Properties of bezier curves - modeling and rendering, Computer Graphics

Properties of Bezier Curves - modeling and rendering

A very helpful property of a Bezier curve is that it always passes via the first and last control points. Such the boundary conditions at the two ends of the curve are as:

P(0) = p 0

P(1) = p n

Values of the parametric first derivatives of a Bezier curve on the end points can be computed from control-point coordinates as

P' (0) = -np0  + np1

P' (1) = -npn -1 + npn

Hence, the slope at the starting of the curve is beside the line joining the first two control points and the slope on the ending of the curve is beside the line joining 2end-point in the last. As the same, the parametric second derivatives of a Bezier curve on the endpoints are computed as:

p' ' (0) = n)n - 1)[(p2  - p1 ) - (p1 - p0 )]

p' ' (1) = n)n - 1)[(pn -2  - pn -1 ) - (pn -1 - p n)]

The other important property of any Bezier curve is which it lies inside the convex hull as convex polygon boundary of the control points. Such follows from the properties of Bezier blending functions as: They are all +ive and their sum is all the time 1,

1408_Properties of Bezier Curves - modeling and rendering.png

Hence any curve position is merely the weighted sum of the control-point positions. The convex-hull property for a Bezier curve makes sure about the polynomial will not have erratic oscillations.

856_Properties of Bezier Curves - modeling and rendering 1.png

Figure (a): Shows closed Bezier curve generated by specifying the first and last control points at the same location

1294_Properties of Bezier Curves - modeling and rendering 2.png

Figure(b): Demonstrates a Bezier curve can be made to pass closer to a given coordinate by assigning multiple control points to that position.

Posted Date: 4/4/2013 5:27:04 AM | Location : United States







Related Discussions:- Properties of bezier curves - modeling and rendering, Assignment Help, Ask Question on Properties of bezier curves - modeling and rendering, Get Answer, Expert's Help, Properties of bezier curves - modeling and rendering Discussions

Write discussion on Properties of bezier curves - modeling and rendering
Your posts are moderated
Related Questions
Exceptional cases - Orthographic Projection 1)   We have an Orthographic projection, if f=0, then cot (β) =0 that is β=90 0 . 2)   β =cot-1 (1)=450 and this Oblique projec

Explain the merits and demerits of Plasma panel display. Advantages: Refreshing is not needed. Produce a very steady image free of Flicker. Less bulky than a CRT.


3-D Modeling and Animation Tools: By 3-D modeling software, objects rendered in perspective show more realistic. One can produce stunning scenes and wander by them, choosing just

Can you list at least three important applications of computer graphics? There are lots of interesting applications of computer graphics. Three common applications are compute

Example 1 : Determine the number of memory bits essential for three bit plane frame buffer for the 512 x 512 raster Solution: Whole memory bits needed are 3 x 512 x 512 = 786,

Discuss various Techniques for Character Generation

Animator Studio and Elastic Reality - Computer Animation Animator Studio It is a cell animation program from AutoDesk. Its predecessor was Animator Pro for PC DOS. Anima

Explain about the Computer Based Training CBT makes use of a computer system to train people in numerous applications. It makes use of self-assessment and multimedia with minim

Determine the transformation matrix for the reflection about the line y = x. Solution: The transformation for mirror reflection regarding to the line y = x, comprises the subs