Create a set of solid primitives as you did in Problem 15.1. Using 20-lb white (copy) paper, cut out the paper so that it just fi ts around a solid. The result should be one piece of paper, and the creases and cuts in the paper should clearly delineate all of the faces of the solid. This is a development of the solid. a. Create a development of a rectilinear prism. Make a number of photocopies of the resulting development for later parts of the exercise. Can you create another development of the prism that looks completely different from the fi rst when laid out fl at?
b. Cut an oblique face on the prismatic solid. Modify a photocopy of the original development to match the new solid.
c. Cut a notch out of the original prismatic solid. Modify a photocopy of the original development to match the new solid.
d. Create a development of the cone. Make a number of photocopies of the resulting development for later parts of the exercise. How many faces are defi ned on the development? Is there a clearly defi ned connection point between the base and the side of the cone?
e. Cut an oblique face in the solid cone. Modify a photocopy of the original development to match the new solid.
f. Imagine drilling a skew hole through the cone. Modify a photocopy of the original development to match the new solid.
g. Make developments of the other solid primitives. What would a development of a sphere look like? Does the shape of the face in the fl attened development match its shape on the sphere? How is this development similar to that of the cone? How is it different?
Problem 15.1:
Construct foam or clay models of primitive solid shapes, such as cylinders, cones, rectilinear prisms, cubes, etc.
a. Intersect the primitive solid with a plane: Cut the primitive into two pieces, and place a (preferably transparent) plane between the two halves. By lifting off one half, you can see the cross section of the intersection, along with the orientation of the cutting plane. Using the cone, create a variety of conic sections by intersecting the plane at different orientations. What is the relationship of the orientation of the intersecting plane to the shape of the intersection? When is the shape and size of the intersection the same as the base? Repeat these tests with the other primitive solids. Also, compare the intersections of different solids with the planes in similar orientations.
b. Intersect two solids with each other: Keep one solid whole and cut the other solid into distinct pieces representing volumes internal and external to the first solid. Create a cone and a rectilinear prism for which one dimension is twice the diameter of the base of the cone and the other two dimensions are two-thirds the diameter. Cut the prism to represent intersecting half-way through the cone at its base. You should end up with the prism in three pieces: two external to the cone and one internal. Cut up three other identical prisms representing the intersections as the prism moves up vertically toward the tip of the cone. Compare the internal and external pieces of the various intersections. When do the internal pieces look more like pieces of the cone? Pieces of the prism? How would the pieces have to change if none of the sides of the prism were parallel to the base of the cone? Try intersecting the prism with the cylinder and then the cube, at both normal and skew angles.