Important Points about the Curve segment - properties of bezier curves
Note: if P (u) → = Bezier curve of sequence n and Q (u) → Bezier curve of sequence m.
After that Continuities in between P(u) and Q(u) are as:
1) Positional continuity of 2 curves
That is p_{n} = q_{0}
2) C^{1} continuity of 2 curve P (u) and Q (u) as that point p_{n - 1}, p_{n} on curve P(u) and q_{0}, q_{1} points upon curve Q(u) are collinear that is:
n( p_{n} - p_{n-1} ) = m(q_{1} - q_{0} )
n q_{1} = q_{0} +( p_{n} - p_{n -1} ).(n/m)
⇒ (d p/du)_{u=1} = (d q/dv)_{v=0}
G^{(1) } continuity of two curves P(u) and Q(u) at the joining that are the end of P(u) along with the beginning of q(u) as:
p_{n } = q_{0}n( p_{n} - p_{n -1} ) = kn(q_{1} - q_{0} ),
Here k is a constant and k > 0
⇒ p_{n -1} , p_{} = q_{0} , q_{1} are collinear
3) c2 continuity is:
a) C^{(1)} continuity
b) m (m - 1) (q_{0} - 2q_{1} + q_{2})
= n (n - 1) (p_{n} - 2p_{n - 1} + p_{n - 2})
That points are as: p_{n - 2}, p_{n - 1}, p_{n} of P(u) and points q_{0} , q_{1}, q_{2} of Q(u) should be collinear further we can verify whether both second and first order derivatives of two curve sections are similar at the intersection or not that is:
(d p)/( d u)_{ u=1} = (d q) /(d v )_{v=0}
And (d^{2} p)/( d u^{2})_{ u=1} = (d^{2} q) /(d v^{2} )_{v=0}
Whether they are similar we can as we have C^{2 }continuity
Note: as the same we can explain higher order parametric continuities