Assignment 2 (Answers): Tessellation of shapes
Please refer to the handout for description of the size and location of each of the four shapes (in
the Shape Specification section). When computing surface normals, the results don’t have to be
unit vectors.
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Cylinder
Question 1 The caps of the cylinder are regular polygons with N sides, where N is equal to the first
parameter. You will notice in the demo they are cut up like a pizza with N slices each of which is
an isoceles triangle. The vertices of the N-gon lie on a perfect circle. Where do these vertices lie,
on the top cap and on the bottom cap? (e.g., what are their 3D coordinates?)
xi = 0.5 ∗ cos(i2π/N ) zi = 0.5 ∗ sin(i2π/N ) yi = ±0.5
Question 2 What is the angle at the tip of each isoceles triangle on the cap when the first tessellation
parameter is N?
There will be N triangles meeting in the middle, which divides the circle into N pieces,
so each angle is 2π/N .
Question 3 In solid mode, you can notice that the triangles on the barrel of the cylinder are shaded
with a gradient; this is done automatically when you create them with three surface normals, one for
each point. What is the surface normal of an arbitrary point (x, y, z) on the barrel of the cylinder,
whose size/location is described in the handout?
The x and z directions are the same as the point calculation, while the y direction is 0.
That is, (x, 0, z).
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Cone
Question 4 What is the surface normal of an arbitrary point (x, y, z) on the side of the cone, whose
size/location is described in the handout?
A little more complex than the cylinder case, the cone normal involves a non-zero y
component. Using simple trigonometry, the normal is:
p
(x, 0.5 x2 + z 2 , z)
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3
Sphere
Question 5 What is the surface normal of an arbitrary point (x, y, z) on the side of the sphere,
whose size/location is described in the handout?
(x, y, z)
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Cube
Question 6 Take a look at one face of the cube. In fact, take a look at one edge of one face in
wireframe mode. Change the tessellation parameter. How do the number of small squares against
one edge correspond to the tessellation parameter?
The number of squares is the same as the tessellation parameter.
Question 7 You need to tessellate the cube face with triangles. You know that the edges of the cube
span from −0.5 to +0.5. Let’s make the tessellation parameter = 2. Write down the coordinates of
the vertices for the eight triangles which make up the y = −0.5 plane. (Draw the triangles out and
label the vertices with their 3D coordinates).
Question 8 What are the normal vectors that correspond to each of these eight triangles?
All the normal vectors are (0, −1, 0).
Question 9 A good strategy to use when creating the cube is to first create one side. Once you
have the side described above, you can put up the one diametrically opposite by changing just two
constants. What are they? ( Hint: consider both positions and normals)
Change the y value for all the points from −0.5 to 0.5 and change the normal from
(0, −1, 0) to (0, 1, 0)
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Extra credit
Question 10 For extra fun and extra credit: Can you name all five platonic solids? How about
their duals? What does it mean to be a platonic solid? What does dual mean?
• Tetrahedron - Tetrahedron
• Cube - Octahedron
• Octahedron - Cube
• Dodecahedron - Icosahedron
• Icosahedron - Dodecahedron
The platonic solids are built from polygons that are all the same shape, and where the
number of faces at each vertex is the same.
What does “dual” mean? It means you can place one shape inside of the other such that
the vertices of one shape lie in the center of the faces of the other shape.
Question 11 The other method to tessellate the sphere is the geodesic method. How can you use
the platonic solids to approximate a sphere? Which solid would give you the best results?
The tetrahedron/octahedron/icosahedron are the best ones to use, since they start off
with nice equilateral triangles. Each triangle can then be split to make the next level of
triangles.
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