2.5: Curved edges and surfaces in orthographic projections

Learning Objectives

By the end of this section, learners will be able to

1. Identify curved surfaces and edges in orthographic projections and multiview drawings.
2. Describe how curved surfaces are represented across different orthographic views.
3. Create multiview drawings of objects that feature curved edges and surfaces.

Curved edges and surfaces in orthographic projections can be challenging to interpret because they appear as straight lines in some views. This section explores why a curved edge may project as a straight line on an orthographic plane.

Consider the example of a cylinder, as shown in the figures below. As with previous examples, the object is imagined to be enclosed within a transparent glass box. Each face of this box represents one of the principal planes of projection. The object is projected onto these planes using parallel projectors perpendicular to the surfaces of the box.

To help describe and visualize this example in three dimensions, a Cartesian coordinate system is included. The x-axis represents the width, the y-axis represents the depth, and the z-axis represents the height. Any point in space can be described by its three coordinates (x,y,z).

As explained in previous book sections, each orthographic plane provides information about two of the three dimensions. For example, the frontal plane represents the width (x-axis) and the height (z-axis), but it does not provide any information about the depth (y-axis).

Figure \(\PageIndex{1}\): Cylinder contained in glass box.

The figures below illustrate how the front view of the cylinder is constructed. Four points along the curved top edge are highlighted, and their projections onto the frontal plane are shown. Observing how these points align on the plane helps explain why a curved edge appears as a straight line in an orthographic view.

Let us examine how the front view of a cylinder is constructed.

Four points along the curved top edge of the cylinder are marked in the images below. These points are then projected onto the frontal plane.

Figure 2.5.2 shows points 1 and 2 projected onto the frontal plane. On this plane, point 1 is defined by its width (coordinate x1​) and height (coordinate z1)​. Point 2 has a different width (x2)​ but shares the same height as point 1, with z1 = z2.

Figure \(\PageIndex{2}\): Projection of points 1 answer 2 on the frontal plane.

Similarly, point 3 is projected on the frontal plane, with the width coordinate x3 located between x1 and x2, and the height coordinate z3 equal to z1 and z2, as shown in Figure 2.5.3. Point 4 overlaps with point 3 in the front view, with x4 = x3, and Z4 = Z3,2,1.

Figure \(\PageIndex{3}\): Projection of point 3 on the frontal plane.

Connecting points 1, 2, and 3 on the frontal plane, we obtain a straight line, since the front view does not capture the depth (y-values) of the original points located in a 3D space, as shown in figure 2.5.4. The curved edge appears straight due to the limitations of the two-dimensional representation.

Figure \(\PageIndex{4}\): Projection of curved edge as a straight line on frontal plane.

Figure 2.5.5 shows the projections of the cylinder onto the top, front, and right-side planes. In the top view, which captures width (x) and depth (y), the circular shape of the cylinder becomes visible. In the front view, the cylinder shows as a rectangle, since the view does not provide information about the depth. Similarly, in the right-side view, the cylinder appears as a rectangle, since the width is not represented in the right-side view.

Figure \(\PageIndex{5}\): Top, front, and right-side views of the cylinder.