Question

Derive the conditions for smooth joining of two bezier curve segment of degree three

Answer 1

Joining two or more curve segments together forms

a continuous composite curve. We start the discussion of

composite curves and their continuity by investigating how to

blend a new curve between two or more existing curves,

forming a composite curve consisting of two or more curve

segments together. In geometric modeling, curves are joined

with smooth transitions at the junction. When joining two or

more curves, the end points must coincide. In order to

maintain smoothness, their first parametric derivatives must

be equal at that point and this will create a condition of

continuity called

parametric continuity. In this

paper conditions for smooth connection of interval Bezier

curve segments are presented. The conditions for joining two

or more interval Bezier curve segments at a common point are

converted into the conditions for smooth connections of the

four fixed Kharitonov's polynomials (four fixed Bezier curves)

associated with the first interval Bezier curve with the

corresponding four fixed Kharitonov's polynomials (four fixed

Bezier curves) associated with the second interval Bezier curve

at common points

for . Finally, the

interval control points of the second interval Bezier curve are

obtained from conditions for smooth connections of the four

fixed Kharitonov's polynomials (four fixed Bezier curves)

associated with the first interval Bezier curve with the

corresponding four fixed Kharitonov's polynomials (four fixed

Bezier curves) associated with the second interval Bezier

curve. A numerical example is included in order to

demonstrate the effectiveness of the proposed method

Answer 2

give us a chance to begin the talk of composite bends and their congruity by researching how to mix another bend between at least two existing bends, shaping a composite bend comprising of at least two bend portions together. In geometric displaying, bends are joined with smooth changes at the intersection. When joining at least two bends, the end focuses must harmonize. So as to look after smoothness, their first parametric subordinates must be equivalent by then and this will make a state of progression called  parametric congruity. In this paper conditions for smooth association of interim Bezier bend sections are displayed. The conditions for joining at least two interim Bezier bend sections at a typical point are changed over into the conditions for smooth associations of the four settled Kharitonov's polynomials (four settled Bezier bends) related with the primary interim Bezier bend with the comparing four settled Kharitonov's polynomials (four settled