Answers to the Exercises - 3D Math Primer for Graphics and Game Development - page 798

For the second control point, we do the last round using the fraction 0.5.

b 0 = (3.84, 3.64)
b 1 = (4.04, 1.80)
b

′

1 = 0.50 b 0 + 0.50 b 1 = 0.50(3.84, 3.64) + 0.50(4.04, 1.80) = (3.94, 2.72)

For the third control point, we do the last two rounds using the fraction 0.5.

b 0 = (3.60, 4.20)
b 1 = (4.80, 1.40)
b 2 = (1.00, 3.40)
b 0 = 0.50 b 0 + 0.50 b 1 = 0.50(3.60, 4.20) + 0.50(4.80, 1.40) = (4.20, 2.80)
b 1 = 0.50 b 1 + 0.50 b 2 = 0.50(4.80, 1.40) + 0.50(1.00, 3.40) = (2.90, 2.40)
b ′

2 = 0.50 b 0 + 0.50 b 1 = 0.50(4.20, 2.80) + 0.50(2.90, 2.40) = (3.55, 2.60)

For the final control point, we do all the rounds using 0.5 as the fraction.

b 0 = 0.50 b 0 + 0.50 b 1 = 0.50(3, 5) + 0.50(6, 1) = (4.50, 3.00)
b 1 = 0.50 b 1 + 0.50 b 2 = 0.50(6, 1) + 0.50(0, 3) = (3.00, 2.00)
b 2 = 0.50 b 2 + 0.50 b 3 = 0.50(0, 3) + 0.50(5, 5) = (2.50, 4.00)
b 0 = 0.50 b 0 + 0.50 b 1 = 0.50(4.50, 3.00) + 0.50(3.00, 2.00) = (3.75, 2.50)
b 1 = 0.50 b 1 + 0.50 b 2 = 0.50(3.00, 2.00) + 0.50(2.50, 4.00) = (2.75, 3.00)
b ′

3 = 0.50 b 0 + 0.50 b 1 = 0.50(3.75, 2.50) + 0.50(2.75, 3.00) = (3.25, 2.75)

(b) q 0 = b 0 = (3.0, 5.0)
q 1 = b 0 /2 + b 1 /2 = (3, 5)/2 + (6, 1)/2 = (4.5, 3.0)
q 2 = b 0 /4 + b 1 /2 + b 2 /4 = (3, 5)/4 + (6, 1)/2 + (0, 3)/4 = (3.75, 2.5)
q 3 = r 0 = b 0 /8 + 3b 1 /8 + 3b 2 /8 + b 3 /8
= (3, 5)/8 + 3(6, 1)/8 + 3(0, 3)/8 + (5, 5)/8 = (3.25, 2.75)
r 1 = b 1 /4 + b 2 /2 + b 3 /4 = (6, 1)/4 + (0, 3)/2 + (5, 5)/4 = (2.75, 3.0)
r 2 = b 2 /2 + b 3 /2 = (0, 3)/2 + (5, 5)/2 = (2.5, 4.0)
r 3 = b 3 = (5.0, 5.0)
(c) Using Equation (13.38) with n = 3:

1− 0
4

0 = 0

?
?

3
5

3.00
5.00

′

b

4 b −1 +

b 0 = 0 + 1 =

1 = 1

1− 1
4

b 1 = 1
4

3
5

+ 3
4

6
1

5.25
2.00

′

b

4 b 0 +

=