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262 11. Navigation and Movement in VR
or equivalently (to speed up its calculation),
p(
) = ((K
3
+ K
2
) + K
1
) + K
0
.
The unknown vector constants K
0
, K
1
, K
2
and K
3
, as well as the parameter ,
have to be determined. To do this, we initially impose four conditions on
Equation (11.1):
1. The spline passes through the point P
i
at the start of the curve. At this
point, = 0.
2. The spline passes through the point P
i+1
at the end of the curve. At
this point, = 1
3. The derivative of the spline at P
i
is given. That is P
i
.
4. The derivative of the spline at P
i+1
is given. That is P
i+1
.
The parameter
in the equations for the spline is related to the distance
traveled ...