AML710 CAD LECTURE Parallel Projections a) Orthographic Projections b) Axonometric Projections 2. Perspective Transformations and Projections

Transcription

1 AML7 CAD LECTURE 8 PROJECTIONS. Parallel Projections a) Orthographic Projections b) Aonometric Projections. Perspective Transormations and Projections PROJECTIONS Aine, Rigid-bod/Euclidian Vs Perspective Both aine and perspective transormations are - dimensional The are transormations rom one -D space to another Viewing D transormations (results) on a - Dimensional surace(screen) and requires projections rom -Space to -Space. This is known as plane geometric projection

2 PROJECTIONS Projections are a necessar part o Graphics Pipeline Modeling Transormations Rendering/ Shading Viewing Transormations Geometrical Model Visual Realism Orthographic/ Perspective Clipping Displa Rasteriation PROJECTION Graphics Pipeline PROJECTIONS - Classiication Plane Geometric Projections Parallel Perspective Orthographic Aonometric Oblique Single Pt Two Pt Three Pt Trimetric Cavelier Cabinet Dimetric Isometric

3 PROJECTIONS Parallel Vs Perspective Image screen Ininit Projectors Object Parallel Projection COP Projectors Object Image screen Perspective Projection Generalied 4 4 transormation matri in homogeneous coordinates a d T g l b e i m c j n p q r s Translations l, m, n along,, and ais Linear transormations local scaling, shear, rotation relection Perspective transormations Overall scaling

4 Orthographic projection matrices T ; T Orthographic Views ; T View C.O.Projection Front On +ve ais Right Side On +ve ais Top On +ve ais Rear On -ve ais Let Side On -ve ais Bottom On -ve ais Proj. Plane Z () X () Y () Z () X () Y () Ortho graphic views Top Y Ininit Projectors Right X Front Z 4

5 5 Eample Auiliar View Consider the position vector X Direction ines are T P T Concatenated matri c c c β α and AXONOMETRIC PROJECTIONS The limitations o orthographic projections are overcome An aonometric projection is obtained b manipulating the object, ug rotation and translations such that at least adjoining aces are shown. The result is then projected rom COP at ininit onto one o the coordinate planes,usuall on Features Unless the plane is parallel to the POP, an aonometric projection does not show its true shape Parallel lines are equall oreshortened and the relative lengths o parallel lines remain constant

6 TRIMETRIC PROJECTIONS Arbitrar rotations in arbitrar order about an or all o the coordinate aes, ollowed b parallel projections on plane. The ratios o lengths are obtained as: T T U length true length projected oreshortening actor DIMETRIC & ISOMETRIC PROJECTIONS Just as in the case o trimetric projections, similar transormations + projections cause dimetric and isometric projections with ollowing conditions: Isometric Dimetric same are o An Trimetric

7 7 Eample: Trimetric projections Consider the ollowing cube rotated b about ais and 45 about -ais ollowed b a parallel projection onto the plane. The position vectors or the cube with one corner removed are X ϑ ϑ ϑ ϑ R R P T Eample (contd.): Trimetric projections The concatenated matri is : X 4 4 R R P T

8 Calculation o angles Let us consider the aonometric projection o unit vectors X T U projected length oreshorte ning actor true length Calculation o angles For dimetric projections (sa) then + The second equation is obtained in terms o and solving or theta 4 ( ) ( ± / ) and ( ± / ) 8

9 Calculation o angles For Isometric projections then + and From the above two equations, solving or theta ± ± 5. Substituting this in the above eqn., we obtain ± 45 Foreshortening actor.85 Calculation o angle that the projected -ais makes with the horiontal in isometric case U* U T projected length oreshorte ning actor true length The angle between the projected -ais and horiontal is given b * tan α ± as 45 * α tan ( ± 5.) ± 9

10 Perspective Transormations