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Disk Bezier curves

Disk Bezier curves. Slides made by:- Mrigen Negi Instructor:- Prof. Milind Sohoni. A disk in the plane R 2 is defined to be the set <P> := {x | | x - c | . r, c ,r }. we also write <P> = <c ; r>,

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Disk Bezier curves

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  1. Disk Bezier curves Slides made by:- Mrigen Negi Instructor:- Prof. Milind Sohoni.

  2. A disk in the plane R 2 is defined to be the set <P> := {x | | x - c |. r, c ,r }. we also write <P> = <c ; r>, The following operations are defined for disks <c ; r>, <ci ; ri > a<c ; r> = <ac ; |a |r >, <c1 ; r1> + <c2 ; r2> = <c1 + c2 ; r1 + r2 > We get equations And

  3. A planar disk Bezier curve is then defined as the center curve of the disk Bezier curve <Q>(t) is a Bezier curve with control points { ck } the radius of <Q> (t) is the weighted average of the radii { rk: } of the control points.

  4. The disk Bezier curve is a fat curve with variable width which is a function of the parameter t given by • The disk Bézier curve <Q>(t) can also be written as <Q(t)>=(x (t), y(t)) r(t) and r(t) are called the center curve and the radius of the disk Bézier curve (Q)(t)respectively. A disk Bézier curve can be viewed as the area swept by the moving circle with center C(t) and radius r(t)

  5. De Casteljau algorithm. • For any t0 [0,1], (P)(t0) can be computed as follows: • Set ,i=0,1,2,…n • For k = 1, 2, . . . , n do • For l= 0, 1, . . . ,n - k do • End l • End k • Set an obvious generalization of the real de Casteljau algorithm.

  6. Envelop of disk Benzier curves • Let the Bezier curve be thought of as the envelope of a set of curves parametrized by t. If this is written as F ( x , y, t) = 0 then the envelope is found by solving F(x,y,t)=0 and Since We have

  7. R=r(t), , (1) (2) Now substituting (2) to (1) we get We have assumed ||c’||>|R’| for real solutions

  8. solution of the above system of equations is • We might have a particular case when r0 = 'rl . . . . . . . rn= constant. Then r(t) = r in which case

  9. Let T • Then c ' (t) q (t) = 0, II q (t)|| = 1. This means that the two envelopes of the disk Bezier curve can be written as Q1(t) = c(t) + rq(t), Q2(t) = c(t) - rq(t).

  10. Subdivision • Let c ∈ (0, 1) be a real number. Then the disk Bézier curve can be subdivided into two segments: for 0<=t<=c for c<=t<=1.

  11. Degree elevation • The degree n disk Bézier curve can be represented as a degree n + 1 disk Bézier curve as follows where the control disks for the degree elevated curve are i=0,1,2,…,n+1

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