Isometric drawing versus true isometric projection for a drilled bracket
A manufacturer supplies a bracket whose front face is an 80 × 50 mm rectangle with a Ø30 mm through hole centred in it; the bracket is 15 mm thick. A diagonal stiffener runs across the front face from a point 10 mm above the lower-left corner to a point 10 mm below the upper-right corner, making 35° with the horizontal, and must be drawn in the axonometric view. The draughtsman sets an isometric drawing with the three axes at 120° and uses full true lengths along the axes, ignoring the 0.816 factor, because the drawing is not to scale. The hole is drawn as an ellipse using the familiar 1.22D major axis and 0.71D minor axis. The client, however, asks for the true isometric projection of the diagonal stiffener and wants to know whether the hole ellipse is correct under that projection. A second option is a caballera projection with the front face true shape and the depth axis at 45° with a 0.5 reduction.
1. Distinguish between a full-scale isometric drawing and a true isometric projection. Compute the ellipse axes for the Ø30 hole in each case and say whether the proposed 1.22D/0.71D ellipse is correct.
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In a true isometric projection, the three coordinate axes are foreshortened by the factor
0.816 ≈ √(2/3). A circle of diameter D lying in a coordinate plane projects as an ellipse whose major axis equals the true diameter D and whose minor axis equals D/√3 ≈ 0.577D. For D = 30 mm:
major = 30.0 mm, minor = 17.32 mm.
In a full-scale isometric drawing, every projected length is multiplied by 1/0.816 ≈ 1.2247 so that the axes can be drawn at true scale. The ellipse axes therefore become
major = 1.2247D ≈ 36.74 mm, minor = 0.7071D ≈ 21.21 mm for D = 30 mm.
The proposed 1.22D and 0.71D values are correct only for a full-scale isometric drawing. They are not the axes of the true isometric projection. If the client asks for a true projection, either use 30.0 mm and 17.32 mm, or state clearly that the drawing is an isometric drawing enlarged by 1.2247 and label the scale accordingly. Mixing true lengths along the axes with true-projection ellipses would be inconsistent.
2. The stiffener is 60 mm long at 35° to the horizontal on the front face. Compute its length in the full-scale isometric drawing and in the true isometric projection. Explain why applying 0.816 to 60 mm is wrong.
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Resolve the stiffener into components in the front face:
h = 60 cos 35° = 49.15 mm,
v = 60 sin 35° = 34.41 mm.
In the isometric drawing the two front-face axes appear at 120°, not 90°. The projected vector is therefore
L² = h² + v² + 2hv cos 120°
= 49.15² + 34.41² − 49.15 × 34.41
= 2416 + 1184 − 1691 = 1909,
L = 43.69 mm.
For the true isometric projection, multiply the drawing length by 0.816:
L_true = 0.816 × 43.69 = 35.65 mm.
Applying 0.816 directly to 60 mm gives 48.96 mm. That would be valid only if the stiffener were parallel to one of the isometric axes; a general line in a face is foreshortened non-uniformly because the two projected axes are not perpendicular. The correct drawing method is to mark h along one axis and v along the other, then join the endpoints; do not scale the 60 mm length directly.
3. Recommend caballera or isometric for this bracket, considering the hole, the stiffener and the client’s request. Give a reasoned judgement rather than a simple preference.
Read the worked answer
Choose caballera if the priority is clear, measurable front-face geometry. With the front face true shape, the Ø30 hole remains a true circle, the 35° stiffener is drawn at its true angle and 60 mm length, and the 12 mm end offsets are easy to check. The depth is 15 mm along the 45° axis with a 0.5 reduction, i.e. 7.5 mm on the drawing, which still gives a readable pictorial view.
Choose isometric if the priority is equal treatment of the three directions and a standard axonometric presentation. It shows the thickness and the stiffener in a more balanced three-dimensional way, but the hole must be an ellipse and the 35° angle is distorted. It also forces the client to accept either a true projection, with all axes at 0.816, or an enlarged drawing at 1.2247; the two cannot be mixed.
For checking the front-face hole and stiffener, caballera is the better engineering judgement. For a general assembly illustration, isometric is preferable. Whichever is chosen, the drawing must state the projection and scale, and the ellipse axes must match that choice.