the tangent plane. In this way the tangent vector
to the surface s(u, v) at an arbitrary point in any
arbitrary direction is related to the partial derivatives of the parametric representation of the
curved surface at that point.
The following example provides geometric
interpretations for the quantities du/dt and dv/dt
in (3.66) and further insight about the tangent
vector T. Consider a straight line in the parameter
plane through the arbitrary point p(u o , v o ) and
having an arbitrary slope, m (Fig. 3.19c). The equation for this line, given in the standard point-slope
form, is:
(3.67)
To write this equation in parametric form consider the parameter, t, to be the coordinate measured along the line from the arbitrary point
p(u o , v o ). The parameters u and v are related to the
parameter t by noting that the slope m ϭ tan ␪,
where ␪ is the angle from the positive u-axis to the
line, so:
(3.68)
By varying the angle, ␪, lines with any orientation
can be used to specify the direction of the tangent
vector, T, at a point on the surface. The derivatives
of u(t) and v(t) as used in the definition of the
tangent vector (3.66) are:
(3.69)
These are the direction cosines of the angles
between the arbitrary line and the two coordinate
axes in the parameter plane. In (3.66) these two
quantities scale the tangent vectors for the u- and
v-parameter curves to determine the tangent
vector, T, for the curve c[u(t), v(t)]. The ratio
(dv/dt)/(du/dt) ϭ tan ␪ determines the direction of
the tangent line to this curve.
Returning to the example of the elliptic paraboloid (3.60), the straight line in the parameter
plane (Fig. 3.19c) maps onto this surface as the
curve:
(3.70)
The tangent vector anywhere along this curve is:
ϩ 2v o t sin ␪ ϩ t 2 )e z
ϩ (u 2
o ϩ 2u o t cos ␪ ϩ v 2
o
c[u(t), v(t)] ϭ (u o ϩ t cos ␪)e x ϩ (v o ϩ t sin ␪)e y
du
dt
ϭ cos ␪,   
dv
dt
ϭ sin ␪ ϭ cos ΂
␲
2
Ϫ ␪ ΃
u ϭ u o ϩ t cos ␪,  v ϭ v o ϩ t sin ␪
v Ϫ v o ϭ m(u Ϫ u o )
(3.71)
At the arbitrary point p(u o , v o ) the curve and its
derivative are found by setting t ϭ 0 in these equations. The tangent vectors anywhere on the elliptic paraboloid (3.60) in the direction specified by
the direction cosines of the arbitrary line are calculated using (3.66) as:
(3.72)
This is equivalent to the tangent vector calculated
for the curved line in the preceding equation,
given our definitions of the parameters u and v as
functions of t.
The orientation of the tangent plane (3.63) is
uniquely determined by either of the two unit
normal vectors to that plane. The choice between
these two oppositely directed vectors is determined by a right-hand rule: the unit normal
vector, N, makes a right-handed orthogonal
system with the two tangent vectors, Ѩs/Ѩu and
Ѩs/Ѩv (Fig. 3.19b) and is defined as (Lipschutz, 1969,
p. 158):
(3.73)
Recall that the vector (cross) product of two arbitrary vectors, v ϫ w, is normal to the plane containing v and w and that the thumb of your right
hand points in the direction of v ϫ w when your
fingers curl from v toward w. Using (3.73) the unit
normal at any point on a surface can be calculated
from its parametric representation s(u, v).
As an example consider the unit normal vector
for the elliptic paraboloid (3.60). Using (3.28) the
vector product of the two tangent vectors is:
(3.74)
Therefore, the unit normal vector is:
(3.75)
N ϭ (Ϫ2ue x Ϫ 2ve y ϩ 1e z )ր(4u 2 ϩ 4v 2 ϩ 1) 1ր 2
ϭ Ϫ2ue x Ϫ 2ve y ϩ 1e z
Ѩs
Ѩu
ϫ
Ѩs
Ѩv
ϭ det
΂
e x 1
0
e y 0
1
e z 2u 2v ΃
N ϭ
Ѩs
Ѩu
ϫ
Ѩs
Ѩv
|
Ѩs
Ѩu
ϫ
Ѩs
Ѩv |
T(u, v) ϭ (e x ϩ 2ue z ) cos ␪ ϩ (e y ϩ 2ve z ) sin ␪
ϩ (2u o cos ␪ ϩ 2v o sin ␪ ϩ 2t)e z
dc
dt
ϭ ( cos ␪)e x ϩ ( sin ␪)e y
3.2 THE CONCEPT AND DESCRIPTION OF CURVED SURFACES
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