change. The error introduced by this
approximation at each step decreases as the
rotation angle decreases. But even with small
rotat~on angles, the accumulated
error over many steps can become quite
large. We can control the accumulated
error by estimatin
coordinates are translated. A more efficient
approxh would be to combine the
transformations so that the final coordinate
pnsitions are obtained directly from
the initial coordinates, thereby eliminating the
calculation of intermediate coordinate
values.
concatenation, the md~c:dual transformations
would bt applied one at a time
and the number of calnrlations could be
significantly rncrrascd. Ail cff~c~enint:
plementation for the trar~sformatiun
operations, therefor*, is to formulate
transformation
matric
which transforms coordinate positions as
Any real number can be assigned to the
shear parameter sh,. A coordinate position
(.u, y) is then shifted horizontally by an
amount proportional to its distance (y
value) from the x axis (y = 0). Setting sh, to 2,
the explicit calculations for the transformed
coordinates are
Composite Transformations
Chapter 5
Twdlimensional Geometric
Transformations
Final
- - - - - - . . . . . .- . . .- - - - - - - - - - - - -. Figure 5-13
Reversing the order in which a sequence o
We can rewrite these scaling transformations
to separate. the mdtiplicative and
additive terms:
where the additive terms r,(l - s,) and y,(l - s,)
are constant for all points in the
object.
Including coordinat~?fo~r a hxed point in the
scalin~:e quations
of a specified translation vector to move an
object from one position to another.
Similar methods are used to translate curved
objects. To change the position
of a circle or ellipse, we translate the center
coordinates and redraw the figure in
the new loc
the elements of u' to the first row of the
rotation matrix and the elements of v' to
the second row. This can be a convenient
method for obtaining the transfonnation
matrix for rotation within a local (or "object")
coordinate system when we
know the final
General Composite Transformations and
Computational Efficiency
A general two-dimensional transformation,
representing a combination of translations,
rotations, and scalings, can be expressed as
The four elements rs, are the multiplicative
rotation-scaling
about the coordinate origin, we can generate
rotations about any selected pivot
point (x, y,) by performing the following
sequence of translate-rotatetranslate
operations:
1. Translate the object so that the pivot-point
position is moved to the coordinate
X ' = X, + (a - x,) cos V - (y - y,) sin 0
y = , + (1 - v,) sin H + (y - y,) cos B (5-9)
These general rotation equations differ from
Eqs. 5-6 by the inclusion of additive
terms, as well as the multiplicative factors on
the coordinate values. Thus, the
ma
to the origin of the xy system. A unit vector in
the y' direction can then be
obtained as
And we obtain the unit vector u along the x'
axis by rotating v 90" clockwise:
-4 Figure 5-27
Position of the reference frames
shown in Fig. 5-26 after translating
t
and I/, such as Iix, y) = 0, become
homogeneous tytations in the three
parameters
x, y, and 11. 'This just means that if each of
thtl three parameters is replaced
by any value n times that parameter, the
value 7; c,ln he factored out of the equations.
Exp
plan, with multiple copies of the chairs and
other items in different positions. In
other applications, we may simply want to
reorient the coordinate reference for
displaying a scene. Relationships between
Cartesian reference systems and some
c%mrnon non-
4-20. Write a procedure to fill the interior oi a
given ellipse with a specified pattern.
4-21. Write a procedure to implement the
serPa:ternR.epresentation function.
C1uplt.r 4 4-22. Ddine and implerne~l,t i
procedure for rhanging the sizr r ~ atn
exlstl
cussed in the previous sections can be
reformulatej so that such transformation
iOmo~eneous Coordinates
sequences can be efficiently processed.
We have seen in Section 5-1 that each of the
basic transformations can be expressed
in the general matrix form
s, am to be applied in We rotate the diagonal
onto they axis and double its length with the
transformaorthogonal
directions
defined by the angular
tion parameters 8 = 45O, s, = 1, and s2 = 2.
displacement 6. In Eq. 535, we assumed that
scaling was to be p
Essay 1 for English B: Arguing a Position
Due: Wednesday 1/25
The Topic
In Essay 1 you will argue a reasonable position over a debatable issue. You must learn more
about the issue through research, take a distinct and readily identifiable position, presen
Alternative Topics (choose one)
1. Is college worth it?
Do the costs outweigh the benefits of going to college? Write an argumentative-persuasive essay for
whether or not college is valuable today. Do not take for granted that it is or not; rather, prove
Essay 2 for English 1B: Proposing a Solution
The Topic
In this essay you will be defining a problem faced by your community and then proposing a
solution. Choose any aspect of your life such as academics, school clubs, workplaces,
neighborhoods, sports te
MATH 1A TENTATIVE SPRING COURSE CALENDAR (MW)
MONDAYWEDNESDAY CLASS
Week
Monday
April 4
MOD 1 CP DUE
Introductions
Website overview
1.1 (Boelkins book)
Review Problems
Assigned (optional but
recommended)
Comments:
Tuesday
4/5
Wednesday
4/6
MOD 2 CP
ACTIVE CALCULUS
ACTIVITIES WORKBOOK CHAPTERS 1-4 2015 Edition
Matthew Boelkins
David Austin
Steven Schlicker
2
Active Calculus
Activities Workbook, chapters 1-4
2015 edition
Matt Boelkins, David Austin, Steven Schlicker
Department of Mathematics
Grand Val
STAPLE
PREPARATORY WORK (CP) COVER PAGE
(For credit, a copy of this cover sheet must be attached by staple to each (CP) work submitted.)
1. NAME (First, Last) _
2. NAME (First, Last) _
3. NAME (First,
4.13. Define and implement a function for
setting the w~dttoi f displayed ellipses
4 14. Write a routlne to display a bar graph in
anv specfled screen area. Input is to include
Ihc data set, labeling for the coordmate ,ixes,
and th,' coordinates for the s