Graphics Reference
In-Depth Information
¢=
-
(
)(
)
¢
T
cos , sin
qqy
qqy
J
J
•
b
b
q
T
(
)(
)
sin , cos
•
a
¢=
¢=
cos
cos .
q
1
(15.7)
a
q
2
We shall return to this equation in Section 15.4.
Now what we have described so far are just local conditions for geodesics. The
curves must satisfy the differential equations shown above in a neighborhood of any
point through which they pass. Furthermore, there are of course many solutions to
these equations and to get unique solutions one needs to specify additional con-
straints. The most common constraints, and the ones handled most easily with stan-
dard numerical techniques for solving differential equations, are initial conditions.
Typical initial conditions would be a start point of the desired curve and a direction
vector. However, this does not solve the problem of finding a shortest curve between
two points because we would not know the initial direction of the curve. The short-
est curve problem is a boundary value problem and much more difficult. A solution
to the discrete version of this problem is described in the next section.
One area where one has to deal with geodesics is in the design and manufacture
of composite materials. See Section 15.4 below. In this case one wants to generate
geodesics given a start point and an initial direction. A common approach is to tes-
sellate the surface and generate geodesics on the resulting polygonal surface. The
paper [KSHS03] describes differential equations for a geodesic obtained from a vari-
ational approach and compares the numeric solution to these equations to the dis-
crete geodesics one can generate on the approximating polygonal surface using two
different algorithms. It turns out that the deviation of the discrete geodesics from the
smooth geodesic is not always proportional to the error caused by the tessellation but
depends also on the complexity of the surface.
We finish this section with an example. Unfortunately, just as very few curves have
a simple formula for their length, very few geodesics have a simple formula. Never-
theless, the following may help the reader understand the mathematics.
15.3.1.1
Example.
Consider the paraboloid of revolution
S
defined by the
equation
(
)
=- -
2
2
fxyz
,,
z x
y
and parameterization
(
)
(
)
=
2
2
(
)
Œ
2
j uv
,
uvu
, ,
+
v
,
uv
,
R
.
We want to compute the equations that define the geodesics for this surface.
Solution.
Let a(t) = (u(t),v(t)) be a curve in the domain of j. First of all, observe
that
∂
∂
j
∂
∂
j
=
(
)
=
(
)
102
,,
u
and
012
,,
v
,
u
v
