What object is a circle in plan, front and side views?(Besides a sphere)orWhat is the resulting geometry from the intersection of three cylinders?
Written by Paul Bourke
Question.
Is a 3D object uniquely described by a plan, front and side view?
The object described here has the same plan, front and side view as a sphere. Both the sphere and this object have circles as projections in these three views. 
The following image was generated using the RayShade rendering package. It is of three semitransparent orthogonal intersecting cylinders coloured red, green, and blue. 
The following shows the intersection solid of these three cylinders in plan, and two elevations as well as an isometric view. 
The volume of the intersection is 8 (2 - sqrt(2)) r3, where r is the radius of the cylinder. What is the volume of the whole object? Knowing the volume of a cylinder (pi r2 l) and the volume of the steinmetz solid (16 r3 / 3)) then the volume can be calculated using the three dimensional version of the inclusion-exclusion principle, namely that:
So the volume is 3 pi r2 l - 8 sqrt(2) r3 where l is the length of the cylinders. I've had one of these made by carefully lathing a block of wood three times. The result is an interesting object to hold, it contains 8 "corners" and yet if you put it on a slope it will roll! 
Credit to the staff in the workshop at the Auckland University School of Architecture, Property, and Planning for building this model for me. I originally designed this object because Architecture students would ask me why the computer couldn't scan their plans and elevations and automatically build a 3D model. This was the simplest demonstration I could come up that there isn't enough information about a 3D object in its plane projections. |
