Fatih GÜNDÜZ 030060144 9th week Answers Part 3

Bezier Surface : (Previous)
This is a synthetic surface similar to the bezier curve ans is obtained by the transformation of a bezier curve.It permits twists and kinks in the surface.The surface does not pass through all the data points.A Bezier surface is defined by a twoo-dimensional set of control poins Pi,j where i is in the range of 0 and m, and j is in the range of 0 and n.Thus, in this case , we have m+1 rows and n+1 columns of control points and the control point on the ith row and jth column is denoted by Pi,j.Note that have (m+1)(n+1) control points.
( Alavala,Cad/Cam: Concepts And Applications,page 207)



Bezier Surface : (New) (Better) (Surface Representation)
A tensor product surface patch is formed by moving a curve through space while allowing deformations in that curve. This can be thought of as allowing each control point bi to sweep a curve in space. If this surface is represented using Bernstein polynomials, a Bezier surface patch is formed, with the following formula:

 Here, the set of straight lines drawn between consecutive control points bij is referred to as the control net. It is easy to see that boundary iso parametric curves have the same control points as the corresponding boundary points on the net.
(Shape Interrogation for Computer Aided Design and Manufacturing, Springer, 2010, p.19)

B-spline surface : (Previous)

The same tensor product method used with Bezier curves can extend B-splines to describe B-spline surfaces. A rectangular set of data (control) points creates the surface. this set forms the vertices of the characteristic polyhedron that approximates and controls the shape of the resulting surface. 

(CAD/CAM Theory and Practice, İbrahim Zeid, McGraw Hill. 1991, p300.)

 

B-spline surface : (New) (Better) (Surface Representation)

The surface analogue of B-spline curve is the B-spline surface (patch). This is a tensor product surface defined by a topologically rectengular set of control points

 

 

    Parametric lines on a B-spline surface are obtained by letting u=const, or v=const. A parametric line with u=u0 is a B-spline curve in v with V as its knot vector and verticles 

Some of the properties of the B-spline curves can be easily extended to surfaces such as:

- Geometry invarience property.

- End points geometric property.

- Convex hull property.

- B-spline to Bezier property.

   However, no variation diminishing property is known for B-spline surface patches.

(Shape Interrogation for Computer Aided Design and Manufacturing, Springer, 2010, p.30)