Past exam of the computer science course of the University of Oxford / 2015 / Geometric modelling / 2 / f

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- Solution f

Solution

Assign one unit parameter interval to each piece. Then

S (u) = {F (u), E (u - 1), 0 \leq u \leq 1, 1 \leq u \leq 2.

There is no ambiguity at

u = 1

, since both pieces give the origin. For a single B-spline representation take degree two, the four de Boor points

(D_{0}, D_{1}, D_{2}, D_{3}) = ((- 4, - 1), (0, 1), (0, - 1), (4, 1))

and the equally spaced spline knot vector

τ = (- 2, - 1, 0, 1, 2, 3, 4)

. With

N_{i, 2}

denoting the normalized degree-two B-spline on this vector, the complete valid parameter interval is

[τ_{2}, τ_{4}] = [0, 2]

and

S (u) = \sum_{i = 0}^{3} D_{i} N_{i, 2} (u) .

On

[0, 1]

the active handles are

D_{0}, D_{1}, D_{2}

, and on

[1, 2]

they are

D_{1}, D_{2}, D_{3}

. Shared handles and a simple interior spline knot are what merge the two pieces into one uniform B-spline. The exterior spline knots support the basis; they do not extend the interval being drawn.

In particular, the requested horizontal pieces and their vertical counterparts are

S_{x} (u) = {- 2 (1 - u)^{2}, 2 (u - 1)^{2}, 0 \leq u \leq 1, 1 \leq u \leq 2, S_{y} (u) = {2 u (1 - u), 2 (u - 1) (u - 2), 0 \leq u \leq 1, 1 \leq u \leq 2.

Figure 1.
Two quadratic pieces, their Bézier handles, the shared uniform B-spline control polygon, and the vertical tangent at the join
. Details.

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