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

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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

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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

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Classical Projections

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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)

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Parallel Projection

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17 Perspective Projection

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Taxonomy of Planar Geometric Projections

planar geometric projections

parallel perspective

1 point2 point 3 point multiview axonometric oblique orthographic

isometric dimetric trimetric

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18 Orthographic Projection

Projectors are orthogonal to projection surface

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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

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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

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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

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20 Types of Axonometric Projections

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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

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21 Oblique Projection

Arbitrary relationship between projectors and projection plane

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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

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22 Perspective Projection

Projectors converge at center of projection

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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

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23 One-Point Perspective

• One principal face parallel to projection plane • One vanishing point for cube

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Two-Point Perspective

• On principal direction parallel to projection plane • Two vanishing points for cube

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24 Three-Point Perspective

• No principal face parallel to projection plane • Three vanishing points for cube

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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!)

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25 Computer Viewing

• Need to build transformation that defines the projection plane based on the chosen projection

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Orthogonal Projection

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26 Orthogonal Projection

•Set z =0 • Equivalent homogeneous coordinate transformation:

Morth  px'  0001  px  py'  0010  py   =    0   0000  pz      1   1000 1 

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Oblique Projections

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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

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28 Shear

•Helpful to add one more basic transformation •Equivalent to pulling faces in opposite directions

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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

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29 Oblique Projection

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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

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