Q)
(chatquestion)...The process of generating primitive shapes in computer-aided design (CAD) software is often
mathematical-based. Case in point: a torus surface (commonly known as the 'doughnut'). This
has the shape and Cartesian equation as follows:
(x2 + y2 -R)+ z? = r2
Figure 1: Description of a torus surface (source: web resource)
In CAD software, this surface is usually generated by the following procedures:
(1) The values of major radius, R and minor radius, rare fixed by the user.
(2) Revolve a point on the x-axis for one round until a circle (line) is formed.
(3) Revolve the circle (line) for one round about the z-axis until the torus (surface) is formed.
Figure 2: Common process flow of generating a torus surface
1
Question (a): [4 marks]
With the introduction as a hint, parameterize a torus surface with two parameters, u and v.
State the parameterization as x(u, v), y(u, v), z(u, v), along with the range of u and v.
Question (b): [6 marks]
A surface integral has the form of fS f(x,y, z) dS. At its simplest, f(x,y, z) = 1, this integral
simply gives the area of the entire surface S. Use the surface integral method to determine
the surface area of a torus. Refer to the information below for values of R and r:
Table 1: Substitution of Rand r based on last two digits of student number (e.g. KIX1900 R
Digit
Rorr 1
6
3
4
5
10
...
