Graphic Design and Mathematical Concepts (CSC219)
Computer Science - COS
Semester: First Semester
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 mathematical concepts
OPTION: FCS DATE: February, 2016 HALL: TIME: 2:00 NATURE: Exam
Instructions:…… Answer all questions…………..
Exercise 1: The mouse (considered as a material point) moves in the basis
from a point O(0,0,0) to another point with the vector
. One can
write
′
.
1) Give the expression of in function of
2) Demonstrate that this translation is given by the expression
where
the matrix is giving by the expression
to complete.
3) The same mouse continues to the point ’’’’ by a rotation with the angle
(with ) and the radius as shown in the figure below:
3.1) Give the expression of if possible, in terms of R and or
3.2) Give also the expression of ′′′ if possible, in terms of R and or
3.3) Give the expression of
′
′
′
in terms of
and or or
3.4) Demonstrate that the matrix for the rotation can be given by:
3.5) Then movement of the mouse from the point O(0,0,0) to the point ’’’’
can be represented by the matrix
.
3.5.1) Give the relation between
,
and
3.5.2) Give the expression of
3.5.2) Conclude
Exercise 2: Demonstrate that for a scaling, the homogenous matrix [H] with 0(a,b,c)
as the center could be written in the form:
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2
=
1 0 0 0
k)-c(1k 0 0
k)-b(1 0k 0
k)-a(1 0 0 k
H
where k is the ratio.
Exercise 3: Breseham algorithms
Let us consider the points
,
,
where , are positive real constant numbers. Let us consider the distances
and
. Drawing a line for the computer, is to decide from pixel (i,i), what pixel
should be illuminated between (i+1,i) or (i+1,i+1).
3.1) What pixel will be illuminated when
?
3.2) What pixel will be illuminated when
?
3.3) Give the expression of
and
in term of
and
3.4) Give the expression of
and
in term of
and
3.5) Let us assume that
and
3.5.1) Demonstrate that
and give the expression of C
3.5.2) Demonstrate that
)(22
11 iiii
yyxyPP −−+=
++
3.5.3) Demonstrate that
3.5.4) What the pixel to illuminate when
? Give the expression of
in function of
3.5.5) What the pixel to illuminate when
? Give the expression of
in function of
3.5.6) Deduce the Bresenham algorithm.
3.5.7) Write an appropriate C program for this algorithm
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