Always construct the true scale at $45^{\circ}$ and the isometric scale at $30^{\circ}$ on the same origin $P$ to form a precise, standard projection triangle.
Step 1: Construction of the Isometric Scale:
An isometric scale is a special construction used to obtain the foreshortened isometric length from any given true length. The mathematical relation is:
$$\text{Isometric Length} = \text{True Length} \times \cos 45^{\circ} / \cos 30^{\circ} = \text{True Length} \times \sqrt{\frac{2}{3}} \approx 0.816 \times \text{True Length}$$
Step 2: Construction Steps: • Draw a horizontal baseline $PQ$ of any convenient length.
• From point $P$, draw a line $PM$ at an angle of $45^{\circ}$ to the baseline $PQ$. This line is the True Scale Line. Mark actual centimeter/millimeter divisions on this line up to $90\text{ mm}$ ($0, 10, 20, \dots, 90$).
• From point $P$, draw another line $PA$ at an angle of $30^{\circ}$ to the baseline $PQ$. This line is the Isometric Scale Line.
• From each marked division point on the $45^{\circ}$ true scale line $PM$, draw vertical projectors perpendicular to $PQ$ downward to intersect the $30^{\circ}$ isometric scale line $PA$.
• Label the intersection points on the line $PA$ as $0', 10', 20', \dots, 90'$. The length $P-90'$ represents the isometric length of $90\text{ mm}$ to be used in the projection.
