Important point about the De casteljeau algorithm

1)     

Bezier Curve: P (u) =

   ................     (1)

Here B_n,i (u) = ⁿc_i uⁱ (1 - u) ^n-i       ..............        (2)

C (n, i) = nC_i = n!/i!(n - i)!      0 ≤ u ≤ 1

2)      Cubic Bezier curve has n = 3 as:

------------------(3)

= p₀ B_{3, 0} (u) + p₁ B_{3, 1} (u) + p₂ B_{3, 2} (u) + p₃ B_{3, 3} (u)

Now, let's find B₃, 0 (u), B₃, ₁ (u), B_{3, 2} (u) , B_{3, 3} (u) by using equation as in above:

Bn, i (u) = nci u (1 - u)

a)  B3, 0(u) = 3C0   u0   (1 - u)3 - 0

=          3!       . 1. (1 - u)3 = (1 - u)3

       0!(3 - 0)!

 

b)  B3, 1(u) = 3C1   u1   (1 - u)3 - 1

=          3!       . u. (1 - u)2 = 3u (1 - u)2

1!(3 - 1)!

c) B3,2(u) = 3C2   u2   (1 - u)3 - 2

=          3!       . u². (1 - u) = 3u²(1 - u)

       2!(3 -2)!

B3,3(u) = 3C3   u3   (1 - u)3 - 3

=          3!       . u³. (1 - u) 

         3!(3 -3)!

By using (a), (b) , (c) and (d) in (5) we find here:

P (u) = p₀ (1 - u)³ + 3p₁u (1 - u)² 3p₂u² (1 - u) + p₃ u³

Posted Date: 4/4/2013 5:13:58 AM | Location : United States

Ask an Expert

Related Discussions:- Important point about the de casteljeau algorithm, Assignment Help, Ask Question on Important point about the de casteljeau algorithm, Get Answer, Expert's Help, Important point about the de casteljeau algorithm Discussions

Write discussion on Important point about the de casteljeau algorithm
Your posts are moderated

Write your message here..

Related Questions

Area-subdivision method, Area-Subdivision method This method is a ty... Area-Subdivision method This method is a type of an image-space method although uses object-space operations re-ordering or sorting of surfaces as per to depth. Area sub-div

Delta-delta arrangement and in-line arrangement, Delta-Delta Arrangement an... Delta-Delta Arrangement and In-Line Arrangement There are two types of shadow masks available, delta-delta arrangement and in-line arrangement. The in-line arrangement refers t

What do you mean by perspective projection, What do you mean by Perspective... What do you mean by Perspective projection?  Perspective projection is one in which the lines of projection are not parallel. Instead, they all converge at a single point known

Proof of subsequent properties of bezier curves, Proof of subsequent proper... 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 (p 1 - p 0 ) P

Line drawing - points and lines, Line Drawing - Points and lines Line ... Line Drawing - Points and lines Line drawing is accomplished through computing the intermediate point coordinates along the line path between two given end points. Since, scre

Interactive 3d computer graphics, Describe interactive model of computer gr... Describe interactive model of computer graphics and application areas of interactive computer graphics.

Analog sound vs. digital sound, Analog Sound vs. Digital Sound Sound e... Analog Sound vs. Digital Sound Sound engineers have been debating the respective merits of digital and analog sound reproduction ever if the form of digital sound recordings.

Raster scan Display, Draw and explain the diagram of a Raster scan system w... Draw and explain the diagram of a Raster scan system with a display processor. Explain each unit of the diagram.

Microsoft Word 2010, what are three features to use to create your flyer to... what are three features to use to create your flyer to market?

What colour depths are basically used, Q. What Colour depths are basically ... Q. What Colour depths are basically used? Basically the number of colours are an exponential power of 2 as for Colour Depth n, colours can be showed. The most popular colour mo