Question 1B.
Show how the equation of a general Bernstein polynomial may be used to give a Bezier curve with four control points.

Derive Bezier curves to design an output curve representing the letter `S'. The first point on the curve (at the top) has the coordinates (1.0,1.8), the mid-point has the coordinates (0.5,1.0) and the end-point has the coordinates (0.5,0.2). Use two 4-point Bezier curves to produce the curve through those points and ensure that the gradient does not change at the mid-point where the curves join. (20%)

Estimate suitable positions for the control points and calculate at least three points along each section of the curve (at t=0.25, t=0.5 and t=0.75) to check the shape of your profile. (15%)

(This should take about 45 mins.)

Question 2B (January 1998)
The constellation shown below is sometimes referred to as "the saucepan". The saucepan has a curved handle and a three-dimensional body. Assign suitable coordinates to the image of the stars and then show how to derive interpolating curves passing through the stars to draw a saucepan. (35%)

           *           *
  *                 
                                   *
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(This should take about 50 mins.)

Question 3B.
Explain how the general equation of a B-spline curve may be used to generate a curve of order k with n+1 control points P0, P1, ..., Pn.

Now consider the particular example of a third-order curve with the following six control points in the plane z=0. P0 = (10,20), P1 = (30,80), P2 = (50,10), P3 = (70,70), P4 = (90,50) and P5 = (110,20). Evaluate a number of points along the curve and give a rough sketch of the resulting curve.

Now assume the point P3 is moved to a new position P3 = (70,-10,20). State how much of the curve is altered and confirm your statement by evaluating some points along the section which is changed. Again illustrate your answer with a rough sketch.

(A 40-minute question. )

Question 4B (June 1997)
When producing cartoons, much time may be saved by drawing a limited number of outlines and using computer generated curves for the intermediate drawings. If the example below shows part of a drawing, explain how you would generate intermediate curves moving from the one profile to the other. (15%)

          A1                                  A5




                                                        B5
                B1




                                                                      C5
                      C1
                                                                D5

          D1

The initial points are A1(10,70), B1(24,60), C1(30,40) and D1(20,20). The final points are A5(18,70), B5(32,62), C5(46,44) and D5(40,30).

Assume you use a set of curves (probably cubics) to interpolate the sections between successive points along the profile and assume that the points (e.g. A1 to A5) may be linearly interpolated to get successive curves. Calculate the points A2, A3 and A4 for successive positions of the point A. (5%)

Now calculate the points A3, B3, C3 and D3 along this curve and use suitable intepolation formulae to calculate three points in the segment B3C3 (You may assume that this segment has gradients of -0.5 at B3 and -0.3 at C3). Explain how you decide which form of curve to represent each section. Will the same form of curve and the same conditions at the knot points apply in each case. Hence sketch the curve from A3 to D3. (15%)

(This should take about 50 mins.)