Whether occurring in nature or in the mind of a designer,
curves and surfaces that are pleasing to the eye are not necessarily
easy to express mathematically. In Solidworks 2014, Solidworks
introduced a new entity called Style
Spline.
I
would like to share
about
mathematical concepts in style spline.
Style
spline is something differs from spline which we are using currently,
Because Spline curve is a
piecewise
cubic curve,
made of pieces of different cubic curves glued together. Style spline
is a Bezier
curve.
A Bezier curve is one of the parametric
curve frequently used in computer
graphics and related fields. But Bezier curves differ from
other types of parametric curves by the type of basis polynomials
used to form them
The
study of these curves was however first developed in 1959 by
mathematician Paul
de Casteljau using de
Casteljau algorithm
at Citroën. The idea of this algorithm is plotting the curve through
repeated linear interpolation by using given control points (P0,
P1,
P2,
P3...
Pn
).
The following discussion will explain how Bezier curve has been
derived mathematically. For an example, Lets we discuss about
methodology of deriving Quadric Bezier curve (i.e. Parabola).
Generic
Linear interpolation formula between two points P0P1
is
P(u) = (1-u)P0
+ uP1,
for
0
≤ u ≤ 1
For
better understanding, Lets we take "u" = 0.2, 0.4, 0.6, 0.8
,between the limit 0 to 1
We
know that the value of starting (P0),
control (P1)
and ending (P2)
points. Firstly , we have to do linear interpolation between P0
and
P1
as
well
as
P1
and P2
(for u=0.2), so we get P01
and
P02
points respectively . And again we have to do linear interpolation
between P01
and P02
, finally we will get P(u) point (for u=0.2). So for Quadric Bezier
curve, we need three iteration to find the curve points. We have to
repeat these three iteration with changing value of "u".
Note:
The tangent vector formed by the starting point is tangent to the
curve at point (P0). The derived lines from calculations at every
stage (like P0P1 ) is tangent to the curve at point P(u). Likewise,
Tangency of the curve is controlled.
Lets we see the animation of curve formation in Higher order (4 point) Bezier curve which is created... Click Here
Team EGS
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