10  
                                                                    Axonometric Projections 
                            10.1  AXONOMETRIC VIEWS 
                                                        G
                                                               F,H
                                              C
                                                                      E
                                                      B,D
                                                            A
                             C,G             B,F
                                                              S
                            D,H              A,E
                                              R                      F
                                                          B                                   Isometric Drawing
                                                                        Y
                            D,C              A,B
                                                                             E Isometric View
                                                                 A,G                      R = 45.00°
                                                     C                                    S = 35.26°
                                                                           X              X = 30.00°
                            H,G              E,F                                          Y = 30.00°
                                                                        H
                                                             D                                                   
                            10-1  
                            An isometric (and axonometric view) of a cube 
                       
                      Axonometric projections are parallel projections onto an oblique plane. Axonometric 
                      projections  have  the  advantage  that  they  give  a  pictorial  view  of  the  object,  yet 
                      dimensions are measurable.  
                      Manually, axonometric views can be constructed from orthographic views. This is best 
                      illustrated by an example. The construction in Figure 10-1 shows a cube in plan and 
                      elevation, from which an axonometric view of the cube is constructed in a direction 
                      parallel to one of its diagonals.   
                      Notice that in this view each of the sides of the cube has been foreshortened equally (to 
                                                                    √2
                      0.8165 of the actual length, or more precisely,  /√3) and that the indicated angles X and Y 
                      are each 30°. Such a projection is also called an isometric projection, meaning equal 
                      measure.  Isometric views can be drawn directly, as shown in Figure 10-2 where the 
                      view has been rotated until the vertical edge of the cube appears vertical.   
                                                     ordinary scale
                                                                                   width
                                                            isometric  scale             height
                                                     is = 30.00°
                                                              os = 45.00°
                                                                                               width
                                                                                               height  = 1.73
                                                                                                           
                      10-2  
                      Isometric scale for the cube in 10-1 
                      The 30° isometric projection has a height to width ratio of 1:√3. Two other common 
                      isometric views are shown in Figures 10-3 and 10-4. There are popular projections, 
                      which, however, are not true axonometric projections. 
                       
                       
                      The projection shown on the right is a                        height
                      27°    isometric  projection   (actually,                     width  = 0.50
                      26°34'12")  also  known  as  a  1:2 
                      projection as this is the height to width 
                      ratio of the top rhombic face.  
                                                           10-3  
                                                   1:2 projection 
                                                                        X = 27°                   Y = 27°
                                                                                                           
                      294   
                              
                              
                             The  one  on  the  left  based  is  a  45° 
                             isometric view, also known as a military 
                             projection.  It has a unit height to width                          height
                                                                                                  width  = 1.00
                             ratio. 
                                                                    10-4  
                                                         Military projection 
                              
                                                                                       X = 45.00°            Y = 45.00°
                                                                                                                         
                             10.1.1  Axonometric scales 
                             By adjusting the angles X and Y, views of the cube can be created according to a variety 
                             of axial scales. Notice that in some drawings two directions are equally scaled and one 
                             differently (called a diametric projection) and in other drawings all three directions are 
                             scaled  differently  (called  a  trimetric  projection).  These  non-isometric  axonometric 
                             projections tend to be more realistic in their depiction. In fact, Chinese scroll paintings 
                             tend to use diametric projections.  See Figure 10-6 for an example.  
                             Equally as is shown in Figure 10-1, every axonometric projection corresponds to a line 
                             of sight whose bearing is indicated by angle R and altitude (true angle of inclination) by 
                             angle  S.    Correspondingly,  we  can  specify  the  axonometric  scale  by  specifying  the 
                             angles for the line of sight.  
                             Table 10-1 gives the angles for the line of sight for the axonometric scales shown in 
                             Figure 10-5. 
                              
                                    1             1                                                3/4         3/4
                                                                  1            1
                                                                                                           1
                                             3/4
                                                                         1/2
                                 36°50'           36°50'      41°25'           41°25'     13°38'                     13°38'  
                             10-5  
                             Various axonometric scales 
                             Sides along the same axial direction have the same scale.  Unmarked sides have unit value 
                                                                                                                       295 
                                
                                          1               3/4                                      1/2
                                                                                    1                                          1             1/3
                                                      1                                         1                                          1
                                    16°20'                   36°50'    7°11'                           41°25'   3°11'                              43°24'
                                                                                                                                                           
                                                                              7/8                3/4                    7/8              2/3
                                                         1
                                         1
                                                                                           1                                        1
                                                 3/4
                                                             62°44'
                                     13°38'                                 17°                    24°46'      12°28'                        23°16'  
                                
                               10-5 (continued) 
                               Various axonometric scales 
                                
                               Table 10-1  Line of sight for the axonometric scales shown in Figure 10-5 
                                                                                    Direction of the            Angle of                    True fore-
                               Type of        Scale ratios                          sight angles                drawing axes                shortening 
                               drawing                                                                                                      ratio 
                                                                                    R              S            X             Y 
                               Isometric      X = 1         Y =1         Z = 1      45º            35º16'       30º           30º           0.8165 
                                                                              3
                               Dimetric       X = 1         Y =1         Z =  /     45º            48º30'       36º50'        36º50'        0.8835 
                                                                               4
                                                                              1
                               Dimetric       X = 1         Y =1         Z =  /     45º            61º52'       41º25'        41º25'        0.9428 
                                                                               2
                                                  3             3
                               Dimetric       X =  /        Y =  /       Z = 1      45º            14º2'        13º38'        13º38'        0.9701 
                                                    4             4
                                                                3
                               Dimetric       X = 1         Y =  /       Z = 1      32º2'          27º56'       16º20'        36º50'        0.8835 
                                                                  4
                                                                1
                               Dimetric       X = 1         Y =  /       Z = 1      20º42'         19º28'       7º11'         41º25'        0.9428 
                                                                  2
                                                                1
                               Dimetric       X = 1         Y =  /       Z = 1      13º38'         13º16'       3º11'         43º24'        0.9733 
                                                                  3
                                                                3             3
                               Dimetric       X = 1         Y =  /       Z =  /     19º28'         43º19'       13º38'        62º44'        0.9701 
                                                                  4            4
                                                  7             3
                               Trimetric      X =  /        Y =  /       Z = 1      39º8'          22º3'        17º0'         24º46'        0.9269 
                                                    8             4
                                                  7             2
                               Trimetric      X =  /        Y =  /       Z = 1      35º38'         17º57'       12º28'        23º16'        0.9513 
                                                    8             3
                                
                               296