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Descriptive Geometry• Descriptive geometry the branch of geometry which allows the representation of three-dimensional objects in two dimensions, by using a specific set of procedures. 2
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Descriptive Geometry • Gaspard Monge is usually considered the "father of descriptive geometry". • He first developed his techniques to solve geometric problems in 1765 while working as a draftsman for military fortifications, and later published his findings 3
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Orthographic Projection• Orthographic projection (or orthogonal projection) is a means of representing a three-dimensional object in two dimensions.• It is a form of parallel projection, where all the projection lines are orthogonal to the projection plane. 5
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Orthographic Box• Space is defined as the infinite extension of the three dimensional region in which all matter exists.• In order to study an object in space, it is necessary to “enclose” the object so that we can make use of the enclosure as a reference point.• This is done with the use of the orthographic box. 6
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Orthographic Box • An orthographic box is an imaginary glass box that encloses the object in space being studied. 7
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Orthographic Box • The sides of the box are called planes of projection. • By projecting the points of the object perpendicularly to the planes of projection we are able to form an IMAGE of the object 8
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Orthographic Box • Depending on their position relative to the person viewing the orthographic box, the images are called either TOP VIEW image, FRONT VIEW image, or SIDE VIEW image 9
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Planes of Projection• There are two main types of planes of projection • Principal planes of projection • Auxiliary planes of projection 11
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Principal Planes of Projections• Principal planes of projection are the main “faces” of the orthographic box.• They are labelled depending on their relative position to the viewer.• There are basically three principal planes of projections: • Frontal Plane • Horizontal Plane • Profile Plane 12
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Frontal Plane • The plane facing the viewer is called the Frontal Plane (or F-plane). • This is where the front view images are projected 13
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Horizontal Plane • The plane perpendicular to but horizontal to the viewer is the Horizontal Plane (or H-plane). • This is where the top view images are projected. 14
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Profile Plane • The plane perpendicular to but vertical to the viewer is the Profile Plane (or P- plane). • This is where the side-view images are projected 15
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Alternatives• It is also possible to project an image backwards, to the left, and downwards to get the rear view, left side view, and the bottom view respectively.• In most problems, however, we normally consider its two dimensional images in at least two views.• Usually the top view and front view images will suffice.• Following the American system for technical drawings, the right side view may be used when a side view is necessary. 16
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Auxiliary Plane of Projection• The auxiliary planes of projections are any planes which are neither parallel to nor perpendicular to the viewer.• Their orientation is arbitrary.• Their labels are also arbitrary, i.e. they may use any letters other than the letters used to label the principal planes of projections 17
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The Auxiliary Plane • In this figure, the object has an inclined surface that does not appear in its true size and shape in any regular view. The auxiliary plane is assumed parallel to the inclined surface P 18
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The Auxiliary Plane • The auxiliary plane is then perpendicular to the frontal plane of projection and is hinged to it. The inclined surface is shown in its true size & shape in the auxiliary view. The long dimension of the surface is projected directly from the front view and the depth from the top view 19
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Points in space • A point is a theoretical location in space having no dimension. A point in space may be defined by its coordinates from a fixed reference. • P = (x,y,z). 21
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Projection of a point • Given a point P = (x,y,z), projecting P into the planes of the glass box shows its x and z coordinates in the front plane, its x and y coordinates in the top plane and its y and z coordinates in the profile plane. 22
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Projection of a point • Unfolding the glass box provides a complete description of the point’s location in the 3D space. 23
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The Folding Line • The folding line is the intersection between two projection planes 24
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The Folding Line • The folding line TF between the top and the front views is the intersection of the horizontal and frontal planes. 25
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The Folding Line • Folding line FR, between the front and the right side views is the intersection of the frontal & the right side planes. 26
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Point projection constraints • Since the x coordinate of a point appears in both the frontal view and the top view, its projections in those two views must be horizontally aligned. 27
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Point projection constraints • Since the z coordinate of a point appears in both the frontal view and the right side view, its projection in those two views must be vertically aligned. 28
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Point projection constraints • Since the y coordinates of a point appears in both the top view and the right side view, the distance of the projections from the folding lines must be the same. 29
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Rule 1: The alignment rule • The projections of a point in two consecutive views are aligned with respect to the normal to the folding line. 30
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