Projections Projectors A Center of Projection A B B Projection Plane Perspective
Projections Projectors A A B At Infinit B Projection Plane Parallel
Parallel Projections Orthographic 3D Viewing Top View Y Z X Front View Side View
Parallel Projections Orthographic On plane 3D Viewing Multiviews ( or or planes), one View is not adequate True sie and shape for lines P
Parallel Projections Orthographic 3D Viewing
Parallel Projections Aonometric Additional rotation,translation and then projection on plane [ ][ ] [ ] T T U ; ; f f f
Parallel Projections Aonometric 3D Viewing Three tpes Trimetric: No foreshortening is the same. Dimetric: Two foreshortenings are the same. Isometric: All foreshortenings are the same.
Parallel Projections Aonometric 3D Viewing f Trimetric f f Dimetric f f Isometric f f f
Parallel Projections Isometric Let there be rotations a) about -ais φ b) about -ais θ T
Parallel Projections Isometric 3D Viewing Let there be rotations a) about -ais φ b) about -ais θ T
Parallel Projections [ ][ ] T U Isometric
Parallel Projections f f f Isometric
Parallel Projections Isometric 3D Viewing f f f f f f f f Solving equations find θ, φ and f
Parallel Projections Oblique 3D Viewing Non-perpendicular projectors to the plane of projection True shape and sie for the faces parallel to the projection plane is preserved
Parallel Projections Oblique 3D Viewing
Parallel Projections 3D Viewing Oblique p L φ p L φ P(,,) P ( p, p ) L α φ P (,) tan α /L or L cot α
Parallel Projections 3D Viewing Oblique P(,,) P ( p, p ) L α φ P (,) When α 45 o > Cavalier Lines perpendicular to the projection plane are not foreshortened When cot α ½ > Cabinet Lines perpendicular to the projection plane are foreshortened b half φ is tpicall 3 or 45
Perspective Projections Center of Projection Projectors A A B B Projection Plane Parallel lines converge Non-uniform foreshortening Helps in depth perception, important for 3D viewing Shape is not preserved
Perspective Projections
Perspective Projections Matri Form [ ] [ ] [ ] r r r r r
Perspective Projections Matri Form Projection on plane [ ][ ] [ ] r r r P P P T r r
Perspective Projections Geometricall Z P(,,) X Y P(,) c l l c c c l l l l l l,
Perspective Projections Geometricall Y c P(,) l l c Z P(,,) X c c When r - / c this becomes same as obtained in matri form c
Perspective Projections Vanishing Point Set of parallel lines not parallel to the projection plane converge to Vanishing Point Y VP Z X
Perspective Projections Vanishing Point Point at infinit on Z ais : (homogenous) [ ] [ ] [ ] [ ] [ ] [ ] ' ' ' r r r w Recall r -/ c, the vanishing point is at c
Perspective Projections Single Point Perspective [ ] [ ] [ ] p p p p p COP on X-ais COP (-/p ) VP (/p )
Perspective Projections Single Point Perspective [ ] [ ] [ ] q q q q q COP on Y-ais COP ( -/q ) VP ( /q )
Perspective Projections Two Point Perspective P pq [ P ][ P ] p q p q
Perspective Projections Three Point Perspective P pqr [ P ][ P ][ P ] p q r p q r
Perspective Projections
Generation of Perspective Views Additional transformation and then gle point perspective transformation Simple Translation: Translation (l,m,n),cop c,projection plane rn m l r r n m l T
Generation of Perspective Views Y X Translation along line:
Generation of Perspective Views Translation in Z > Scaling COP Projection plane
Generation of Perspective Views Rotation [ ][ ] r P R T r Rotation about Y-ais b φ
Generation of Perspective Views Rotation Rotation about Y-ais b φ T [ R ][ P ] r r r > Two Point Perspective Transformation
Generation of Perspective Views Rotation Two Rotations a) about Y-ais b φ b) about X-ais b θ [ ][ ][ ] r P R R T r
Generation of Perspective Views Rotation Two Rotations a) about Y-ais b φ b) about X-ais b θ T r r r > Three Point Perspective Transformation