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**Engineering Graphics I**

Chapter 8 Engineering Graphics I Dr Simin Nasseri Southern Polytechnic State University © Copyright 2010 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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Objectives: 1.Describe the importance of engineering geometry in the engineering design process. 2. Describe coordinate geometry and coordinate systems and apply them to CAD. 3. Explain the right-hand rule. 4. List the major categories of geometric entities. 5. Explain and construct the geometric conditions that occur between lines. 6. Construct points, lines, curves, polygons, and planes. 7. Explain and construct tangent conditions between lines and curves. 8. Explain and construct conic sections, roulettes, double-curved lines, and freeform curves. 9. List and describe surface geometric forms. 10. Explain and construct 3-D surfaces. 11. Describe engineering applications of geometry.

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INTRODUCTION Graphics is used to represent complex objects and structures that are created from simple geometric elements, such as lines, circles, and planes. Current 3-D CAD programs use these simple geometric forms to create more complex ones, through such processes as extrusion, sweeping, and solid modeling Boolean operations. Therefore, to fully exploit the use of CAD, you must understand geometry and be able to construct 2-D and 3-D geometric forms. ENGINEERING GEOMETRY Geometry provides the building blocks for the engineering design process. Engineering geometry is the basic geometric elements and forms used in engineering design. Engineering and technical graphics are concerned with the descriptions of shape, size, and operation of engineered products. The shape description of an object relates to the positions of its component geometric elements in space. To be able to describe the shape of an object, you must understand all of the geometric forms, as well as how they are graphically produced.

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Figure 8.2 The Cartesian coordinate system, commonly used in mathematics and graphics, locates the positions of geometric forms in 2-D and 3-D space.

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Figure 8.3 Example:

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Figure 8.4 In a 3-D coordinate system, the origin is established at the point where three mutually perpendicular axes (X, Y, and Z) meet. The origin is assigned the coordinate values of 0,0,0.

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Figure 8.5 A rectangular prism is created using the 3-D coordinate system by establishing coordinate values for each corner.

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Figure 8.6 Display of coordinate axes in a multiview CAD drawing. Only two of the three coordinates can be seen in each view.

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Figure 8.8 The right-hand rule is used to determine the positive direction of the axes. The right-hand rule defines the X, Y, and Z axes, as well as the positive and negative directions of rotation on each axes.

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**Y x Distance= 4.5 Angle = 30o X x = 4.5 cos(30) y = 4.5 sin(30)**

Figure 8.9 Y Polar coordinates are used to locate points in the X-Y plane. Polar coordinates specify a distance and an angle from the origin (0,0). x Distance= 4.5 Angle = 30o X x = 4.5 cos(30) y = 4.5 sin(30)

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Figure 8.10 Cylindrical coordinates locate a point on the surface of a cylinder by specifying a distance and an angle in the X-Y plane, and the distance in the Z direction. x = 4.5 cos(30) y = 4.5 sin(30) z = 7 30o

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Figure 8.11 Spherical coordinates locate a point on the surface of a sphere by specifying an angle in one plane, an angle in another plane, and one height. For more info, click here.

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Figure 8.22 Tangencies: A tangent condition exists when a straight line is in contact with a curve at a single point. Figure 8.23 In engineering drawing, lines are used to represent the intersections of nonparallel planes, and the intersections are called edges.

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**Drawing a line parallel to a given line at a given distance.**

Figure 8.24 Tangencies: Drawing a line parallel to a given line at a given distance.

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Figure 8.25 Tangencies: Drawing a line parallel to a given line at a given distance from that line, alternative method.

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**Bisecting a line, using a compass and a triangle.**

Figure 8.28 Bisecting a line, using a compass and a triangle.

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**Bisecting a line using two triangles.**

Figure 8.29 Bisecting a line using two triangles.

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Figure 8.31 Tangencies: In planar geometry, a tangent condition exists when a straight line is in contact with a curve at a single point; that is, a line is tangent to a circle if it touches the circle at one and only one point. Two curves are tangent to each other if they touch in one and only one place.

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Figure 8.32 Tangencies: In 3-D geometry, a tangent condition exists when a plane touches but does not intersect another surface at one or more consecutive points.

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Figure 8.33 Another tangent condition exists where there is a smooth transition between two geometric entities. However, a corner between two geometric entities indicates a nontangent condition.

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**Locating tangent points on circles and arcs.**

Figure 8.35 Locating tangent points on circles and arcs.

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**Drawing an arc tangent to a line at a given point on the line.**

Figure 8.36 Drawing an arc tangent to a line at a given point on the line.

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**Constructing arcs tangent to two lines.**

Figure 8.37 Constructing arcs tangent to two lines.

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**Constructing an arc tangent to a line and another arc.**

Figure 8.38 Constructing an arc tangent to a line and another arc.

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Figure 8.39 Construct an arc tangent to two given arcs or circles by locating the center of the tangent arc, then drawing the tangent arc.

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Figure 8.40 Construct an arc tangent to two given arcs by locating the centers of the given arcs, then drawing the tangent arc.

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**Construction of an arc tangent to a given arc.**

Figure 8.41 Construction of an arc tangent to a given arc.

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Figure 8.42 Constructing lines tangent to two circles or arcs, using a straightedge.

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Figure 8.43

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**Center. The 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. Secant. A straight line that goes through a circle but not through the-center. Arc. A continuous segment of the circle. Semicircle. An arc measuring one-half the circumference of the circle. Minor arc. An arc that is less than a semicircle. Major arc. An arc that is greater than a semicircle. Central angle. An angle formed by two radii. Sector. An area bounded by two radii and an arc, usually a minor arc. Quadrant. A sector equal to one-fourth the area of the circle. The radii bounding a quadrant are at a right angle to each other. Segment. An area bounded by a chord and a minor arc. The chord does not go through the center. Tangent. A line that touches the circle at one and only one point. Concentric circles. Circles of unequal radii that have the same center point. Eccentric circles. Circles of unequal radii that have different centers, and one circle is inside the other.

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Figure 8.64 An ellipse is a single-curved-surface primitive and is created when a plane passes through a right circular cone oblique to the axis, at an angle to the axis greater than the angle between the axis and the sides.

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**A circle when viewed at an angle, appears as an ellipse.**

Figure 8.65 A circle when viewed at an angle, appears as an ellipse.

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Figure 8.66 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. The major diameter (major axis) of an ellipse is the longest straight-line distance between the sides and is through both foci. The minor diameter (minor axis) is the shortest straight-line distance between the sides and is through the bisector of the major axis. The foci are the two points used to construct the perimeter and are on the major axis.

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**Constructing an ellipse.**

Figure 8.67 Constructing an ellipse.

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**Constructing an ellipse using the trammel method.**

Figure 8.69 Constructing an ellipse using the trammel method.

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Figure 8.73 The viewing angle relative to the circle determines the ellipse template to be used. The circle is seen as an inclined edge in the right view and foreshortened in the front view. The major diameter is equal to the diameter of the circle.

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**Trapezium. No sides parallel or equal in length, no angles equal.**

Figure 8.102 Square. Opposite sides parallel, all four sides equal in length, all angles equal. Rectangle. Opposite sides parallel and equal in length, all angles equal. Rhombus. Opposite sides parallel, four sides equal in length, opposite angles equal. Rhomboid. Opposite sides parallel and equal in length, opposite angles equal. Regular trapezoid. Two sides parallel and unequal in length, two sides nonparallel but equal in length, base angles equal, vertex angles equal. Irregular trapezoid. Two sides parallel, no sides equal in length, no angles equal. Trapezium. No sides parallel or equal in length, no angles equal.

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**Constructing an inscribed square.**

Figure 8.104 Constructing an inscribed square.

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**Constructing an circumscribed square.**

Figure 8.105 Constructing an circumscribed square.

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Figure 8.106 A polygon is a multisided plane of any number of sides. If the sides of the polygon are equal in length, the polygon is called a regular polygon. Regular polygons are grouped by the number of sides.

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Figure 8.107

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Figure 8.108 Right triangles are acute triangles and can also be isosceles triangles, where the other two angles are 45 degrees. The other right triangle is the 30/60 triangle, where one angle is 30 degrees and the other is 60 degrees.

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**Constructing equilateral triangles.**

Figure 8.109 Constructing equilateral triangles.

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**A pentagon has five equal sides and angles.**

Figure 8.110 A pentagon has five equal sides and angles.

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Figure 8.111 A pentagon is constructed by dividing a circle into five equal parts, using a compass or dividers.

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Figure 8.112 A hexagon has six equal sides and angles. Hexagons are constructed by using a 30/60-degree triangle and a straightedge to construct either circumscribed or inscribed hexagons around or in a given circle. A circumscribed hexagon is used when the distance across the flats of the hexagon is given

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Figure 8.113 and the inscribed method is used when the distance across the corners is given.

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**An octagon has eight equal sides and angles.**

Figure 8.114 An octagon has eight equal sides and angles.

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Figure 8.115

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