Projection Lecture Series

Transcription

1 Projection Lecture Series Prof. Gary Wang Department of Mechanical and Manufacturing Engineering The University of Manitoba

2 Overview Coordinate Systems Local Coordinate System (LCS) World Coordinate System (WCS) Viewing Coordinate System (VCS) Screen Coordinate System (SCS) Projection Lab Parallel Projection Perspective Projection 2

3 Local (Working) Coordinate System (LCS) Attached to the modeled object Defines the size and shape of the object Facilitates Geometric Construction B A X A X B A X 3

4 World (Global or Model) Coordinate System (WCS) A Cartesian coordinate system independent of viewing or display Default coordinate system for a CAD package Known as the scene universe All geometrical data of modeled objects are saved with respect to it. Y w X w Z w 4

5 Transformation Translation, rotation, and reflection preserve the lengths of line segments and the angles between segments. Uniform scaling preserves angles but not lengths. Nonuniform scaling and shearing do not preserve angles or lengths; Translation Rotation Uniform Scaling Nonuniform Scaling Reflection Shearing 5

6 Modeling Transformation From LCS to WCS (3D 3D) Y L Z L Y w X L X w Z w 6

7 Transformation Pipeline A sequence of transformations from the infinite and continuous threedimensional WCS to the finite and discrete two-dimensional screen coordinate system 7

8 Projection Transforms a point in n-space to m-space (m < n), e.g. 3D 2D Terms Center of projection (C) Projection plane Projectors Parallel Projection and Perspective Projection 8

9 Parallel Projection If the center of projection is at an infinite distance from the projection plane, all the projectors become parallel (meet at infinity) and parallel projection results. Parallelism preserved Dimensions and shape preserved A Useful in engineering drawings. C (at infinity) Projector B A Projection Plane Object B 9

10 Perspective Projection If the center of projection is at a finite distance from the projection plane, perspective projection results and all the projectors meet at the center of projection. A Parallelism not preserved Dimensions and angles changed Applied to the artistic effect Projector A B Object B C Projection Plane

11 Orthographic Projections (Parallel) Orthographic If the direction of projection is normal to the projection plane, this type of parallel projection is orthographic projection. For engineering drawings - projection plane perpendicular to one of the principal axes of the WCS; that is, direction of projection coincides with one of principal axes of the WCS. Angles preserved but not necessarily lengths. Isometric projection

12 Isometric projection Three principal axes (WCS) equally foreshortened on the projection plane Measurements along the axes of the WCS made with the same scale Projection (Center of Projection and Projection Plane) Iso -> equal Metric -> measure Perspective Projection Parallel Projection (Projection Direction v.s. Projection Plane) Orthographic Projection (Projection Plane v.s. WCS axes) Oblique Projection Engineering Drawing Isometric Projection 2

13 Viewing Coordinate System (VCS) 3D coordinate system (right-handed or left-handed) Viewpoint (eye or camera) corresponds to the center of projection View plane corresponds to the projection plane Z v defines the viewing direction (projection direction), which is normal to the view plane View Plane Zv Eye at infinity Yv Viewing direction Window Xv 3

14 Screen/Device Coordinate System (SCS) 2D system to show the image on the display eventually Device coordinate system Measured by pixels for raster graphics displays Y Pixel (7, 7) y s X A normalized SCS is called Virtual Device Coordinate System O s x s 4

15 Viewport Mapping Map a 2D image to the Viewport on the Normalized SCS, and finally, the 2D image will be mapped from the Normalized SCS to the SCS. Viewport A viewport is an area of the display screen on which the window data is presented. (Kunwoo Lee, 999) 5

16 Transformation Pipeline Coordinate Values in Local Coordiante System 3D -> 3D Modeling Transforamtion Yw X w Coordinate Values in World Coordiante System Z w WCS 3D -> 3D Viewing Transforamtion Yw Yv Z w WCS X w Coordinate Values in Viewing Coordiante System Xv Z v VCS 3D -> 2D Projection Transforamtion Coordinate Values in Window y s 2D -> 2D Viewport Mapping O s x s Coordinate Values in Normalized Screen Coordinate System (Normalized Viewport) Coordinate Values in Screen Coordinate System (Viewport) 6

17 A Graphic Illustration of 4 Coordinate Systems Local Coordinate System (Kunwoo Lee, 999) 7

18 Lab Z V Table Point coordinates of the object Points x y z 3 Y V X V

19 Yv Yw Align the VCS and the WCS along their corresponding axes and origins, and Zv defines the viewing direction. Use Y v -X v plane as the view plane Top Front Right Xw Zv Zw Xv Rotate properly and then project it Yv Zw Top Xv, Xw Zw Yw Yv Zv Front Top Right Xw Xv Yv,Yw Yv, Yw Front Right Xv, Xw Zw Xv 9

20 Y Top Front Right X Z Yv,Y Front Pv P Xv, X 2

21 2 Front Top Right X Y Z Top Xv, X Yv Z ( ) P P Pv ) (9 9 ) (9 ) (9 9(CCW)

22 22 Front Top Right X Y Z Right Xv Yv, Y Z ( ) P P Pv ) 9 ( 9 ) 9 ( ) 9 ( -9(CW)

23 Isometric Projection (Ry( --> > Rx ) Three principal axes (WCS) equally foreshortened on the viewing plane Use unit vector along each direction representing the length for principal axes Yw Zw Top Front P Right Xw Zw Yw Yv Zv Front Top Right Xw Xv Projection plane: Yv-Xv 23

24 24 P P Ty Tx Pv ] ][ [ P Isometric Projection ( Isometric Projection (Ry Ry -- --> Rx ) > Rx )

25 25 v v v v v v y x y x y x ± ± , 45 Contd.

26 Other possible rotation paths Rx --> Ry r x ± 45, ry ± Rz --> Ry(Rx) r z ± 45, r y( x ) ± Rx(Ry) --> Rz ry ( x) ± 45, rz ANY ANGLE 26

27 Notes The viewing coordinate system is different from the ones in the notes, i.e., you cannot simply plug in the equation to create the orthographic views. Add a line between two points (x,y) and (x2, y2) Plot([x, x2],[y, y2]); Erase a lineplot([x, x2],[y, y2], w ); Rz45 o, Rx o for the case 27

28 Notes (cont d) You have to move the views to the right spot as indicated in Figure 4 (translation?) How to manage the points? Why? Arrange them sequentially in a matrix, then remove/add the necessary ones Group them to sub-matrices and transform them individually Create an index matrix letting the system know which ones are connected 28

29 Summary Graphical Coordinate Systems Various Transformations Orthographic Projection Isometric Projection Lab 29