Computer Aided Engineering Design

128 COMPUTER AIDED ENGINEERING DESIGN

(b) If the end tangents have the magnitudes as α and β, show some results of the variation in curve shape

due to the changes in α,β.

8. The geometric matrix G of a parametric cubic curve defines a straight line segment if

G = [P 0 P 1 α (P 1 – P 0 ) β (P 1 – P 0 )]T

Express the equation of the straight line as a cubic function in u. Tabulate and draw the points on the straight

lines at intervals of Δu = 0.01 from u = 0 to u = 1 in the following cases:

(a)α = β = 1 (b) α = β = – 1

(c)α = 2, β = 4 (d) α = –2, β = –4

At what values of u the trace of the line changes directions in each of the cases?

9. Write a procedure to truncate a parametric Ferguson segment curve at two specified values of u and
   subsequently reparametrize it. Test your program for a parametric cubic curve with a given set of end
   pointsP 0 (1, 1, 1) and P 1 (4, 2, 4) and the end tangents ru(0) = (1, 1, 0) and ru(1) = (1, 1, 1) truncated at:
   (a)u = 0.25 and u = 0.75, (b) u = 0.333 and u = 0.667.


10. Write a procedure for blending a Ferguson segment between two given such segments. Create a 2-D
    numerical example to test your algorithm. Show the effect of changing the magnitudes of the tangent
    vectors at curve joints.


11. Find the expressions for the curvature at a point on a Ferguson and Bézier segment. Calculate the curvatures
    at the end points of a Bézier segment having the control points (1, 1), (2, 3), (4, 6), (7, 1). Plot the Bézier
    curve along with its convex polygon.


12. A composite Bézier curve is to be obtained by joining two Bézier curves with control points at P 0 ,P 1 ,P 2 ,
    P 3 and Q 0 ,Q 1 ,Q 2 ,Q 3. Develop a procedure and check your results by taking a 2-D example. Modify your
    results by taking Q 0 ,Q 1 ,Q 2 ,Q 3 ,Q 4 as control polyline for the second curve.


13. Enumerate conditions to obtain a closed, C^1 continuous Bézier curve.


14. Write a computer program implementing de Casteljau’s algorithm for cubic curves, over some interval u 1
    andu 2. Test your program with points P 0 = (6, – 5), P 1 = (–6, 12), P 2 = (–6, –14), and P 3 = (6, 5). Use de
    Casteljau’s algorithm to find the coordinates of points on the curve at u= 0.25, 1/3, 0.5, 2/3, 0.75 and 1. Plot
    the cubic curve.


15. Show, through an example, that a Bézier curve is affine under both translation and rotation. You can choose
    the control points in Exercise 15 and rotate the axes by 45 degrees or translate the origin to (–2, –2) for
    demonstration.


16. Given a set of control points P 0 ,P 1 ,P 2 ,P 3 explain what happens to a Bézier segment when two of the
    control points are coincident. Give an example. Does the degree of the curve drop? Does the curve have a
    cusp at some control point? Does the curve have an inflexion at some control point?


17. Show that the curvature of a planar curve is independent of the parametrization, that is, if r(u) = [x(u)y(u)]
    is the curve, then a change of variables u = φ (v), where φ ̇() 0v ≠ does not affect the curvature.


18. LetP 0 ,P 1 ,P 2 ,P 3 be given control points. Construct two quadratic segments Q 1 Q 2 (r 1 (u)) and Q 2 Q 3 (r 2 (u))
    such that, for u∈ [0, 1] as shown in Figure P4.1.

Figure P4.1

P 1

P 2

P 3

P 0

Q 1

Q 2

Q 3