Graphic Design and Mathematical Concepts (CSC219)
Computer Science - COS
Semester: First Semester
Level: 300
Year: 2015
1
HTTTC Bambili Computer Science Department
Continuous Assessment
COURSE: R graphic design mathematical concepts
Level 200 FCS
2014-2015 academic year
Exercise 1: Let us consider the scheme 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).
a) What pixel will be illuminated when
?
b) What pixel will be illuminated when
?
c) Give the expression of
and
in term of
and
d) Give the expression of
and
in term of
and
e) Let us assume that
and
e.1 Demonstrate that
and give the expression of C
e.2 Demonstrate that
)(22
11 iiii
yyxyPP −−+=
++
e.3 Demonstrate that
e.4 Deduce the Bresenham algorithm.
Exercise 2: One defines a Bézier curve with 4 points
1
P
,
2
P
,
3
P
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)
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