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a0
P0
B a1 C
BC
B C = A B P1 C
@ a2 A
@ P2 A
a3
P3 c Given that the spline patch is one of several making up a more complex curve,
and the points are deﬁned as
✓◆
✓◆
✓◆
✓◆ a2 , a1 and a0 are vector constants that deﬁne the shape of the curve.
0
P i +1 = 3 a 3 + 2 a 2 + a 1
a Marks: that the patch is to be drawn between two points Pi and Pi+1 and the
Given
0
0
gradients at the ends are to be Pi and Pi+1 respectively, write down four
00
equations connecting Pi Pi+1 Pi Pi+1 and a3 a2 a1 a0 .
b Solve the above equations to ﬁnd the values in matrix A in the equation:
01
01
Pi = 0a0
a)!
a
P0
0
BC
PiBa1a1
=C
B C = A B P1 C
@a A
@P A
Pi+1 = 2a3 + a2 + a12 + a0
a
0
Pi+1 = 33 a3 + 2 a2 P3 a1
+ 4 Splines" b)Marks: equations in part a can be written
! The
0 10 1 0 1
1000
a0
P0
B to 1 the C Ba1 C B A in
b Solve the above equations 0 ﬁnd 0 0values in matrix P1 C the equation:
B
CB C=B C
@1 1 1 1 A @ a2 A 1 @ P2 A
1
0
0
0 1 a0 3
2
a3 P0
P3
B a1 C
B P1 C
B C = AB C
@ give
and the above matrix needs to@a2 A
be inverted to P2 A the matrix A:
a3
P
0
11 03
1
1000
1
0
0
0
The equations in partB0 1be 0 0C
a can written
B
1
0
0C
B
C =B 0
C
A = @0
A1 0 @ 3 0 2 1 3
1
1A
1111
1000
a
P
0 1 2 3 C B 0 C 2 B 10 C 2
1
B0 1 0 0
a1 C B P1 C
B
CB
@1 1 1 1A @ a2 A = @ P2 A
Marks:
0123
a3
P3 4 6 00
equations connecting Pi Pi+1 Pi Pi+1 and a3 a2 a1 a0 . b c Solve the above equations to ﬁnd the values in matrix A in the equation:
01
01
Splines"
a0
P0
B a1 C
BC
B C = A B P1 C
@ a2 A
@ P2 A
a3
P3 Given that the spline patch is one of several making up a more complex curve,
and the points are deﬁned as
✓◆
✓◆
✓◆
✓◆
0
1
5
4
Pi 1 =
Pi =
P i +1 =
P i +2 =
0
3
3
2
0
0
calculate the values of the gradients Pi and Pi+1 using the central difference
approximation and hence calculate the values of a3 a2 a1 a0 . d Calculate the coordinate and the gradient direction at the midpoint of the spline
patch. The four parts carry, respectively, 20%, 30%, 30%, and 20% of the marks. calculate the values of the gradients Pi and Pi+1 using the central difference
approximation and hence calculate the values of a3 a2 a1 a0 .
1
0
Pi =
2 Splines" ✓ ◆
5
3 ✓◆
✓
◆
0
5/2
=
0
3/2 1
=
2 ✓ ◆
4
2 ✓◆
✓
◆
1
3/2
=
3
1/2 0
P i +1 c)! Using the entries in the matrix A from part b, we have
✓◆
1
a0 =
3
a1 = ✓ 5/2
3/2 ◆ ✓◆
✓
◆
✓◆
1
5/2
5
a2 = 3
2
+3
3
3/2
3
✓
◆
11/2
=
5/2
✓◆ ✓
◆
1
5/2
a3 = 2
+
3
3/2
✓◆
4
=
1
Marks: ✓ 3/2
1/2 ◆ ✓◆ ✓
◆
5
5/2
2
+
3
3/2 6 Department of the coordinate and the gradient direction at the midpoint of the spline
Conﬁdential
d Calculate Computing Examinations – 2011  2012 Session patch. MODEL ANSWERS and MARKING SCHEME Splines"Putting µ = 2 into
First Examiner: Paul Aljabar
d)! Paper: C317  Graphics Second Examiner: Daniel Rueckert
Question: 2 Page 7 of 16 P ( µ ) = a3 µ3 + a2 µ2 + a1 µ + a0 gives the coordinate as ✓◆
1
1
1
1
P
= a3 + a2 + a1 + a0
2
8
4
2
and substituting the values found for a3 a2 a1 a0 gives
✓◆
✓◆
✓
◆
✓...
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 Spring '14

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