A Modified BEZIER Algorithm For Controlling and Generating A Third Degree Curves

This paper presents a new method to generate curves, which allows the designer to produce a curve in combinational way allowing him to get the shape that he had in his imagination with keeping the four control point for the curve design. Basing on the modified BEZIER curve, the upgraded algorithm used de Casteljau algorithm on interval [r, s] as a mathematical base to upgrade the suggested method. The suggested method shows a great flexibility at the curve controlling area with changing one, more or no need to change the control points of Bezier curve as it shows in the figures presented in the paper.


Introduction
Curve generating algorithms by using de Casteljau algorithm.In this proposed work that we modified for controlling and generating Bezier curves of degree three.Polynomial curve is defined in terms, of polar forms.Because the polynomial curve involved containing one variable, natural way to polarize polynomial curve.
The approach yields polynomial curve.It shows versions of the de-Casteljau algorithm can be turned into subdivisions, by giving an efficient method of performing subdivision.It is also shown that it is easy to compute a new control net from given net.For this the paper develops an algorithm which should include:-1-Original Bezier curve.We will use [4], [5], [6], [7].2-Gallier Modified Bezier curve.We A linear combination of m vectors V 1 , V 2 , ..., V m is a vector, say W, of the form W=a 1 V 1 +a 2 V 2 +…+a m V m , …(1) Where a 1 , a 2 … a m are scalar.A linear combination (1) of vectors is called an affine combination if a 1 +a 2 +...+a m =1 The coefficients of an affine combination of two vectors V 1 and V 2 are often forced to have scalar t for one of them and (1-t) for the other in the form: Affine space A n is called the real affine space of dimension n.In most case, we will consider n= 1,2,3.A is defined over the field R (where R is real numbers) Polar Form of a Polynomial Curve [7], [8].
A method of specifying polynomial curves that yield very nice geometric constructing the curves is based on polar polynomial form.To define the polar from of a polynomial of degree three, let  8) It is easy to notice that f 1 (t 1 , t 2 , t 3 ) in ( 7) is polar form of F 1 =X (t) in (5), and f 2 (t 1 , t 2 , t 3 ) in ( 8) is polar form of Using two control points say P 0 and P 1 shows that P (t) is on the line determined by P 0 and P 1 , P (t) is indeed between P 0 and P 1 , at (1-t) of the way from P 0 , every point P (t) on the line P 0 P 1 is obtained by a single interpolation step, P (t)=(1-t) P 0 + tP 1 .Which linear interpolation at P (0)= P 0 and P (1)= P 1 ,for 0 ≤ t ≤ 1.

De-Casteljau Algorithms
De-Casteljau algorithm is an algorithm uses a sequence of control points to construct a well defined curve P(t) at each value of t from 0 to 1.This provides a way to generate a curve from set of points.Changing the points will change the curve.Let us illustrate the algorithm of degree three in Table (1).The de Casteljau algorithm uses a sequence of four control points P 0 , P 1 , P 2 and P 3 to construct a well-defied curve P (t) at each value of t from 0 to 1 P(t) is defined as:  14) Substitution of equations (10,11,12,13),and ( 14) in ( 9) gives The coefficients of the control points in the cubic curve ( 15) are called Bernstein Polynomials, gives as: This equation immediately yields an important property of these polynomials.They add to unit at every t, mathematically.

Gallier Modified cubic Bezier Curves [7], [8].
The modified de-Casteljau algorithm is an algorithm that uses a sequence of control points To construct a well defined curve f(t)=f(t, t, t), on a given affine frame [r, s] for which r ≠ s, t ∈ [r, s], and (r, s∈ A).Let us begin with straight lines, given any affine frame [r, s] for which r ≠ s for t ∈ A can be written and …(19) De Casteljau algorithm uses two control points say F(r) and F(s) as in equation ( 19).
Treat the coordinates of each point as a two-component vector and using the symbols p 0 , p 1 , p 2 and p 3 for control points.Let p i =(x i , y i ) for i=0,1,…,m.And The set of points, in parametric form is The pressing equation ( 39) represents the pair of equations PDF created with pdfFactory Pro trial version www.pdffactory.comThe set of points, in parametric form is To explain the above new cubic Bezier curve (39) by the following example is given Example: -Let the four control points as: P 0 =(x 0 , y 0 ) = (100,300), P 1 =(x 1 , y 1 ) = (40, 50), P 2 =(x 2 , y 2 ) = (430, 50), P 3 =(x 3 , y 3 ) = (270,300).
With respect to the interval r=-5, s=5.For parameter t where , where t is increases by (t=0.0005).The following cases have been discussed.

Case 1
Let λ 6 =λ 5 =λ 4 =λ 3 =λ 2 =λ 1 .Then equation ( 39) reduces to the original Bezier curve in equation ( 15) see figure 3. Equation ( 39) is building by mathematical developing of de Casteljau algorithm.This case can be seen by simply mathematical changing of equation ( 39) which reduces to modified equation ( 27), that easily reduces to original equation ( 30) by take special case for r=0 and s =1.This gives the base classical populates of Bezier curve that are interpolations at the first point P 0 and the last point P 3 .The design obtained does not change only from change at least one or more of control points of Bezier curve.

Case 2:
Let λ 6 =λ 5 =λ 4 =λ 3 =λ 1 , and (+λ 2 ) is take to be increases (t 2 =0.02).In this case it is found that the design can be moved to right in an interior and exterior manner with no need to change any of the control points, see figure 4. New formula of equation (39).In this case which is build by a new change easily done by changing only part λ 2 (i.e.t 2 =0. 02),which gives the new design and move it to right as see in figure 4. Case 3: Let λ 6 =λ 5 =λ 4 = λ 3 =λ 1 , and (-λ 2 ) is taken to be decreased.It is found in this case that the design can be moved to left in an exterior and interior manner with no need to change any of the control points, see figure 5.In this case as in case 2 building by new condition in this which give new formula of equation ( 39). by changing only part λ 2 (i.e.t 2 = -0.02) can change the design to left, see figure 5.

Case 4:
Let λ 6 =λ 5 =λ 3 = λ 2 =λ 1 , and (+λ 4 ) increased.In this case, it is found that the design can be moved right to (exterior manner) with no need to change any of the control points.See figure 6.By variation in last equation only part λ 4 (i.e.t 4 =0.02) the design can be moved to right with out changing any of the control points.See figure 6. Case 5: Let λ 6 =λ 5 =λ 3 =λ 2 =λ 1 , and ( -λ 4 ) decreased.In this case, it is found that that the design can be moved to the left to ( interior manner) with no need to change any of the control points.See figure 7. By changing the formula of equation ( 39) as the condition of this case.

Figure ( 4 )
Figure (4) Bezier cubic when (+ λ2) increases the design can be moved to right in an interior and exterior manner with no need to change any of the control points.

Figure( 5 )
Figure(5) Bezier cubic when (-λ 2 ) decreases the design can be moved to left in an exterior and interior manner with no need to change any of the control points

Figure ( 7 )
Figure (7) Bezier cubic when (-λ 4 ) decreases the design can be moved to the left to (interior manner) with no need to change any of the control points .

Figure ( 8 )
Figure (8) Bezier cubic at all cases the designer controlling the generating curve keeping the control points unchanged.

com Eng. & Tech. Journal, Vol.28, No.21, 2010 A Modified BEZIER Algorithm for Controlling and Generating a Third Degree Curves 6265
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& Tech. Journal, Vol.28, No.21, 2010 A Modified BEZIER Algorithm for Controlling and Generating a Third Degree Curves 6267
PDF created with pdfFactory Pro trial version www.pdffactory.comEng.

Table ( 2) de Casteljau algorithm
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