Graphic Design and Mathematical Concepts (CSC219)
Computer Science - COS
Semester: Resit
Level: 300
Year: 2016
1
SHOOL: H.T.T.T.C DEPARTEMENT: CS LECTURER: Dr. DADA Jean-Pierre
COURSE CODE: CSC219 COURSE TITLE: Graphic design mathematic concepts
OPTION: FCS DATE: September, 2016 HALL: TIME: 1:30 NATURE: Resit
Instructions:…… Answer all questions…………..
Exercise 1: Demonstrate that for a rotation around OX axis, the homogenous matrix could be
written in the form:
=
1 0 0 0
0 cos sin 0
0 sin- cos 0
0 0 0 1
ox
R
where
is the rotation angle.
Exercise 2: Let us consider Hermitian curves define with two points
1
P
et
2
P
and the
tangents at those points
/
1
P
et
/
2
P
. By taking a degree 3 Hermitian curve, demonstrate that the
coordinates of a point M can be written on the form:
( )
=
/
2
/
1
2
1
123
0 0 0 1
0 1 0 0
1- 2- 3 3-
1 1 2- 2
1 ),,(
P
P
P
P
tttzyx
And give the expression of Hermitian polynomials
)(tH
i
(i=1,2,3,4).
Exercise 3: Let us consider a Bézier curve with three points , et
3
P
. Demonstrate by
using the De Casteljau recursive definition that the coordinates of a point M can be written as:
( )
=
3
2
1
012
0 0 1
0 2 2-
1 2- 1
t ),,(
P
P
P
ttzyx
An give the expression of the Bertstein polynomials
)(
2
tB
i
(i=0,1,2).
Exercise 4: 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 Berstein 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
)!(!
!
)(
. We should take :
=
+
=
n
i
i
i
n
PtBtM
0
1
)()(
c) Demonstrate that
=
=
3
0
1)(
i
i
n
tB
1
P
2
P
1
P
2
P
3
P
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