Mathematics 308 — midterm examination solutions
1. Let
If
is considered as the matrix of a projective transformation, what quadrilateral does it take the unit square
with corners at
and
to? Draw the figure below (
grid), and indicate exact coordinates.
% &' ( *)
!"$#
&* *)
+/( ,. 10 ' 2 %,+, 10 *) %,,+ *3 +, 2. A certain affine function
is equal to
between these two points is it equal to ?
%,+-.
of the way from
to
at
and at
. At what point on the line segment
, which is
3. Write a complete PostScript program that produces the following picture:
y = x2
464,458E&7-=.9-F,:.G&;-<I<-=?HJG->K'@-E:'=A"LNBDC MOCPCRQTS'CPUV
W SPSYX,Z&[?G]\^'[X-@
WQ AYW <APF,<I_`HTb2\,a.\.[-@^'X&@ ^HcG@.d]He<'X'f
g.g.Lhkl:IQTC,QiCm<,<-@'j@'j
g<'LnAYkn<IHTbP<-@'j
GKI:I@.\.Qod'[.Bp_b&Q[@ CX'f q]=b&@X-=
Points will be given for the simplicity and readability of your code. Put in only what is necessary.
1
rorospsprurPtwr~]v&xc|y€&zz{&{-||
} rosprnrv&xcyz{&|
ttƒ}ros‚s‚rr~Iv&|xcy€&zz{&{&||
}} rrP}ros‚s‚r„r~]v|xcy€&z.z.{&{&||
rrP}ros‚s‚rr~Iv&|xcy€&zz{&{&||
ˆ {,†&†-z |… ‡&{&z|†&z
yz.‰'ŠŠ ‹{'Œ Š~ v~]|.€&z{&|
w‘2Ž.’  Ž. 
“ PwŠ Ž' ‹ŠŽ,~ŽYŽvnz'”vxJyz{&|
w• †-Ž.z.Šz,Ž.‹ { 
{,†-|‡&z
t"“.–zs‚{,z,rc”-tv.€&|.…y-z—?{]{ ‹-xv'—.z'‹ ”-}?|?˜ y-|,{ v Ž ”™xcy Ž ”-|?y-{
t"žŸns›š' œPt"sDŒ~]|.|.‰€&z{&|
tnž¥¤ t¢s›t]Œ£Y¡m|?†?‰ ~]… |€&z.{&|
¡m
4. What does the following fragment of PostScript code draw? Assume units in inches, origin centred. Show it
below. Include calculations so I can see what you’ve done.
š ’ y z.‰,Š&‹{'Œ
¤t} r mt¤ } †† rvv~]xcxcyy|.€&z.z.{&{&z{&|| |
}—v'rY| r… z.v Š xc‹y&{'zŒ {&|
£&t"… {,s›†& †&|.|‡&{&z Š‹ {-z .— ‹-vz
• †-z.Šz,‹{ Ž
Indicate features of the drawing carefully.
2
5. On the next few pages, draw a sequence of diagrams, with as little text as possible, explaining how to construct
a regular pentagon inside a given circle with ruler and compass. Explain as you do this why the construction
works.
The easiest construction is the one attributed to H. M. Taylor at the end of the selection I handed out.
3