C. Questions on Surfaces.

Question 1C.
The following data points have been recorded over a surface:- A(6,12,3), B(8,9,4), C(3,8,5), D(6,4,8), E(9,3,4), F(1,2,3) and many others elsewhere on the surface. The others need not concern us.

(a) Calculate the Delaunay triangulation for these six points.

(b) Using linear interpolation, calculate the contours for the heights 4 and 6 and sketch to resulting surface.

(This should take about 40 mins.)





Question 2C.
Part of an array of spot-heights contains the following values: 

          20    25    35    40 

          30    40    40    36

(a) Using Bi-linear patches, calculate and draw the profile of the slice along v=0.75. (10%)

(b) Fit cubics to each of the rows (you can find the coefficients of a cubic through any four points) and then use the equations for a lofted surface to calculate the profile of the same slice. (20%)

(c) Comment on any differences between these profiles. (5%).

(THis should take about 50 mins.)





Question 3C.
(a) Give a brief description of the methods used to present a three-dimensional surface as a drawing in two dimensions. (10%)

(b) A surface has been recorded as a series of spot-heights at the vertices of a square mesh, and a small area within this surface corresponds to the values given below. 

           30    35    33    28 

           26    32    24    20 

           19    24    15    10
Show how you would represent this surface by drawing the contours for the values c=20 and c=30. (20%)

(c) Comment on the overall shape of the surface. (5%)

(This should take about 45 mins).





Question 4C.
(a) Derive the equation for a Bezier surface and describe how these surfaces may be used with a control mesh to model the shape of a surface over a given area. What conditions must be satisfied for two adjoining surfaces to meet smoothly ? (15%)

(b) The values given below specify the heights of the control points of a Bezier surface. Calculate enough surface points to draw the profile of a diagonal slice from B(1,1) to B(3,4). (15%)

      B(1,1)=10  B(1,2)=12  B(1,3)=16  B(1,4)=10 
      B(2,1)= 8  B(2,2)=10  B(2,3)=18  B(2,4)=14
      B(3,1)=12  B(3,2)=16  B(3,3)=14  B93,4)=16

(c) Discuss the effect of changing the value of B(2,2) to 20. (5%)

(This should take about 50 mons.)