For Leaving Certificate Pupils

Oblique Surfaces in Isometric Projection

Oblique surfaces in isometric may be drawn by establishing the intersections of the oblique surface with the isometric planes. Firstly, lets use a cube to explain isometric planes. In an isometric projection or drawing of a cube, the faces of the cube, and any planes parallel to them, are called isometric planes. Oblique surfaces in isometric may be drawn by establishing the intersections of the oblique surface with the isometric planes.

Take the incomplete isometric drawing of this object which has been cut by an oblique plane. The oblique plane contains points 'A', 'B' and 'C'. Extend line 'AB' until they intersect the edges of the enclosing box. These points, 'VW', are points of intersection between the oblique plane and the isometric planes. Point 'C' is on both these isometric planes and the oblique plane. So, if we join 'C' to points 'V' and 'W' we get lines which lie on both the isometric planes and the oblique plane. To complete the isometric drawing simply draw lines parallel to these lines. Carry 'A' parallel to 'XC' onto the edge directly 'A'. Draw a line from 'D' to 'E' parallel to line 'VW'. Continue this process until the oblique surface is complete.

Irregular Objects

Some objects do not conform to a rectangular shape. Take the object to the left. It is made up of several curves. This object is drawn in isometric through the use of sections or cutting sections.

The object is cut into slices by cutting planes which are evenly spaced. The traces of these cutting planes are drawn in both plan and elevation. Points are indexed in both the plan and elevation, where the cutting planes intersect the outline of the object. Points on the object are transferred to the isometric drawing by the offset location measurements method. Pick a point on the outline of the object in the elevation which lies on a cutting plane. Now find this point in plan. This point can be easily found in the isometric drawing by using the horizontal and vertical measurements from the orthographic views and stepping them parallel to isometric lines in the isometric drawing. Every point in the isometric drawing is found in this way. A sufficient number of sections should be taken to give an acceptable representation of the object. The curved parts of the object can be drawn in by hand (as in the third last frame of the animation above) or by irregular curves (the completed drawing in the last frame above).

Take point 'A'. It's horizontal distance from one corner of the object is measured from the plan and stepped of along an isometric line in the isometric drawing. The height for 'A' is taken from the elevation and again stepped off in the isometric drawing parallel to an isometric line. See if you can follow how points 'B', 'C', 'D' and 'Q', 'R', 'S' and 'T' are obtained?

Intersections

Take a square based prism and drill a hole down through the top and out the bottom. Now cut the top off the prism at some arbitrary angle. The object should look similar to the object below.

So, how do we set about drawing this in isometric. As always, start by drawing the enclosing box. Draw as much of the object as possible. The next step is to find the lines of intersection between the object and the oblique plane. If you have difficulty finding these edges refer to Oblique Surfaces in Isometric Projection above.

Once you have all the edges of the object drawn in draw the ellipse on the top of the enclosing box. Take cutting planes through the object. Draw the traces of these cutting planes on the top of the box. Where the cutting plane traces intersect the ellipse will give a series of points. Lets take point 'A'. Carry the cutting plane line on which 'A' lies onto one of the vertical isometric planes. Next carry it onto the top edge of the object. From the top edge of the object carry the line parallel to the side edges onto the oblique surface until it meets the opposite edge. This is the actual trace of 'A's cutting plane. Draw a line from 'A' parallel to the isometric axis onto 'A's cutting plane trace on the oblique surface. This gives us one point on the ellipse on the oblique surface. Take as many cutting planes as necessary to give sufficient points to draw a smooth ellipse. Take note of how point 'C' is found.

Cutting planes can also be used to find points on the line of intersection between two objects.

Given the orthographic views of the intersection of two cylinders. Take cutting planes parallel to their axes. Now find points of intersection on these cutting planes in all three orthographic views. The horizontal distance of a point can be transferred from the plan to the isometric drawing. The vertical height of any point can be transferred from the elevation to the isometric drawing.

As normal start by drawing the enclosing box. Draw as much of the object as possible. The cutting plane for points '1' and '2' is drawn on top of the vertical cylinder. Locate and carry the points where the top of the cylinder and the cutting plane intersect onto the surface of the cylinder parallel to the axes of the cylinder. Find the corresponding points ('1' and '2') on the horizontal cylinder and again carry them onto the surface of the cylinder parallel to the axes of the horizontal cylinder. Where these lines meet gives points of intersection between the horizontal and vertical cylinders. Take as many cutting planes as is necessary to give enough points to draw a smooth line of intersection between the two cylinders.