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Orthographic Projection

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Orthographic Projection Introduction to Viewing 14 Objectives • Introduce viewing •Compare and contrast image formation by computer with how images have been formed by architects, artists, and engineers • Learn the benefits and drawbacks of each type of view 29 Classical Viewing • Viewing requires three basic elements - Object(s) to be viewed - Projection surface (image plane) - Projectors: lines that go from the object(s) to the projection surface •Classical views are based on the relationship among these elements - Must orient the object as it should be viewed • Object assumed to be constructed from flat principal faces - Buildings, polyhedra, manufactured objects 30 15 Classical Viewing • Projectors are lines that either - converge at a center of projection (COP) - or are parallel • Standard projections project onto a plane • Such projections preserve lines - but not necessarily angles • Nonplanar projections are needed for applications such as map construction 31 Classical Projections 32 16 Parallel vs Perspective • Computer graphics treats all projections the same and implements them with a single pipeline • Classical viewing developed different techniques for drawing different types of projections • Fundamental distinction is between parallel and perspective viewing (even though mathematically parallel viewing is the limit of perspective viewing) 33 Parallel Projection 34 17 Perspective Projection 35 Taxonomy of Planar Geometric Projections planar geometric projections parallel perspective 1 point2 point 3 point multiview axonometric oblique orthographic isometric dimetric trimetric 36 18 Orthographic Projection Projectors are orthogonal to projection surface 37 Advantages and Disadvantages • Preserves both distance and angle - Shapes are preserved - Can be used for measurements • Building plans • Manfactured parts • Cannot see what object really looks like because many surfaces hidden from view - Often we add the isometric 38 19 Multiview Orthographic Projection • Projection plane parallel to principal face • Usually form front, top, side views isometric (not orthographic) top in CAD and architecture, we often display three multiviews plus isometric front side 39 Axonometric Projections Move the projection plane relative to object classify by how many angles of a corner of a projected cube are the equal to each other three: isometric 1 two: dimetric 2 3 none: trimetric 40 20 Types of Axonometric Projections 41 Advantages and Disadvantages • Used in CAD applications • Parallel Lines preserved but angles are not • Can see three principal faces of a box-like object • Some optical illusions possible - Parallel lines appear to diverge • Does not look real because far objects are scaled the same as near objects 42 21 Oblique Projection Arbitrary relationship between projectors and projection plane 43 Advantages and Disadvantages • Angles emphasize a particular face - Architecture: plan oblique, elevation oblique • Angles in faces parallel to projection plane are preserved while we can still see “around” side • No physical analogue, cannot create with real- world camera 44 22 Perspective Projection Projectors converge at center of projection 45 Vanishing Points • Parallel lines on the object not parallel to the projection plane converge at a single point in the projection (called the vanishing point) • Can draw simple perspectives (by hand) using the vanishing point vanishing point 46 23 One-Point Perspective • One principal face parallel to projection plane • One vanishing point for cube 47 Two-Point Perspective • On principal direction parallel to projection plane • Two vanishing points for cube 48 24 Three-Point Perspective • No principal face parallel to projection plane • Three vanishing points for cube 49 Advantages and Disadvantages • Objects further from viewer are projected smaller than the same sized objects closer to the viewer - Looks realistic, gives sense of depth • Equal distances along a line are not projected into equal distances (nonuniform foreshortening) • Angles preserved only in planes parallel to the projection plane • More difficult to construct by hand than parallel projections (but not more difficult by computer!) 50 25 Computer Viewing • Need to build transformation that defines the projection plane based on the chosen projection 51 Orthogonal Projection 52 26 Orthogonal Projection •Set z =0 • Equivalent homogeneous coordinate transformation: Morth px' 1 0 0 0 px py' 0 1 0 0 py = 0 0 0 0 0 pz 1 0 0 0 11 53 Oblique Projections 54 27 Oblique Projections Direction of projection dop = (dopx,dopy,dopz) z dop x Lets say we project onto z = 0 plane 55 Oblique Projections Direction of projection dop = (dopx,dopy,dopz) z px' 1 0 -dopx/dopz 0 px py dop py' 0 1 - dopy/dopz 0 = 0 0 0 0 0 pz 1 0 0 0 1 1 note: tan = dopz/dopx x 56 28 Shear •Helpful to add one more basic transformation •Equivalent to pulling faces in opposite directions 57 Shear Matrix Consider simple shear along x axis x’ = x + y shy y’ = y z’ = z 1 shy 0 0 0 1 0 0 H() = 0 0 1 0 0 0 0 1 58 29 Oblique Projection 59 Computer Viewing There are three aspects of the viewing process: - Selecting a lens Setting the projection matrix - Positioning the camera Setting the view-orientation matrix - Clipping Setting the view volume 60 30.
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