Graphic Design and Mathematical Concepts (CSC219)
Computer Science - COS
Semester: First Semester
Level: 300
Year: 2017
1
SCHOOL: H.T.T.T.C DEPARTEMENT: CS LECTURER(S): Dr. DADA Jean-Pierre
COURSE CODE: CSC219 COURSE TITLE: Graphic design math. Concepts OPTION: FCS 200
DATE: December ……., 2017 HALL: …….… TIME: 2 hrs NATURE: Exam
Instructions:……… Answer all questions………..
Exercise 1: Let us consider the figure below, where
,
,
. and
are positive real constant numbers. Let us consider the distances
and
. Drawing a line is
for the computer, to decide from pixel (i,i), what pixel
should be illuminated between (i+1,i) or (i+1,i+1).
1. Complete the figure by adding points A, B, M and the line
2. What pixel will be illuminated when
?
3. What pixel will be illuminated when
?
4. Give the expression of
and
in term of
and
5. Give the expression of
and
in term of
and
6. Let us assume that
and
6.1 Give the expression of
in terms of
,
, , and other constants
6.2 Give the relation between
and
6.3 Give the expression of
6.4 Deduce the Bresenham algorithm.
6.5 From the algorithm, write a program using Turbo Pascal or C language
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2
Exercise 2: One defines a Bézier curve with 4 points , , et
4
P
.
a) Demonstrate by using the De Casteljau recursive definition that the coordinates of a point M can be
written as:
( )
−
=
4
3
2
1
0123
0 0 0 1
0 0 3 3-
0 3 6- 3
1 3- 3 1
t ),,(
P
P
P
P
tttzyx
An give the expression of the Bertstein polynomials
)(
2
tB
i
(i=0,1,2).
b) Do again this demonstration by using Bernstein polynomial defined as:
( )
in
ii
n
tt
ini
n
tB
−
−
−
= 1
)!(!
!
)(
. You should take:
=
+
=
n
i
i
i
n
PtBtM
0
1
)()(
b) Demonstrate that
=
=
3
0
1)(
i
i
n
tB
c) Give the linear expression of x(t), y(t) and z(t)
1
P
2
P
3
P
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