# Presentation Slides to be used in conjunction with the Developing Spatial Thinking Curriculum.

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1 Presentation Slides to be used in conjunction with the Developing Spatial Thinking Curriculum. Module 1 Solids of Revolution Module 2 Combining Solids Module 3 Isometric Sketching Module 4 Orthographic Projection Module 5 Incline and Curved Surfaces Module 6 Flat Patterns Module 7 Rotation of Objects about 1 Axis Module 8 Rotation of Objects about 2 Axes Module 9 Object Reflections and Symmetry Module 10 Cutting Planes Accessible at

2 Surfaces and Solids of Revolution Module 1 HigherEdServices.org

3 Session Topics Surfaces and Solids of Revolution Degree of Revolution Hollow Objects Visualizing Revolution

4 Surfaces and Solids of Revolution Surfaces and Solids of Revolution are formed when a 2-D shape is revolved about an axis Surfaces result if the shape is open Solids result if the shape is closed

5 Solids of Revolution The resulting 3-D object depends on the axis about which the 2-D shape was revolved

6 Resulting 3-D object also depends on the degree of angular revolution Angle of Revolution

7 Hollow Objects If a 2-D shape is located "away from" the axis of revolution, a solid of revolution with a cylindrical hole in it will result

8 Visualizing Revolutions To visualize a revolution, first think about mirroring the shape about the axis of revolution and then forming a cylindricallyshaped object from the two shapes

9 Course Software... Work through the surfaces and solids of revolution software module

10 Surfaces and Solids of Revolution Homework... Complete the following pages in Module 1 rev

11 Combining Solids Module 2 HigherEdServices.org

12 Session Topics Combining Overlapping Objects Volume of Interference Cutting Joining Intersecting Multiple Combinations Visualizing Combinations

13 Combining Solids Two overlapping objects can be combined by cutting, joining, or intersecting

14 Volume of Interference The Volume of Interference is defined as the overlapping volume between two objects

15 Joining Objects When two objects are joined, the volume of interference is absorbed into the final object

16 Cutting Objects When two objects are cut, the volume of interference is removed from the object being cut.

17 Intersecting Objects When two objects are intersected, the volume of interference becomes the new object.

18 Multiple Combinations Complicated objects can be created through cutting, joining, or intersecting parts.

19 Visualizing Combinations When presented with two overlapping objects, try to visualize which edges will remain after a combining operation has been performed

20 Course software Work through the combining solids software module

21 Combining Solid Objects Homework Complete the following pages in Module 2 ob/int Problem 1 on page 2 is not a typo the resulting object is nothing for pages 13 & 14 show only visible lines

22 Isometric Drawings & Coded Plans Module 3 HigherEdServices.org

23 Session Topics Isometric Axes Coded Plans Objects from Multiple Viewpoints Isometric Sketching

24 Isometric Sketching Used to portray a 3-D object on a 2-D sheet of paper. The relationship between the 3-D axes appears differently when projected onto a 2-D surface, i.e., perpendicular angles no longer appear perpendicular.

25 Isometric Axes

26 Isometric Sketch of Cube Isometric sketches are made as if you were looking down a diagonal of a cube The surfaces of the cube appear to be rhomboidal Note that all surfaces of an object sketched in isometric will appear distorted!

27 Isometric Axes Isometric grid paper or dot paper is a useful tool for constructing isometric drawings.

28 Coded Plans Pictorial sketches can be made from coded plans. Coded plans define the shape of a building made of blocks. Coded Plan Building

29 Coded Plans Isometric sketch from corner C.

30 Different Viewpoints The object will appear differently depending on the corner you are viewing it from

31 Isometric Sketching C Steps 1 & 2 Step 3 Step 4... Step 4... C Guidelines for constructing isometric drawings: 1.Draw edge "C". 2.Sketch surface to the right or left of edge C. 3.Draw a surface that shares an edge with the surface just drawn. 4.Continue drawing one surface at a time until the object is complete. NOTE: Do not show the individual blocks in the building. Just show edges where surfaces intersect.

32 Course Software Work through the Isometric Drawings and Coded Plans software module

33 Homework Complete the following pages in Module 3 iso

34 Orthographic Drawings Module 4 HigherEdServices.org

35 Session Topics Orthographic Projection Normal Surfaces Hidden Lines Isometric Sketching from Orthographic Projections

36 Orthographic Projection Imagine an object is surrounded by a glass cube. The object's surfaces are projected onto the faces of the glass cube. The projection rays are perpendicular to the panes of glass

37 Orthographic Projection Unfold the cube so that it lies in a single plane Three views of the object are now visible on the same plane in space Fold lines

38 H H D Orthographic Projection When the glass cube is unfolded each view shows two dimensions of the object: W Front view: Height and Width Top view: Width and Depth Right view: Depth and Height D W

39 Orthographic Projection Views are aligned with one another (features project from one view to the next)

40 Normal Surfaces A B Parallel to one of the six glass panes of our transparent cube Perpendicular (normal) to the projectors to/from that plane Shown true size and B Edge 3 shape in the view that they are parallel to Seen as edges (lines) Edge 2 in the other principal Edge 1 views A Surface A appears as edge 1 and 4 in top & Edge 4 right views Surface B appears as edges 2 and 3

41 Hidden Lines Some object have edges which cannot be seen from certain viewing angles Showing these edges provides valuable graphical information Visible edges are continuous (solid) lines (object lines) Hidden edges are dashed lines (hidden lines) to avoid confusing them with visible edges

42 Hidden Lines Visible Line (Solid lines, can be seen in a given view, aka: Object Lines) Hidden Line (Dashed lines; can't be seen in that view)

43 Hidden Lines and Object Lines Together If a solid line and a hidden line are on top of each other, show only the solid line Coded Plan

44 Isometric Sketches from Orthographic Views Sometimes you are asked to construct Isometric sketches from Orthographic views to develop visualization skills. The box method is one way to do this. For some problems, the box method may not be very helpful.

45 H D Isometric Sketches from Orthographic Views 1. Find the object's overall dimensions from the orthographic views and sketch that size box on isometric dot paper. W

46 Isometric Sketches from Orthographic Views 2. Sketch the top, front, and right side views in their appropriate locations on the box.

47 Isometric Sketches from Orthographic Views 3. Add/remove lines until the view is complete.

48 Course Software.. Complete the software module on Orthographic Projections

49 Orthographic Drawings Homework Complete the following pages in Module 4 ortho

50 Inclined and Curved Surfaces Module 5 HigherEdServices.org

51 Session Topics Orthographic projections of inclined and single-curved surfaces Drawing isometric sketches from orthographic views of objects with inclined and single-curved surfaces

52 Review: Orthographic Projection: Normal Surfaces Normal surfaces appear as a surface in one view, an edge in the other two views. Normal surfaces are shown true size and shape in the view they are parallel to.

53 Inclined Surfaces Are not parallel to any of the principal views Are perpendicular to one of the three views Appear as an area in two views, as an edge in the third view Area views are foreshortened Basic shape is maintained in area views

54 Normal versus Inclined Surfaces

55 Inclined Surfaces An inclined surface maintains its basic shape from view to view basic U-shape is maintained from view to view

56 Orthographic Projection Recall: Align views with each other (features project from one view to the next) Properly aligned orthographic views Improperly aligned orthographic views

57 Orthographic Projection Recall: Use hidden (dashed) lines to show edges of an object that are hidden from a given viewpoint. hidden line

58 Inclined Surfaces in Isometric Constructing an isometric view of an inclined surface: 1. Locate the endpoints of each inclined edge 2. Draw a straight line between them

59 Inclined Surfaces: Constructing Isometric Views Select an orientation that makes the inclined surface appear as a visible area (whenever feasible) Correct Incorrect

60 Single Curved Surfaces

61 Single Curved Surfaces Surfaces having a curvature about one axis. Generated by revolving a line about an axis.

62 Single Curved Surfaces Consider a cylinder: Shown as a circle in one orthographic view Other orthographic views are rectangular Rectangle width is equal to the cylinder diameter and represents the cylindrical boundary or visible extents of the surface

63 Single Curved Surfaces Because a curved surface is rectangular in two views, you MUST indicate the radial center with centerlines. Crossing centerlines are used in the circular view. A single centerline is shown in each rectangular view.

64 Single Curved Surfaces-Holes Holes follow the same rules as external curved surfaces, except the cylindrical boundaries are shown as hidden lines.

65 Drawing Single Curved Surfaces in Isometric A circle appears as an ellipse in an isometric view 1. Locate the center of the circle 2. Locate the radial points 3. Sketch the "bounding box" for the ellipse 4. Sketch the four arcs of the ellipse tangent to the box edges 5. Complete the rest of the object Circle Center

66 Course Software.. Work through the Inclined and Curved Surfaces Software Module

67 Orthographic Projection of Inclined & Curved Surfaces Homework Complete the following pages in Module 5 inc/crv

68 Flat Patterns Module 6 HIgherEdServices.org

69 Session Topics Flat Patterns Fold Lines Multiple Patterns Open Surfaces Closed Surfaces Patterns with markings

70 Flat Patterns Sometimes it is important to visualize how a flat pattern can be folded up to obtain a 3-D object

71 Fold Lines The solid lines on a flat pattern are fold lines When visualizing creating a 3-D object from a flat pattern, think about folding it at the fold lines

72 Multiple Patterns Most objects have more than one pattern that could be folded to form it

73 Open Objects from Patterns No "ends" are included in the pattern, so it is folded up and an open object results Tube from paper towels

74 Closed Objects from Flat Patterns When the pattern includes "ends" a closed surface (object) results

75 Markings on Patterns When there are markings on a pattern that are on adjacent sides Markings must end up adjacent to one another on the object Markings must end up in the same orientation on the object

76 Markings on Patterns (continued)

77 Course Software Work through the Flat Patterns software module

78 Flat Patterns Homework Complete the following pages in Module 6 fp

79 Rotation of Objects about a Single Axis Module 7 HigherEdServices.org

80 Session Topics Object Rotations Right Hand Rule Rotation Notation Single Rotation Multiple Rotations Equivalent Rotations Computer Module

81 Object Rotation A rotation is a turning of an object about a straight line known as the axis of rotation.

82 Direction of Rotation A rotation about an axis can be either positive (counterclockwise) or negative (clockwise)

83 Right Hand Rule If you place the thumb of your right hand down the axis of rotation, your fingers will curl in the direction of the rotation

84 Arrow Coding Object Rotations can be designated by arrow coding A curved counterclockwise arrow is a positive rotation A curved clockwise arrow is a negative rotation The axis for the rotation is included within the notation The increment for the rotation is always 90 o

85

86 For each of the following slides, try to visualize the rotation

87 Y

88 Y

89 Y

90

91

92

93 Multiple Rotations For multiple rotations about the same axis, use an arrow for each rotation of 90º. X X Positive 180 rotation about the x-axis Y Y Y Negative 270 rotation about the y-axis

94

95 Equivalent Rotations Sometimes one set of rotations can be replaced by a simpler set. Y Y Y is equivalent to Y

96

97 To visualize a rotation, think about moving the notation to the positive end of the axis the arrow will show you the direction of the rotation

98 Course Software Work through the Rotation of Objects About a Single Axis computer module

99 Rotation of Objects about a Single Axis Homework Complete the following pages in Module 7 rot

100 Rotation of Objects about Two or More Axes Module 8 HigherEdServices.org

101 Session Topics Rotations about Two Axes Order of Rotations Equivalent Rotations Computer Module

102 Review: Rotation of Objects Rotation: turning an object about a straight line (axis of rotation)

103 Rotation About Two Axes Objects can be rotated about two or more axes the same way they were rotated about a single axis

104 Rotation Origin When objects are rotated about two or more axes, only a single point remains in its original position

105 Arrow Coding in Multiple-Axes Rotations Arrows are placed in the order in which the rotations are performed Z X Y XX Positive 90 rotation about the z-axis followed by a negative 90 rotation about the x-axis. Positive 90 rotation about the y-axis followed by a positive 90 rotation about the x-axis followed by a negative 90 rotation about the x-axis.

106 Order of Rotations Final orientation of the object depends on the order in which the rotations were performed.

107 Order of Rotations Object rotations about two or more axes are not commutative! X Y Y X

108 Equivalent Rotations Two sets of rotations can result in the same final orientation of the object

109 Equivalent Rotations Sometimes one set of rotations can be replaced by a simpler set. X Z Y X X Y

110 Course Software.. Work through the Rotation of Objects about Two or More Axes software module

111 Rotation of Objects about Two or More Axes Homework Complete the following pages in Module 8 rot

112 Object Reflections and Symmetry Module 9 HigherEdServices.org

113 Session Topics Reflection of an object Planes of symmetry Reflections through rotations

114 Reflection Reflection Plane A reflection across a plane displays the object s mirror image. Each point, A, is associated with an image point, A, such that the plane, P, is a perpendicular bisector of the line segment AA.

115 Plane of Symmetry Plane of symmetry occurs if the parts of the object on both sides of the plane are mirror images of each other.

116 Reflection through Rotation For a symmetric object, the mirror image can occur by rotating one side of the object 180 about an axis of rotation that's in the plane of symmetry.

117 Multiple Planes of Symmetry Many objects have multiple planes of symmetry Infinite 5 2

118 Course Software.. Work through the Reflections and Symmetry software module

119 Object Reflections and Symmetry Homework Complete the following pages in Module 9 reflx/sym

120 Cutting Planes and Cross Sections Module 10 HigherEdServices.org

121 Session topics Cutting planes Cross Sections Multiple Cross Sections

122 Cutting Planes A cutting plane is an imaginary plane that slices through an object Cutting Plane Cross-Section

123 Cutting Planes and Cross Sections A cross section is the intersection of a cutting plane with a solid object. The result is a 2-D shape defined by the boundaries of the original object

124 The shape of the resulting cross section depends on the orientation of the cutting plane with respect to the object Cross Sections

125 Multiple Cross Sections Objects can produce several cross sections

126 Cutting Planes As a plane cuts an object, the boundary edges on the cross section that results will be parallel to the edges of the cutting plane itself Rotate the plane into position to view it "straight on"

127 Course Software... Work through the Cutting Planes and Cross Sections software module

128 Cutting Planes and Cross Sections Homework Complete the following pages in Module 10 cp/cs Hint: for pages 14 and 16 all problems have at least two correct answers

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