Q: Explain Bezier curve and properties of Bezier curves.
Ans:-
Bezier curve is discovered by the French engineer Pierre Bézier. These curves can be generated under the control of other points. Approximate tangents by using control points are used to generate curve. The Bezier curve can be represented mathematically as –
Where pipi is the set of points and Bni(t)Bin(t) represents the Bernstein polynomials which are given by –
Where n is the polynomial degree, i is the index, and t is the variable.
The simplest Bézier curve is the straight line from the point P0P0 to P1P1. A quadratic Bezier curve is determined by three control points. A cubic Bezier curve is determined by four control points.
Properties of Bezier Curves
Bezier curves have the following properties −
- They generally follow the shape of the control polygon, which consists of the segments joining the control points.
- They always pass through the first and last control points.
- They are contained in the convex hull of their defining control points.
- The degree of the polynomial defining the curve segment is one less that the number of defining polygon point. Therefore, for 4 control points, the degree of the polynomial is 3, i.e. cubic polynomial.
- A Bezier curve generally follows the shape of the defining polygon.
- The direction of the tangent vector at the end points is same as that of the vector determined by first and last segments.
- The convex hull property for a Bezier curve ensures that the polynomial smoothly follows the control points.
- No straight line intersects a Bezier curve more times than it intersects its control polygon.
- They are invariant under an affine transformation.
- Bezier curves exhibit global control means moving a control point alters the shape of the whole curve.
- A given Bezier curve can be subdivided at a point t=t0 into two Bezier segments which join together at the point corresponding to the parameter value t=t0.
Q: Construct the Bezier curve of Order 3 and 4 polygon vertices A(1,1) , B(2,3) , C(4,3) , D(6,4).
Ans:-
Q: Construct enough points in the Bezier curve whose control points are p0(4,2) , p1(8,8) , p2(16,4) to draw an accurate sketch.
- What is the degree of the curve.
- What is the coordinate at u = 0.5
Ans:-
Q: Calculate and also roughly track the Beizer curve for three control points (1,1) , (2,2) , (3,1).
Ans:-
Numerical :-
Q: Compare and contrast among spline , B-spline and Beizer algorithms for curve generation and write the algorithm for Beizer curve generation.
Ans:-
Ans:-
What is the Bezier Curve?
Bezier curves are parametric curves used frequently in modeling smooth surfaces in computer graphics and many other related fields. These curves can be scaled indefinitely. Linked Bezier curves contain paths that are combinations that are intuitive and can be modified. This tool is also made use of in controlling motions in animation videos. When programmers of these animations talk about the physics involved, they are in essence talking about these Bezier curves. Bezier curves were first developed by Paul de Castlejau using Castlejau’s algorithm, which is considered a stable method to develop such curves. However, these curves became famous in 1962 when French designer Pierre Bezier used them to design automobiles.
What is the B-Spline Curve?
B-Spline curves are considered as a generalization of Bezier curves and as such share many similarities with it. However, they have more desired properties than Bezier curves. B-Spline curves require more information such as a degree of the curve and a knot vector, and in general, involve a more complex theory than Bezier curves. They, however, possess many advantages that offset this shortcoming. Firstly, a B-Spline curve can be a Bezier curve whenever the programmer so desires. Further B-Spline curve offers more control and flexibility than a Bezier curve. It is possible to use lower degree curves and still maintain a large number of control points. B-Spline, despite being more useful are still polynomial curves and cannot represent simple curves like circles and ellipses. For these shapes, a further generalization of B-Spline curves known as NURBS is used.
Q: Let p0(0,0) , p1(1,2), p2(2,1), p3(3,1), p4(4,10) and p5(5,5) be given data control points. If interpolation based on Bezier curve is used to find a curve interpolating these data points. Find parametric midpoint of the gradient and also calculate coordinate of parametric quartiles of the curve.
[AKTU 2014-15 6M]
Ans:-
0 Comments