1 / 28

3D Viewing

3D Viewing. Perspective Projections. Single Point Perspective. COP on X-axis. COP (-1/p 0 0 1) VP x (1/p 0 0 1). 3D Viewing. Perspective Projections. Two Point Perspective. 3D Viewing. Perspective Projections. Three Point Perspective. 3D Viewing. Perspective Projections. Y. X. Z.

moke
Download Presentation

3D Viewing

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 3D Viewing Perspective Projections Single Point Perspective • COP on X-axis • COP (-1/p 0 0 1) VPx (1/p 0 0 1)

  2. 3D Viewing Perspective Projections Two Point Perspective

  3. 3D Viewing Perspective Projections Three Point Perspective

  4. 3D Viewing Perspective Projections

  5. Y X Z 3D Viewing Vanishing Points • Two ways • Intersection of transformed lines • Transformation of points at infinity Y VPz VPx X

  6. 3D Viewing Plane Geometric Projections • Parallel • Perspective • Single Point • Orthographic • Axonometric • Oblique • Two Point • Trimetric • Dimetric • Isometric • Three Point • Cavalier • Cabinet

  7. Y X Z 3D Viewing Implementation Issues More from Interface point of view V Eye U N • Viewing Coordinate System (VCS) • World Coordinate System • (WCS)

  8. 3D Viewing View Coordinate System (VCS) • Viewing coordinate system • Position and orientation of the view plane • Extent of the view plane (window) • Position of the eye • View Plane • View Reference Point (VRP): the origin of VCS specified as (rx , ry, rz) in WCS: center of the scene • Normal to the view plane (nx , ny, nz )

  9. Z  r Y X  3D Viewing View Coordinate System (VCS) • View Plane • Normal Direction (View Plane Normal VPN) n (nx ,ny ,nz) • User may provide normalized vector • e.g. • nx = sin  cos • ny = sin  sin  • nz = cos 

  10. 3D Viewing View Coordinate System (VCS) • View Plane • Direction v • v is a unit vector intuitively corresponding to “up” vector • “up” vector is specified by the user in WCS up’ = up – (up.n)n v = up’ / |up’| up’ up n v • Direction u • u = n x v ( Left Handed)

  11. wr wt n v u e wl wb 3D Viewing View Coordinate System (VCS) • Window and Eye • Window : left, right, bottom,top (wl,wr,wb,wt) • generally is centered at VRP (origin) • Eye : e = (eu,ev,en) • Typically e = (0,0,-E)

  12. 3D Viewing Transformation from WCS to VCS v Y (x, y) O’ u r O X

  13. 3D Viewing Transformation from WCS to VCS • Point object is represented as • (a,b,c) in VCS • (x,y,z) in WCS

  14. 3D Viewing Transformation from WCS to VCS • Conversion from one coordinate system to another • Therefore a=(p-r).u, b=(p-r).v, c=(p-r).n

  15. 3D Viewing Transformation from WCS to VCS • In Homogenous Coordinates • (a,b,c,1) = (x,y,z,1) Awv

  16. 3D Viewing Transformation from WCS to VCS • In Homogenous Coordinates • r’= -rMT = (-r.u,-r.v,-r.n) = (rx’,ry’,rz’) • puvn=pxyzAwv

  17. p t=1 u n p* v p (pu,pv,pn) t=t’ e e t=0 p*(u*,v*) 3D Viewing Transformation from VCS to View Plane Parametrically r(t) = e(1-t)+p.t

  18. 3D Viewing Transformation from VCS to View Plane On u-v plane, r(t)n = 0

  19. 3D Viewing Transformation from VCS to View Plane When eye is on n-axis eu=ev=0 u*=pu/(en-pn), v*=pv/(en-pn) Matrix form (n*=0) Perspective Transformation

  20. 3D Viewing Transformation from VCS to View Plane Using Perspective Transformation Mp

  21. 3D Viewing Transformation from VCS to View Plane If eye is off n-axis we have another matrix p*=(pu,pv,pn,1)MsMp q : in WCS maps to p*=qAwvMsMp

  22. 3D Viewing View Volume Eye View Plane, n=0 Front Plane n=F Back Plane n=B

  23. 3D Viewing View Volume v v wt wt n n F B wb wb F/(1-F/en) B/(1-B/en)

  24. v 1 Vt Vb 0 u Vl Vr 3D Viewing Volume Normalization Transformation

  25. 3D Viewing Volume Normalization Transformation For n no nt F/(1-F/en) B/(1-B/en) 0 1 Scaling sn Translation rn

  26. 3D Viewing Volume Normalization Transformation where Total Transformation: AwvMsMpN

More Related