Raccolta temi 2015

Pagina 1

In this file you will find a collection of the exams given in the year 2015. Most of the topics are covered also this year, but there are some changes. Moreover, not all the solutions are available, and some of the available solutions are not correct. The purpose of this document is to give the idea to the student what to expect during the exam, and to make her/him think about the topic studied during the course. Please use this material wisely, keeping these assumptions in mind! DRAFT ② +1/2/59+ ② Exercise 1 At r i a n g l eh a st h ef o l l o w i n gv e r t i c e si nn o r m a l i z e ds c r e e ncoordinates: A(0.5, 0.5, 0.5) B(0.5, −0.5, 0.5) C(0, −0.5, 2) It is generated by a parallel projection, with the projectio nr a yo r i e n t e da l o n gv e c t o r(0, 0, −1). Vertex are ordered in clo ckwise direction Question 1 The normal vector to the face is: A |0.5, 0.5, 0.5| × |0.5, −0.5, 0.5| = |0.5, 0, −0.5| B (|0, −0.5, 2| − |0.5, −0.5, 0.5|) × (|0.5, −0.5, 0.5| − |0.5, 0.5, 0.5|)= |1.5, 0, 0.5| C (|0.5, 0.5, 0.5| − |0.5, −0.5, 0.5|) × (|0, −0.5, 2| − |0.5, −0.5, 0.5|)= | − 1.5, 0, −0.5| D |0.5, 0.5, 0.5| × |0, −0.5, 2| = |1, −0.75, −0.25| SOLUTION: n =( C − B)x(B − A)=( |0, −0.5, 2| − |0.5, −0.5, 0.5|) × (|0.5, −0.5, 0.5| − |0.5, 0.5, 0.5|)= = | − 0.5, 0, 1.5| × |0, −1, 0| = |1.5, 0, 0.5| Question 2 Is the face visible? A |1.5, 0, 0.5|·| 0, 0, −1| = −0.5 < 0 the face is visible B |1, −0.75, −0.25|·| 0, 0, −1| =0 .25 > 0 the face is visible C | − 1.5, 0, −0.5|·| 0, 0, −1| =0 .5 > 0 the face is visible D |0.5, 0, −0.5|·| 0, 0, −1| =0 .5 > 0 the face is not visible SOLUTION: n · p = |1.5, 0, 0.5|·| 0, 0, −1| = −0.5 < 0 Consider the far plane identified by vector |0, 0, −1, 1|.N o w a n s w e r t h ef o l l o w i n g q u e s t i o n s : Question 3 Is vertex A inside the…