Computer Graphics. Jeng-Sheng Yeh 葉正聖 Ming Chuan University (modified from Bing-Yu Chen s slides)
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1 Computer Graphics Jeng-Sheng Yeh 葉正聖 Ming Chuan Universit (modified from Bing-Yu Chen s slides)
2 Viewing in 3D 3D Viewing Process Specification of an Arbitrar 3D View Orthographic Parallel Projection Perspective Projection 3D Clipping for Canonical View Volume
3 3D Viewing Process Clip against view volume Project onto projection plane Transform into viewport in 2D device coordinates for displa 3D world-coordinate output primitives Clipped world coordinates 2D device coordinates
4 Classical Viewing Viewing requires three basic elements One or more objects A viewer with a projection surface Projectors that go from the object(s) to the projection surface Classical views are based on the relationship among these elements The viewer picks up the object and orients it how she would like to see it Each object is assumed to constructed from flat principal faces Buildings, polhedra, manufactured objects
5 Planar Geometric Projections Standard projections project onto a plane Projectors are lines that either converge at a center of projection are parallel Such projections preserve lines but not necessaril angles Nonplanar projections are needed for applications such as map construction
6 Classical Projections Front elevation Elevation oblique Isometric One-point perspective Plan oblique Three-point perspective
7 Perspective vs. Parallel Computer graphics treats all projections the same and implements them with a single pipeline Classical viewing developed different techniques for drawing each tpe of projection Fundamental distinction is between parallel and perspective viewing even though mathematicall parallel viewing is the limit of perspective viewing
8 Taonom of Planar Geometric Projections planar geometric projections parallel perspective multiview aonometric orthographic point 2 point 3 point oblique isometric dimetric trimetric
9 Perspective Projection
10 Parallel Projection
11 Orthographic Projection Projectors are orthogonal to projection surface
12 Multiview Orthographic Projection Projection plane parallel to principal face Usuall form front, top, side views isometric (not multiview orthographic view) front in CAD and architecture, we often displa three multiviews plus isometric top side
13 Advantages and Disadvantages Preserves both distances and angles Shapes preserved Can be used for measurements Building plans Manuals Cannot see what object reall looks like because man surfaces hidden from view Often we add the isometric
14 Aonometric Projections Allow projection plane to move relative to object classif b how man angles of a corner of a projected cube are the same none: trimetric two: dimetric three: isometric θ 2 θ θ 3
15 Tpes of Aonometric Projections Dimetric Trimetric Isometric
16 Advantages and Disadvantages Lines are scaled (foreshortened) but can find scaling factors Lines preserved but angles are not Projection of a circle in a plane not parallel to the projection plane is an ellipse Can see three principal faces of a bo-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 Used in CAD applications
17 Oblique Projection Arbitrar relationship between projectors and projection plane
18 Advantages and Disadvantages Can pick the angles to emphasie a particular face Architecture: plan oblique, elevation oblique Angles in faces parallel to projection plane are preserved while we can still see around side In phsical world, cannot create with simple camera; possible with bellows camera or special lens (architectural)
19 Specification of an Arbitrar 3D View v VUP n VPN VRP View plane u VRP: view reference point VPN: view-plane normal VUP: view-up vector
20 VRC: the viewing-reference coordinate sstem v (u ma,v ma ) (u min,v min ) CW VRP View plane n VPN u CW: center of the window
21 Infinite Parallelepiped View Volume DOP PRP CW VPN View plane VRP DOP: direction of projection PRP: projection reference point n
22 Truncated View Volume for an Orthographic Parallel Projection Front Clipping plane View plane VRP Back Clipping plane VPN DOP F B
23 The Mathematics of Orthographic Parallel Projection Projection plane p View along ais View along ais p Projection plane P(,, ) P(,, ) p M ort ; p ; p
24 The Steps of Implementation of Orthographic Parallel Projection Translate the VRP to the origin Rotate VRC such that the VPN becomes the ais Shear such that the DOP becomes parallel to the ais Translate and scale into the parallel-projection canonical view volume N par S par T par SH par R T ( VRP)
25 Snthetic Camera Model projector p image plane projection of p center of projection
26 Perspective Projection Projectors coverge at center of projection
27 Truncated View Volume for an Perspective Projection Front Clipping plane View plane VRP Back Clipping plane VPN F B
28 Perspective Projection (Pinhole Camera) P(,, ) P(,, ) d d p p View along ais View along ais Projection plane Projection plane / / ; / ; d M d d d d per p p p p
29 Perspective Division However W, so we must divide b W to return from homogeneous coordinates d d P M W Z Y X per p p p / ( ) d d d W Z W Y W X p p p, /, /,,,,
30 The Steps of Implementation of Perspective Projection Translate the VRP to the origin Rotate VRC such that the VPN becomes the ais Translate such that the PRP is at the origin Shear such that the DOP becomes parallel to the ais Scale such that the view volume becomes the canonical perspective view volume N per S per SH per T ( PRP) R T ( VRP)
31 Alternative Perspective Projection P(,, ) P(,, ) d d p p View along ais View along ais Projection plane at Projection plane at / ) / ( ; ) / ( ; d M d d d d d d per p p p p
32 Vanishing Points Parallel lines (not parallel to the projection plan) on the object converge at a single point in the projection (the vanishing point) Drawing simple perspectives b hand uses these vanishing point(s) vanishing point
33 Three-Point Perspective No principal face parallel to projection plane Three vanishing points for cube
34 Two-Point Perspective On principal direction parallel to projection plane Two vanishing points for cube
35 One-Point Perspective One principal face parallel to projection plane One vanishing point for cube
36 Advantages and Disadvantages Objects further from viewer are projected smaller than the same sied objects closer to the viewer (diminuition) Looks realistic Equal distances along a line are not projected into equal distances (nonuniform foreshortening) Angles preserved onl in planes parallel to the projection plane More difficult to construct b hand than parallel projections (but not more difficult b computer)
37 Canonical View Volume for Orthographic Parallel Projection or - Back plane - Front plane - -, -,,, -
38 The Etension of the Cohen-Sutherland Algorithm bit point is above view volume > bit 2 point is below view volume < - bit 3 point is right of view volume > bit 4 point is left of view volume < - bit 5 point is behind view volume < - bit 6 point is in front of view volume >
39 Intersection of a 3D Line a line from to can be represented as so when t ),, ( P ),, ( P ) ( ) ( ) ( t t t ) )( ( ) )( ( + +
40 Canonical View Volume for Perspective Projection or - Back plane - Front plane -,, - min -, -, -
41 The Etension of the Cohen-Sutherland Algorithm bit point is above view volume > - bit 2 point is below view volume < bit 3 point is right of view volume > - bit 4 point is left of view volume < bit 5 point is behind view volume < - bit 6 point is in front of view volume > min
42 Intersection of a 3D Line so when + + ) ( ) ( ) )( ( ) ( ) ( ) )( (
43 Clipping in Homogeneous Coordinates Wh clip in homogeneous coordinates? it is possible to transform the perspective-projection canonical view volume into the parallel-projection canonical view volume M min, + min + min min
44 Clipping in Homogeneous Coordinates The corresponding plane equations are X -W X W Y -W Y W Z -W Z
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