Content. Coordinate systems Orthographic projection. (Engineering Drawings)

Transcription

1 Projection Views

2 Content Coordinate systems Orthographic projection (Engineering Drawings)

3 Graphical Coordinator Systems A coordinate system is needed to input, store and display model geometry and graphics. Four different types of coordinate systems are used in a CAD system at different stages of geometric modeling and for different tasks.

4 Model (or World, Database) Coordinator System The reference space of fthe model with respect tto which hall of the geometrical data is stored. It is a Cartesian system which forms the default coordinate system used by a software system. z y Y X x Z

5 Viewing Coordinate System View Plane Textbook setup - for Parallel l Projection, the viewer is at infinity. Zv Yv Xv View Window Viewing Direction Viewer AutoCAD Default Setup Z Y X Viewer

6 Viewing Coordinate System (VCS) A 3-D Cartesian coordinate system (right hand of left hand) in which a projection of the modeled object is formed. VSC will be discussed in detail under Perspective or Parallel Projections. Viewing Coor. System Model Coor Sys

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14 Viewer Viewer

15 Parallel Projection Preserve actual dimensions and shapes of objects Preserve parallelism Angles preserved only on faces parallel to the projection plane Orthographic projection is one type of parallel projection

16 Perspective Projection Doesn t preserve parallelism Doesn t preserve actual dimensions and angles of objects, therefore shapes deformed Popular in art (classic painting); architectural design and civil engineering. Not commonly used in mechanical engineering

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18 Geometric Transformations for Generating Projection View Set Up the Viewing Coordinate System (VCS) i) Define the view reference point P = ( P x, P y, P z ) T v ii) Define the line of the sight vector n (normalized) v n = ( N x, N y, N z ) T and N 2 x + N 2 y + N 2 z =1 iii) Define the "up" direction v T V = V x, V y, V z n, V v n v = 0 r u, V v, v n ( ) T v This also defines an orthogonal vector, ( ) forms the viewing coordinates Define the View Window in u v n coordinates v u = V v v n

19 Generating Parallel Projection Problem: for a given computer model, we know its x-y-z coordinates in MCS; and we need to find its u-v-n coordinates in VCS and X s -Y s in WCS. Getting the u-v-n coordinates of the objects by transforming the objects and u-v-n coordinate system together to fully align u-v-n with x-y-z axes, then drop the n (the depth) component to get Xs and Ys Viewing Coor. System Model Coor Sys - Translate O v to O. - Align the n axis with the Z axis. - Fully aligning u-v-n with x-y-z

20 Generating Parallel Projection (1) First transform coordinates of objects into the u-v-n coordinates (VCS), then drop the n component. (n is the depth) i.e. Overlapping u - v n with x -y -z i) Translate O v to O. ) v v n ii) Align the axis with the Z axis. The procedure is identical to the transformations used to prepare for the rotation about an axis. A = N x, B = N y, C = N z L = N x 2 + Ny 2 + Nz 2 y V = N y 2 + Nz 2 [ D] = vx vy vz

21 Generating Parallel Projection (2) θ 1 2 Rotating about X: [R] x ; and Rotating about Y: [R] y θ Fully aligning u-v-n with x-y-z Then, rotate Ө 3 about the Z axis to align ū with X and v with Y

22 Generating Parallel Projection (3) Rotate Ө 3 about the Z axis to align ū with X and v with Y At this point, is given by V x, V y, 0 where V V x V y = R y 0 1 [ ] R x [ ][ Do v,o] ( ) T V x V y V z 1 We need to rotate by an angle Ө 3 about the Z axis L = V x 2 + V y 2, sinθ 3 = V x L, cosθ 3 = V y L [ R z ]= V r L V x L 0 0 V xl V yl Result:

23 Generating Parallel Projection (4) Drop the n coordinate [ ]=] D n , u V = D n 0 1 u V 1 [ ] n In summary, to project a view of an object on the UV plane, one needs to transform each point on the object by: [ T ] = [ Dn ][ Rz][ Ry] [ Rx ][ Do, o] v u V P = 0 1 = [ T ]P = T [ ] x y z 1 Note: The inverse transforms are not needed! We don't want to go back to x - y - z coordinates.

24 Oth Orthographic hi Projection Y Top Front Right X Z Projection planes (Viewing planes) are perpendicular to the principal axes of the MCS of the model The projection direction (viewing direction) coincides with one of the MCS axes

25 Y Top Front Right X Geometric Transformations for Generating Orthographic Projection (Front View) Z Yv,Y Front Pv = P Xv, X Drop Z

26 Y Top Front Right Yv Geometric Transformations for Generating Orthographic Projection (Top View) X Top Xv, X Z Z cos(90) sin( ) P v = P = P sin( 90 ) cos(90) Drop Z 90 [ R] x

27 Y Top Front Right X Yv, Y Geometric Transformations for Generating Oth Orthographic hi Projection (Right View) Right Z Xv Z 1000 cos( 90 ) 0 sin( 90 ) P = = v P P sin ( 90 ) 0 cos( 90 ) Drop Z 90 [ R] y

28 Rotations Needed for Generating Isometric Projection Y Top Front Right Y Yv Top X Front Right Xv Z Z Zv X cosθ 0 sinθ 0 0cos sin P [ ] [ ] v = R φ x R θ yp= φ φ P 0 sinφ cosφ 0 sinθ 0 cosθ

29 Isometric Projection: Equally foreshorten the three main axes θ = ± 45, φ = ±

30 Other Possible Rotation Paths Rx --> Ry Rz --> Ry(Rx) Rx(Ry) --> Rz r x = ± 45, ry = ± r z = ± 45, r y( ( x ) = ± ry ( x) = ± 45, rz = ANY ANGLE