Viewing in 3D (Chapt. 6 in FVD, Chapt. 12 in Hearn & Baker)

Transcription

1 Viewing in 3D (Chapt. 6 in FVD, Chapt. 2 in Hearn & Baker)

2 Viewing in 3D s. 2D 2D 2D world Camera world 2D 3D

3 Transformation Pipe-Line Modeling transformation world Bod Sstem Viewing transformation Front- Wheel Sstem Viewer Sstem Transformation To iewport Projection Transformation

4 Transformation Pipe-Line Modeling Coordinates Modeling Transformation World Coordinates Viewing Transformation Viewing Coordinates Projection Transformation Projection Coordinates Workstation Transformation Deice Coordinates

5 Specifing the Viewing Coordinate Sstem w world w w Viewer Sstem P Viewing Coordinates sstem, P, [,, ].

6 Specifing the Viewing Coordinate Sstem world Viewer Sstem P A iewing plane (projection plane) is set up perpendicular to and aligned with (, ).

7 Specifing the Viewing Coordinate Sstem Camera definition ( option): w w L w N P P=(P,P,P) camera location. L - is a point to look-at. V - is the iew-up ector. V projection onto the iew-plane is directed up.

8 How to form Viewing coordinate sstem : The transformation, M, from worldcoordinate into iewing-coordinates is: Defining the camera in OpenGL glmatrimode(gl_modelview); glloadidentit(); # camera looking down glulookat(p,p,p,l,l,l,v,v,v ) X Z Y Z V Z V X L P L P Z ; ; R T P P P M

9 Projections Viewing 3D objects on a 2D displa requires a mapping from 3D to 2D. Projectors are lines from the center of projection through each point in the object. A projection is formed b the intersection of the projectors with a iewing plane. Center of Projection Projectors

10 Center of projection at infinit results with a parallel projection. A parallel projection preseres relatie proportions of objects, but does not gie realistic appearance (commonl used in engineering drawing).

11 A center of projection at a finite distance results with a perspectie projection. A perspectie projection produces realistic appearance, but does not presere relatie proportions.

12 Parallel Projection Projectors are all parallel. Orthographic: Projectors are perpendicular to the projection plane. Oblique: Projectors are not necessaril perpendicular to the projection plane. Orthographic Oblique

13 If the iewing plane is aligned with (, ), orthographic projection is performed b: Orthographic Projection p p P (,,) (,)

14 Lengths and angles of faces parallel to the iewing planes are presered (Plan View). Problem: 3D nature of projected objects is difficult to deduce.

15 Oblique Projection Projectors are not perpendicular to the iewing plane. Angles and lengths are presered for faces parallel to the plane of projection. preseres 3D nature of an object. ( p, p ) (,,) (,)

16 Seeral Oblique Projections =45 o =3 o Caalinear Projections (=45 o ) of a cube for two alues of angle (angle to the horion) Caalinear projection : Preseres lengths of lines perpendicular to the iewing plane. 3D nature can be captured but shape seems distorted. Can displa a combination of front, side, and top iews.

17 Seeral Oblique Projections.5.5 =45 o =3 o Cabinet Projections (= 63.4 o ) of a cube for two alues of angle (angle to horion) Cabinet projection: lines perpendicular to the iewing plane project at /2 of their length. A more realistic iew than the Caalinear projection. Can displa a combination of front, side, and top iews.

18 Perspectie Projection In a perspectie projection, the center of projection is at a finite distance from the iewing plane. The sie of a projected object is inersel proportional to its distance from the iewing plane. Parallel lines that are not parallel to the iewing plane, conerge to a anishing point.

19 Perspectie Projection A anishing point is the projection of a point at infinit. Z-ais anishing point

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21 Vanishing Points There are infinitel man general anishing points. There can be up to three principal anishing points (ais anishing points). Perspectie projections are categoried b the number of principal anishing points, equal to the number of principal aes intersected b the iewing plane. Most commonl used: one-point and two-points perspectie.

22 One point ( ais) perspectie projection ais anishing point. ais anishing point. Two points perspectie projection

23 Perspectie Projection ( p, p,) (,,) center of projection d d focal length

24 Using similar triangles it follows: d d d d p p ; d - (,,) p ; ; p p p d d d d (,)

25 Thus, a perspectie projection matri is defined: d M per d d d P M per ; ; p p p d d d d

26 Obserations M per is singular ( M per =), thus M per is a man to one mapping. Points on the iewing plane (=) do not change. The homogeneous coordinates of a point at infinit directed to (U,U,U ) are (U,U,U,). Thus, The anishing point of parallel lines directed to (U,U,U ) is at [du /U, du /U ]. When d, M perspectie M ortho

27 What is the difference between moing the center of projection and moing the projection plane? Original Center of Projection Projection plane Moing the Center of Projection Center of Projection Moing the Projection Plane Projection plane Center of Projection Projection plane

28 Summar Planar geometric projections Parallel Perspectie Oblique Orthographic Top Front Caalinear Side Other Cabinet Other One point Two point Three point

29 Demo

30 Viewing Window After objects were projected onto the iewing plane, an image is taken from a Viewing Window. A iewing window can be placed anwhere on the iew plane. In general the iew window is aligned with the iewing coordinates and is defined b its etreme points: (l,b) and (r,t)

31 Viewing Volume Gien the specification of the iewing window, we can set up a Viewing Volume. Onl objects inside the iewing olume will appear in the displa, the rest are clipped.

32 In order to limit the infinite iewing olume we define two additional planes: Near Plane and Far Plane. Onl objects in the bounded iewing olume will appear. The near and far planes are parallel to the iewing plane and specified b near and far. A limited iewing olume is defined: For orthographic: a rectangular parallelpiped. For oblique: an oblique parallelpiped. For perspectie: a frustum. Far Plane Near Plane Far Plane Near Plane

33 The Viewing Plane can be placed anwhere along the Z ais, as long it does not contain the center of projection. Far Plane Near Plane

34 Defining the iewing Volume top right left far near bottom top right left far near bottom

35 Defining the Viewing Volume Definition for Orthographic Projection: glmatrimode(gl_projection) ; glloadidentit(); # camera looks down. glortho(l,r,b,t,n,f); top right left far near bottom

36 Definition for general Perspectie Projection: glmatrimode(gl_projection) ; glloadidentit(); glfrustum(l,r,b,t,n,f); top right left far near bottom Note: near and far alues are quantities, e.g. near=5. means that the near plane passes at Z =-5.

37 Definition for standard Perspectie Projection: glmatrimode(gl_projection) ; glloadidentit(); gluperspectie(iewangle,aspectratio,n,f); h w near far Note: aspectratio=w/h.

38 OpenGL Transformation Pipe-Line Homogeneous coordinates in World Sstem ModelView Matri Projection Matri Clipping Viewing Coordinates Projection Coordinates Viewport Transformation Window Coordinates