DESIGN OF CURVES 127
Exercises
- Consider a parametric cubic curve r(u) where
rPPPP( ) = uF F F 10 + 21 + 30 ′′ + F 41 0 ≤u≤ 1
whereF 1 = 1 – 3u^2 + 2u^3 ,F 2 = 3u^2 – 2u^3 ,F 3 = u – 2u^2 + u^3 ,F 4 = –u^2 + u^3.
In some situations, data about PP 01 ′′ and is not available. Instead, vectors PP 01 ′′ and ′′ are known. In such
cases, derive the expressions for all elements of K for the parametric equation to be written in the form
r(u) = U K C
whereU = [u^3 u^2 u 1], CT = [P 0 P 1 PP 01 ′′ ′′] and Kis the 4 × 4 matrix.
- Given a parametric cubic curve whose geometric coefficients are [ ]PPPP 0101 ′′T snip or trim the curve at
u = 0.7 and reparametrize this segment so that 0 ≤u≤ 1. Find the relationship between the geometric
coefficients of the snipped and original curves.
- Derive the cubic Bézier curve in the matrix form, illustrating the control points, the curve shape, and the
blending functions through sketches. Derive also the expression for the tangent at any given point on the
curve. Write a computer code to display a 3D cubic Bézier curve. The input shall be the control point
coordinates. Shift any one of the given control points to a new location and show the change in shape using
a plot. Output also the tangent at any given u value.
- Consider a Bézier cubic curve obtained by a set of points P 0 ,P 1 ,P 2 and P 3. Assume that it is not possible
to specify P 1 and P 2 but one can specify P, the point of intersection of P 0 P 1 and P 2 P 3. The Bézier curve for
P 0 ,P,P 2 will be quadratic one. What will be the relation between P*,P 0 ,P 1 ,P 2 and P 3 so that the cubic
as well the quadratic Bézier curves are identical.
- A parametric cubic curve is to be fitted to pass through (interpolate) four points P 0 ,P 1 ,P 2 ,P 3. The first and
last points P 0 ,P 3 are to be at u= 0 and u = 1, respectively. Points P 1 and P 2 are at u = 1/3 and u = 2/3,
respectively. The equation of the curve is to be written in the form
rUMP
P
P
P
P
( ) = = [^32 1]
11 12 13 14
21 22 23 24
31 32 33 34
41 42 43 44
0
1
2
3
uuuu
mmmm
mmmm
mmmm
mmmm
p
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
Show that Mp is given by
Mp =
- 4.5 13.5 –13.5 4.5
9.0 – 22.5 18 – 4.5
- 5.5 9.0 – 4.5 1.0
1.0 0 0 0
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
(a) Plot the curve passing through (0, 0), (1, 0), (1, 1), (0, 1).
(b) A circular arc of radius 2 lies in the first quadrant. Write the coordinates of the 4 points that are equally
spaced on this arc. Determine the point on the arc at u =^12 using r (u) above. How far does it deviate
from the midpoint of the true quarter circle?
- In Exercise 5, let P 2 and P 3 be at u = α and u=β(α < β < 1). Re-derive the expression for the basis matrix
Mp.
- A 3-D parametric cubic curve has the start and end points at P 0 (0, 0, 0) and P 1 (1, 1, 1), and the end tangents
are (1, 0, 0) and (0, 1, 0).
(a) Find and draw the parametric equation of the curve segment.
