1
Answers to Quiz 1
Qn 1
a)
𝑋2
𝑎2
+
𝑌2
𝑎2
= 𝑠𝑒𝑐2𝛼
𝑋 = sec 𝛼 cos 𝛽
𝑌 = sec 𝛼 sin 𝛽
𝑍 = 𝑏 tan𝛼
𝛽 ∈ |0, 2𝜋)
𝛼 𝜖 (−
𝜋
2
,
𝜋
2
)
Paper 1: 𝑎 = 2, 𝑏 = 4
Paper 2: 𝑎 = 3, 𝑏 = 6
b) Any reasonable answer, for example,
𝑋 = 𝑎 𝑠𝑒𝑐𝑠1 𝛼 𝑐𝑜𝑠𝑠2𝛽
𝑌 = 𝑎 𝑠𝑒𝑐𝑠1 𝛼 𝑠𝑖𝑛𝑠2𝛽
𝑍 = 𝑏 𝑡𝑎𝑛𝑠1 𝛼
or
𝑋 = 𝑎 𝑐𝑜𝑠−𝑠1𝛼 𝑐𝑜𝑠𝑠2𝛽
𝑌 = 𝑎 𝑐𝑜𝑠−𝑠1𝛼 𝑠𝑖𝑛𝑠2𝛽
𝑍 = 𝑏 𝑡𝑎𝑛𝑠1 𝛼
2
Qn 2
𝑂 = (0, 0, 0)
H height W Width
tan 30𝑜 =
𝐻
2√3
⟹ 𝐻 = 2 = 𝑊
𝑎 = (1,−1,−√3) 𝑏 = (1, 1, − √3)
𝑂𝑎⃑⃑⃑⃑ ⃑ × 𝑂𝑏⃑⃑⃑⃑ ⃑ = |
𝑖 𝑗 𝑘
1 −1 −√3
1 1 −√3
| = (2√3, 0,2)
The set of inequalities
√3𝑋 + 𝑍 < 0 −√3𝑋 + 𝑍 < 0 √3𝑌 + 𝑍 < 0 −√3𝑌 + 𝑍 < 0 −100 < 𝑍 < −√3 a b c d e f g h O X Y Z 3 Qn 3 𝑀𝑃←𝐶𝑇 = [𝑇(10, 10, 10)𝑅𝑧(−135 𝑜)𝑅𝑥(100 𝑜)𝑅𝑧(30 𝑜)]−1 = 𝑅𝑧(−30 𝑜)𝑅𝑥(−100 𝑜)𝑅𝑧(135 𝑜)𝑇(−𝑎,−𝑎,−𝑎) glRotatef ( -30, 0, 0, 1); glRotatef (-100, 1, 0, 0); glRotatef ( 135, 0, 0, 1); glTranslatef (-a, -a, -a); Paper 1: 𝑎 = 10 Paper 2: 𝑎 = 20 Qn 4 VRP = (0, 30, 30) VPN = (0,30, 30) VUP = (0, 1, 0) 𝑍𝑉𝐶 = |VPN| = (0, 1 √2⁄ , 1 √2⁄ ) VUP × VPN = | 𝑖 𝑗 𝑘 0 1 0 0 1 √2⁄ 1 √2⁄ | = (1 √2⁄ , 0, 0) 𝑋𝑉𝐶 = |VUP ×VPN| = (1, 0, 0) 𝑌𝑉𝐶 = 𝑍𝑉𝐶 × 𝑋𝑉𝐶 = | 𝑖 𝑗 𝑘 0 1/√2 1/√2 1 0 0 | = (0, 1 √2⁄ ,−1 √2⁄ ) 𝑀𝑊𝐶←𝐶𝐶 = ( 1 0 0 0 0 1/√2 1 √2⁄ 30 0 −1 √2⁄ 1 √2⁄ 30 0 0 0 1 ) 4 ( 1 0 0 0 0 1/√2 1 √2⁄ 30 0 −1 √2⁄ 1 √2⁄ 30 0 0 0 1 )( 0 1 −1 1 ) = ( 0 30 30 − √2 1 ) The world coordinates are (0, 30, 30 − √2)