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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
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