

15. 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.

16. 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 (90 o ).  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.

17. Lines Finite line Parallel lines Tangent condition Intersecting lines Nonparallel lines

18. 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. + + + + + +

19. 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

20. 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 90 O chord + diameter arc

21. Major axis A D C B AB + BC = AD + CD Ellipse An 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

22. Ellipse A line of sight other than 90 o changes the appearance of a circle to an ellipse.. Line of sight Edge view of circle.. What you see

23. Angles  Angles are formed by the apex of two intersecting line or planes.  Angles are categorized by their degree measurement. Angle Apex Angle 180 o Straight 90 o Right More than 90 o Obtuse Less than 90 o Acute

24. 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.

25. Planes  Planes are formed by three points, two parallel lines, a line and a point, or two intersecting lines. 1+1+ 2 + + 3 Plane 2 parallel lines Plane + 3 points Line and point 2 intersecting lines