Important points about the curve segment, Computer Graphics

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

892_Important Points about the Curve Segment.png

That is pn = q0

2)       C1 continuity of 2 curve P (u) and Q (u) as that point pn - 1, pn on curve P(u) and q0, q1 points upon curve Q(u) are collinear that is:

n( pn  - pn-1 ) = m(q1 - q0 )

n q1  = q0  +( pn  - pn -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:

pn  = q0n( pn  - pn -1 ) = kn(q1  - q0 ),

Here k is a constant and k > 0

⇒ pn -1 , p­  = q0 , q1  are collinear

3)  c2 continuity is:

a)   C(1) continuity

b)   m (m - 1) (q0 - 2q1 + q2)

= n (n - 1) (pn - 2pn - 1 + pn - 2)

That points are as: pn - 2, pn - 1, pn of P(u) and points q0 , q1, q2 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 (d2 p)/( d u2) u=1  =   (d2 q) /(d v2 )v=0

Whether they are similar we can as we have C2 continuity   

 Note: as the same we can explain higher order parametric continuities

Posted Date: 4/4/2013 5:57:14 AM | Location : United States







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