Computer Aided Engineering Design

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)