Introduction to Isometric Projection
In orthographic projections atleast two views are required to define an object completely. But visualization and interpretation of the details of a complete object in orthographic projections is sometimes difficult. In isometric projection the shape as well as size in all the three dimensions of the object can be seen on a single plane.
Definition of Isometric Projection
When the object is so placed that all the three axes make equal angles with the plane of projection, the projection obtained is known as isometric projection. In isometric projection all the edges of the object are equally foreshortened.
Theory of Isometric Projection
If a cube is placed on one of its corners on the ground with a solid diagonal perpendicular to the VP, the front view is the isometric projection of the cube. The step-by-step construction.
To draw the projections of a cube resting on the ground on one of its corners with a solid diagonal perpendicular to the VP, assume the cube to be resting on one of its faces on the ground with a solid diagonal parallel to the VP.
(a) Draw a square abcd in the top view with its sides inclined at 45° to xy. The line ac representing the solid diagonals AG and CE is parallel to xy. Project the front view (Front View-1).
(b) Tilt the front view about the corner g’ so that the line e’ c’ becomes parallel to xy. Project the second top view and front view (Front View-2). The solid diagonal CE is now parallel to both the HP and the VP.
(c) Reproduce the second top view so that the top view of the solid diagonal, viz. e1c1 is perpendicular to xy. Project the required front view.
The front view of the cube in the above position, with the corners named in capital letters. Its careful study will show that:
(a) All the faces of the cube are equally inclined to the VP and hence, they are seen as similar and equal rhombuses instead of squares.
(b) The three lines CB, CD and CG meeting at C and representing the three edges of the solid right-angle are also equally inclined to the VP and are therefore, equally foreshortened. They make equal angles of 1200 with each other. The line CG being vertical, the other two lines CB and CD make300 angle each, with the horizontal.
(c) All the other lines representing the edges of the cube are parallel to one or the other of the above three lines and also equally foreshortened.
(d) The diagonal BD of the top face is parallel to the VP and hence, retains its true length.
Isometric Axes, Lines and Planes
The three lines CB, CD and CG meeting at the point C and making 1200 angles with each other are termed isometric axes. The lines parallel to these axes are called isometric lines. The planes representing the faces of the cube as well as other planes parallel to these planes are called isometric planes.
As all the edges of the cube are equally foreshortened, the square faces are seen as rhombuses. The rhombus ABCD shows the isometric projection of the top surface of the cube in which BD is the true length of the diagonal.
Construct a square BQPD around BD as a diagonal. Then BP shows the true length of BA.
In triangle ABO, BA / BO = 1 / COS300 = 2 / √3
In triangle PBO, BP / BO = 1/ COS450 = √2 / 1
BA / BP = (2 / √3) X (1 / √2) = √2 / √3 = 0.815
Thus the isometric projection is reduced in the ratio √2: √3, i.e. the isometric lengths are 0.815 of the true length.
Therefore, while drawing an isometric projection, it is necessary to convert true lengths into isometric lengths for measuring and marking the size. This is conventionally done by constructing and making use of an isometric scale with any one of the following methods.
(a) Method I. Draw a horizontal line BD of any length. At the end B, draw lines BP and BA, such that DBA = 300 and DBP = 450. Mark divisions of true length on the line BP and from each division point, draw verticals to BD meeting BA at respective points. The divisions thus obtained on BA give lengths on isometric scale.
(b) Method II. The same scale may also be drawn with divisions of natural scale on a horizontal line AB. At the ends A and B, draw lines AC and BC making 150 and 450 angles with AB respectively and intersecting each other at C. From division points of true lengths on AB, draw lines parallel to BC and meeting AC at respective points. The divisions along AC give dimensions to isometric scale.
The lines BD and AC represent equal diagonals of a square face of the cube, but are not equally shortened in isometric projection. BD retains its true length, while AC is considerably shortened. Thus, it is seen that lines which are not parallel to the isometric axes are not reduced according to any fixed ratio. Such lines are called non-isometric lines. The measurements should, therefore, be made on isometric axis and isometric lines only. The non-isometric lines are drawn by locating positions of their ends on isometric planes and then joining them.
Isometric Drawing or Isometric View
If the foreshortening of the isometric lines in an isometric projection is disregarded and instead, the true lengths are marked, the view obtained will be exactly of the same shape but larger in proportion (about 22.5%) than that obtained by the use of the isometric scale. Due to the ease in construction and the advantage of measuring the dimension directly from the drawing, it has become a general practice to use the true scale instead of the isometric scale.
To avoid confusion, the view drawn with the true scale is called isometric drawing or isometric view, while that is drawn with the use of isometric scale is called isometric projection.
The axes BC and CD represent the sides of a right angle in horizontal position. Each of them together with the vertical axis CG represents the right angle in vertical position. Hence, in isometric view of any rectangular solid resting on a face on the ground, each horizontal face will have its sides parallel to the two sloping axes; each vertical face will have its vertical sides parallel to the vertical axis and the other sides parallel to one of the sloping axes.
In other words, the vertical edges are shown by vertical lines, while the horizontal edges are represented by lines, making 30° angles with the horizontal. These lines are very conveniently drawn with the T-square and a 30°-60° set-square.