1
b)
(𝐴𝐴,𝐵𝐵,𝐶𝐶) = 𝑎𝑎𝑎𝑎����⃑ × 𝑎𝑎𝑎𝑎����⃑ = �
𝑖𝑖 𝑗𝑗 𝑘𝑘
0 5 0
−15 0 −12
� = (−60, 0, 75)
The equation of the face � afhd is
−60𝑋𝑋 + 75𝑍𝑍 = 0
(𝐴𝐴,𝐵𝐵,𝐶𝐶) = 𝑐𝑐𝑐𝑐���⃑ × 𝑐𝑐𝑐𝑐����⃑ = �
𝑖𝑖 𝑗𝑗 𝑘𝑘
1 0 −12
0 5 0
� = (60, 0, 5)
To solve for 60𝑋𝑋 + 5𝑍𝑍 + 𝐷𝐷 = 0 , we put (10,0,0) into the equation, 𝐷𝐷 = −600
The equation of the face � cgbe is
60𝑋𝑋 + 5𝑍𝑍 − 600 = 0
The system of linear inequalities are
−60𝑋𝑋 + 75𝑍𝑍 < 0 60𝑋𝑋 + 5𝑍𝑍 − 600 < 0 a c e d a f g b h f f h g b a c d e Answers to Assignment 1 Qn 1 a) 2 −12 < 𝑍𝑍 < 0 0 < 𝑌𝑌 < 5 Qn 2 a) 𝑍𝑍 = 4𝑠𝑠𝑖𝑖𝑠𝑠10 𝛼𝛼 𝑋𝑋 = 2𝑐𝑐𝑐𝑐𝑠𝑠10 𝛼𝛼 𝑐𝑐𝑐𝑐𝑠𝑠5𝛽𝛽 𝑌𝑌 = 2cos10 𝛼𝛼 𝑠𝑠𝑖𝑖𝑠𝑠5𝛽𝛽 b) Super-ellipsoid c) makes the sampling more uniform and avoid square/cubic root, which would cause part of the shape to become missing (either reason acceptable) Qn 3 a) i) 𝑍𝑍2 − [� 𝑋𝑋 2 � 2/𝑠𝑠1 + � 𝑌𝑌 2 � 2/𝑠𝑠1 ] = 1 𝑍𝑍 = sec (𝛼𝛼) � 𝑋𝑋 2 � 2/𝑠𝑠1 + � 𝑌𝑌 2 � 2/𝑠𝑠1 = 𝑡𝑡𝑎𝑎𝑠𝑠2𝛼𝛼 1 tan (𝛼𝛼) � 𝑋𝑋 2 � 1/𝑠𝑠1 = 𝑐𝑐𝑐𝑐𝑠𝑠𝛽𝛽 ⇒ 𝑋𝑋 = 2(𝑡𝑡𝑎𝑎𝑠𝑠𝑠𝑠1𝛼𝛼)(𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠1𝛽𝛽) Similarly, 𝑌𝑌 = 2(𝑡𝑡𝑎𝑎𝑠𝑠𝑠𝑠1𝛼𝛼)(𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠1𝛽𝛽) a) ii) Super-hyperboloid (Two-sheeted Super-hyperboloid) b) c g b f 3 c) (𝐴𝐴,𝐵𝐵,𝐶𝐶) = 𝑎𝑎𝑐𝑐����⃑ × 𝑎𝑎𝑎𝑎����⃑ = (9,−9,−45) × (0,2,0) = � 𝑖𝑖 𝑗𝑗 𝑘𝑘 9 −9 −45 0 2 0 � = (90,0, 18) 5𝑋𝑋 + 𝑍𝑍 + 𝐷𝐷 = 0 Put in a(1, -1, -5) gives 𝐷𝐷 = 0. The plane passes through the origin. The plane equation is 5𝑋𝑋 + 𝑍𝑍 = 0 d) By symmetry, PRP = (0, 0, 0) 𝑡𝑡𝑎𝑎𝑠𝑠 � 𝜃𝜃 2 � = 1 5 ⟹ 𝜃𝜃 = 22.61986495 The command is gluPerspective (22.61986495, 1, 5, 50) c b f g c d a e h d e h a d