246 COMPUTER AIDED ENGINEERING DESIGN
- It is desired to create a closed tubular Bézier surface by using 5 control points (last control point being
the same as the first) at each section, and the control point net created by using 5 such sections. Develop
the program and demonstrate using an example. - Write a procedure to compute the coordinates of a point on a Bézier surface patch. Use this to compute
a rectangular array of points to display a Bézier patch. The program should be generic and not restricted
to cubic. - Develop and discuss the conditions required for C^0 and C^1 continuity between two Bézier patches along
a common boundary. - Write a procedure to compute the coordinates of a point on a B-Spline surface patch. Display the surface
using the code developed. Note that a B-spline patch is a tensor product surface defined as r(u,v) =
jΣΣn
in
=0 =0Np,p+i (u)Nq,q+j (v)rij.- Use the code developed to compute an approximate solution to the minimum distance between two
given parametric B-spline surfaces. First, calculate a rectangular array of points for chosen interval
steps for u and v on both the patches and then proceed. - Write a procedure to compute the intersection between a straight line and a bi-cubic patch. Simplify your
solution by first performing a transformation on both line and the patch so that the line is collinear with
thez-axis. Find the intersection and perform inverse transformation. - (a) Generate a closed tubular surface patch using closed B-Splines. The fundamental aspect is in first
having experience in creating a closed B-spline curve by taking the vertices (for 8 unique control points)
P 1 ,P 2 ,P 3 ,P 4 ,P 5 ,P 6 ,P 7 ,P 8 ,P 1 ,P 2 ,P 3. Uniform knot vector [0 1 2 3 4 5 6 7 8 9 10 11 12 13 14] is to
be used. For a fourth order (k = 4) closed B-spline curve defined by the above polyline and for 0 ≤u≤1,
a point on the curve is calculated from the matrix formulation
rP
P
P
Pjj
j
j
j+1uuuu^32( mod8)+1
(( +1)mod8)+1
(( +2)mod8)+1
(( +3)mod8)+1() =^1
6
[ 1]–1 3 –3 1
3–6 30
–3 0 3 0
14 10⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥⎡⎣⎢
⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥
⎥
wherejmod 8 is the remainder when j is divided by 8 (for example, 10 mod 8 = 2). Let us take the case
of a tubular surface above with 5 axial cross sections, each cross section having the same number of
control points mentioned above. Show the effect of changing the size of different cross sections and also
the effect of relocating any intermediate control point.- Write generic codes for the following:
(a)C^1 continuous composite Ferguson’s surface.
(b)C^2 continuous composite Ferguson’s surface.
(c) Composite Bézier surface with tangent plane continuity. Let all the control points be freely chosen
for each patch so that at least position continuity is addressed. Later, implement interactive relocation
of the control points so that the tangent plane continuity is met. - Write a generic code for the NURBS surface patch.