168 COMPUTER AIDED ENGINEERING DESIGN
x(u,v) = ucosv, y(u,v) = u sin v, z(u,v) = av
Equations in cartesian and parametric form for some known analytical surfaces are given below.
Analytic surface Parametric form
Ellipsoid + + = 1
2
2
2
2
2
2
x
a
y
b
z
c
⎛
⎝⎜
⎞
⎠⎟
r(α,β)≡ [acosα cos β,b cos α sin β,c sin α]
Elliptic Hyperboloid + =
2
2
2
2
x
a
y
b
cz
⎛
⎝⎜
⎞
⎠⎟ r( , ) , ,
uaubu^22 +
c
vv≡⎡ v
⎣⎢
⎤
⎦⎥
Hyperboloid of one sheet + – = 1
2
2
2
2
2
2
x
a
y
b
z
c
⎛
⎝⎜
⎞
⎠⎟ r( , )
cos
cos ,
sin
αβ cos , tan
β
α
β
≡ α α
⎡
⎣⎢
⎤
⎦⎥
ab c
Hyperboloid of two sheets – – = 1
2
2
2
2
2
2
x
a
y
b
z
c
⎛
⎝⎜
⎞
⎠⎟ r( , ) αβ≡ cosα, tan cos , tan sin αβ αβ
⎡
⎣⎢
⎤
⎦⎥
a
bc
Cone + – = 0
2
2
2
2
2
2
x
a
y
b
z
c
⎛
⎝
⎜
⎞
⎠
⎟ r(u,β)≡ [au cos β,bu sin β,cu]
Hyperbolic paraboloid – =
2
2
2
2
x
a
y
b
cz
⎛
⎝⎜
⎞
⎠⎟ r( , ) , ,
(^22) –
uaub
u
vvc
v
≡
⎡
⎣
⎢
⎤
⎦
⎥
Quadric Circular Cylinder Quadric Circular Cylinder
(x–a)^2 + (y–b)^2 = c^2 ,z = h r(θ,h)≡ [a + c cos θ,b + c sin θ,h]
Quadric Parabolic Cylinder Quadric Parabolic Cylinder
(y–a)^2 = bx,z = h r( , ) sinθθ θhabh≡[]^2 , + sin ,
Torus
Torus x = (b + a cos u) cos v
xyz baz ab^222 + + – 2 22 2 2 – = + y = (b + a cos u) sin v
z = asinu, b > a,
0 ≤u≤ 2 π, 0 ≤v≤ 2 π
Asimple sheet of surface r(u,v) is continuous and obtained from a rectangular sheet by stretching,
squeezing and bending but without tearing or gluing. For instance, a cylinder is not a simple sheet,
for it cannot be obtained from a rectangle without gluing at the edges. Similarly, a sphere and a cone
are not simple sheets. A flat sheet with an annular hole is also not a simple sheet. However, a
cylindrical surface with a cut all along or an annular sheet with an open sector, are both simple sheets.
If, for points P on the surface, a portion of the surface containing P can be cut, and if that portion is
a simple surface, then the entire surface is called an ordinary surface.
6.1.1 Singular Points and Regular Surfaces
Letru and rv define the derivatives along the curvilinear coordinates u and vat a point P (r(u 0 ,v 0 ))
on the surface (Figure 6.2), then P is called a regular point if
r
r
r
r
u
uu
u
u
u
u
u
(, ) =
(, )
, ( , ) =
(, )
00
,
00
00 00 ,
v
v
v
v
v v v v
∂
∂
∂
∂ (6.2)