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But the results of this crude method will be very jerky. This is because
the curve joining the points is not smooth. The technical description of this
is that it lacks G1 and C1 continuity. A curve has G0 continuity if it is
connected at the key positions, K n , and has G1 continuity if the tangent at
this point for both the segment K n -1
K n +1 is in
the same direction but not necessarily of the same magnitude. For a curve
to have C1 continuity, the tangents must be in the same direction and of
the same magnitude. For the curve we are considering, we have a value
changing against time. If you are familiar with calculus then you will know
that differentiating a curve gives a new curve that represents the way the
slope of the original curve changes with time. For a smooth curve we need
C1 continuity as it bends through the key positions. There are many
possible contenders for such a curve. The standard technique is to use a
different curve between each pair of key positions. To ensure that the
curve is smooth through the key positions, we need to consider the slope
or tangent to the curve for the end of one segment and the beginning of
the next. The slopes must be in the same direction and of the same
magnitude to ensure a smooth transition. Figure 8.3 shows the result of
using piecewise cubic curves with and without the matching of tangents
for the curve leading up to the key position and the curve following the key
position.
That's all well and good, but how do we define a cubic that is
guaranteed to go through the key positions. The problem comes down to
a familiar problem for computer graphics generally, one of curve fitting.
Some curve-fitting options do not go through the actual key positions so
would not be suitable. A curve that is guaranteed to go through the key
K n and segment K n
Figure 8.3 Key position tangents for piecewise cubic curves.
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