Proof of subsequent properties of bezier curves, Computer Graphics

Proof of subsequent properties of Bezier curves

Note: Proof of subsequent properties of Bezier curves is left as a work out for the students

P' (0) = n (p1 - p0)

P' (1) = n (pn - pn-1) = n (pn - pn - 1)

P" (0) = n (n - 1) (p0 - 2p1 + p2)

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

To ensure about the smooth transition from one part of a piecewise parametric curve or some Bezier curve to the subsequent we can impose different continuity circumstances at the connection point for parametric continuity we match parametric derivatives of adjacent curve section at their ordinary boundary.

Zero order parametric continuity illustrated by C0 continuity means curves are merely meeting as demonstrated in figure 14 whereas first order parametric continuity termed as C1 continuity, implies that tangent of successive curve sections are identical at their joining point. Second order parametric continuity or C2 continuity, implies that the parametric derivatives of the two curve sections are identical at the intersection. As demonstrated in figure 14 as given below; first order continuity have identical tangent vector but magnitude may not be the similar.

772_Proof of subsequent properties of Bezier curves.png

Along with the second order continuity, the rate of modification of tangent vectors for successive section is equivalent at intersection hence result in smooth tangent transition from one section to the other.Initial order continuity is often utilized in digitized drawing whereas second order continuity is utilized in CAD drawings.

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