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The method of characteristics has been used to integrate
the first order partial differential equations that are obtained for the motion
of a surface (or a growth front) when the velocity is a known function of
the surface orientation 
.
A thorough description may be found in Taylor and coworkers[36][32] and
applications may be found in Carter and Handwerker[55].
Let 
be the arrival time of the surface at the position 
.
The level
set 
is the equation for the
position (or shape) of the surface at time 
.
The gradient of 
is along the normal
of the level set and its magnitude must be inversely
proportional to the velocity: 
.
With 
, the PDE is just a statement
that the HD1 function 
is a constant: 
. The characteristics
are straight lines, given by the equation:
and 
is
the
surface at 
.
Letting the initial surface be a point,
the calculation of the shape at a
fixed time (say t = 1) by the
method of characteristics gives the same result as the gradient formulations.