Theory Of Isometric Projection

with wbirh wc can read off dimensions in the same manner that dimensions are read upon an ordinary scale in orthographic. drawings. The above is Parish's method, but the same results maybe obtained in a simpler manner, as shown in Fig. 2. Here two lines are drawn upon the base line A-D, having angles of 450 and 30° respectively between them. The true scale is plotted upon the 45° angle line A-B, and the divisions projected as before upon the 30° line A-C, which becomes the reduced or isometric scale. This method is based on the construction shown in Fig. 3. Where the dotted line C- a is at an angle of 450 with the horizontal diagonal C—b, and the full line c~a' is at an angle of 300 with the horizontal, and, as previously explained, the full line is the isometric equivalent for the real edge c -a, it follows that any portion of the real line would be in like manner represented isometrically by dropping a perpendicular from it to the isometric edge, as shown at /

It is advisable to be acquainted with the above theory of isometric projection for a full understanding of the subject, but in practice it is usual to disregard the fact that an isometric projection is smaller than the object represented, whatever scale may be used, and to mark oft the required dimensions with a rule or an ordinary scale upon the three root lines which represent respectively the length, breadth and thickness of the object. This may be safely done, as every part of the drawing is proportionately reduced. The method now to be described has been variously termed pseudo-isometric, conventional isometric and angular projection. Neither term is entirely distinctive or very illuminating, and the writer suggests the term equal angle projection as being more descriptive of the procedure.

It has already been explained upon page 16 that any circle may be divided into 360 equal parts for the purpose of measuring angles, by observing how many degrees or divisions such angle contains, and, if a circle is divided into three equal parts by lines radiating from its centre, it will be


obvious that earh of these lines will contain 120 degrees,

120. This construction is made the basis of the form of isometric projection now to be described. A vertical line is drawn from the centre of Fig. 4 to the circumference a. Divide the circumference into three equal parts from point a, as at B and C, join these points to A, and we Lave the three " root lines," or isometric axes, upon which the three cardinal dimensions can be measured direct. These lines are the isometric projections of the edges in the object that are perpendicular to each other, and all other edges or surfaces that are parallel to these edges in the object must be made parallel to these axes in the projection. If this requirement is observed, no mistakes are likely to be made in the projection, and the completion of the cube within the circumscribing circle will be easy.

Observe that upon the tangent line, a-c is marked off (to scale) 1 in. long, and that its projector is a-c'; the isometric projection lies in this line also, but is shorter, falling in the circumference of the circle, hut it measures by the same scale 1 in,, which proves that although isometric projection shortens the edges lying in the isometric planes, it does not falsify the dimensions. We may further simplify this method of projection by using the T square and set square, as shown in Fig. 6, to produce the isometric axes, for ii a horizontal line is drawn through point A in Fig. 4—that is, perpendicular to a-A- -the angle contained between such horizontal line and either A-B or A-C will be 300; therefore, if we use the set square of 30°, as shown in Fig. 6, we can draw A- B and A-C with its hypotenuse edge and A-a with its right-angled edge, and subsequently any parallel lire or edge by moving the set square along the T-square to the required point. No difficulty should be experienced in drawing the remaining figures upon this plate if it be remembered that all dimensions are to be ma rked along the three root lines and projected parallel therefrom. Fig. q represents a cube resting upon one edge, and is produced by revolving the circle shown ir.

Isometric Projection Rules

Isometric Projection

Fig. i. A Nest of Shelves. Fig. 2. Details of same. Fig. 3. Corner of a Frame. Fig. 4. Projection of an Octagonal Prism. Fig. 5. Preliminary Construction. Fig. 6. An Octagonal Pyramid. Fig. 7. A Canted Tray. Fig. 3. A Bracket example for practice


No. 4, with its points, until C-c' lies horizontal. A mortise is shown in it to take the student a step farther. Fig. 10 represents the theoretical isometric planes with prisms lying in them. Any plane which contains two isometric axes is called an isometric plane. Figs, i, 2, 3, page 87, are rectangular figures that will not offer much difficulty to the student.

The Nest of Shelves Fig. %, should be commenced by drawing the root lines, a-a', a-b, a -c, to the given dimensions — preferably not less than three times the scale of the copy. When these are drawn, complete the outline of the case by drawing parallels, then mark off the thickness of the sides equal to 1 in., and space out the shelf and divisions. Fig. 2 is an enlarged detail of one side and end of the bottom, showing how the case is jointed and the shelf housed in.

A Corner of a Frame with Mortise and Tenon Joint is shown in Fig. 3. Probably no difficulty will be met with in drawing this until the mortise is reached. The position of this must be located upon the root line by projecting across the inside of the rail to a, and dropping a perpendicular. Mark off a parallel line to the width of the tenon (as shown, it is half the width of rail), then add the wedging; finally draw in the sides of the mortise and project them to the salient angle to project into the adjacent plane, and so obtain the haunching. Finish by indicating the tenon, etc.,. by dotted lines drawn from the visible faces.

To draw Non Rectangular Figures by this method it is first requisite to enclose them within a rectangular figure and, placing this isometrically, we can mark off the points where the contained figure cuts or touches the sides of the rectangle, just as we should dimension points, and, joining up these, obtain an isometric projection of the figure. Fig. 4 is on Isometric View of an Octagonal Prism, and the dotted lines indicate the containing rectangle, produced as shown in Fig. 5. It will be observed that the face of the prism lying below the line b'-b" is shown much wider than

Cupboard Isometric View
Isometric Projection Fig, i. A Dwarf Cupboard. Fig. 2. Enlarged Detail of Joint at A

the corresponding face below c-c'. Tins is a defect inseparable from the method.

An Octagonal Pyramid is shown isometrically in Fig. 6. To produce this place the base. Fig. 5, in the hori zontal isometric plane and erect a perpendicular from the centre X. Locate the apex upon this and draw lines to it from the angles in the base.

The Washing Tray. Fig. 7, has four equally sloping sides, therefore is non-rectangular at either side. To draw it the dotted rectangular prism must be made upon it to the requited size of top, and the height. Then mark off the amount of slope at the bottom, as at a'-i. a'- 2, b'-3, etc., and diaw in the sides to the intersections.

The Gallows Bracket. Fig. 8, shown in orthographic projection is given as an exercise. The student should put it into isometric projection.

The Dwarf Cupboard shown in plan and elevation on page 51 is drawn isometrically on page 89. This is distinctly an example that can be better rendered in isometric than in orthographic projection, so far as giving a clear impression of the construction is concerned.

It is an easy example to draw. Commence as before with the three root lines, making these represent the salient angle of the case, the bottom edges of the front and end respectively. Complete the case by drawing parallels to these, taking the necessary dimensions from page 51. The doors may next be drawn, commencing with the closed one ; no instructions should be needed to locate the face lines of the rails and stiles, these, of course, being measured directly upon the root lines, but a little consideration must be given as to what parts of the recessed surfaces will be visible. In this kind of drawing the observer is always supposed to be standing opposite the near sal'ent angle, therefore he cannot see the edges which face away from him. Obviously the edge of the hanging stile of the door is in this position, and is not shown, whilst that of the meeting stile is ; also that of the bottom rail. The left hand door is

Isometric Projection Construction Detail

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