Assignment 2: Math Review

Question 2: For three points on the plane (x 1 , y 1 ), (x 2 , y 2 ) and (x 3 , y 3 ) show that the determinant
Question of 2: For three points on the plane (x 1 , y 1 ⇥), (x 2 , y 2 ) and (x 3 , y 3 ) show that the determinant
of
x 1 y 1 1
⇧
⌃⇥
⇤
Assignment #2
x 12 y 12 1 ⌅
CS 184: ⇧
Foundations of Computer Graphics page 2 of 3
3 3 ⌃
⇤ x 2 y 2 1 ⌅
Point Value: 15 points
Fall 2011
is proportional to the area of the triangle whose
x 3 y
corners 3 1
are the three points. If these points lie on
Due Date: Sept. 6th, 1:00 pm
Prof. James O’Brien
ais straight proportional line, to what theisarea theof value the triangle of the determinant whose cornersDoes are the thisthree give points. a usefulIftest these to points tell whether lie on
three a straight points line, lie what on a line is theWhy valuedoofyou thethink determinant so Does this give a useful test to tell whether
three Question points lie 3: on The a line equation Why do of you a line think the so plane is ax + by + c = 0. Given two points on the
plane, Question show how3: toThe findequation the values of of a line a, b, inc for the the plane lineis that ax + passes by + c through = 0. Given thosetwo two points. onYou
the
may plane, find show thehow answer to find to question the values 2 useful of a, b, here. c for the line that passes through those two points. You
mayQuestion find the answer 4: Let toequestion 1 = (1, 0, 20), useful e 2 = here. (0, 1, 0), e 3 = (0, 0, 1). Show that if {i, j, k} is {1, 2, 3},
{2, 3, Question 1}, or {3, 1, 4: 2}, Let thene 1 = e i (1, ⇤ e0, j 0), = e k 2 , where = (0, 1, ⇤0), is ethe 3 = cross (0, 0, product. 1). ShowNow thatshow if {i, that j, k} if is {i, {1, j, 2, k} 3}, is
{2, {1, 3, 1}, 2}, {3, or {3, 2, 1} 1, 2}, or {2, then 1, 3}, e i ⇤then e j = e i e⇤ k , ewhere j = e⇤ k . is the cross product. Now show that if {i, j, k} is
{1, 3, 2}, {3, 2, 1} or {2, 1, 3}, then e i ⇤ e j = e k .
Question 5: For four points in space (x 1 , y 1 , z 1 ), (x 2 , y 2 , z 2 ), (x 3 , y 3 , z 3 ), (x 4 , y 4 , z 4 ) show that
the Question determinant5: ofFor four points in space ⇤ (x
x 1 , y
1 x 1 , z
2 x 1 ), (x
3 x 2 , 2 , z 2 ), (x 3 , y 3 , z 3 ), (x 4 , y 4 , z 4 ) show that
the Question determinant5: ofFor four points in space 4
⇤ (x 1 , y 1 , z 1 ), (x 2 , ⌅y 2 , z 2 ), (x 3 , y 3 , z 3 ), (x 4 , y 4 , z 4 ) show that
the Question determinant5: ofFor four points in space y y y y ⇤ x(x 1 1 x, y 21 , zx 13 ), (x x 42
, ⌅y 2 , z 2 ), (x 3 , y 3 , z 3 ), (x 4 , y 4 , z 4 ) show that
⇧
the determinant of
z z z z ⌃
y y y y ⇤
x 1 x 2 x 3 x 4 ⌅
⇧
xy z 1 1 1
1 xy z 2 xy z 3 xy z 4
⌃
is proportional to the volume of the tetrahedron
⇧
yz 1 yz 1 2 whose
yz 1 3 yz 1
4 corners
⌃
⌥
are the four points. If these points
lie on a plane, what is the value of the
⇧
determinant
z 1 z1 2 z1 3 z
Does
1 4 ⌃
is proportional to the volume of the tetrahedron this give a useful test to tell whether
four given points lie on a plane Why do you
1
think
1
whose
1
so
1
corners are the four points. If these points
lie is proportional on a plane, what to theisvolume value of the of the tetrahedron determinant whoseDoes corners thisare give thea four useful points. test to If tell these whether points
four lie is proportional Question given a plane, points what 6: tolie the The on isvolume equation the a plane value of of Why ofathe tetrahedron plane dodeterminant you space think whose is so ax Does corners + by this + are cz give the + d afour = useful 0. points. Given test to three If tell these points whether points
four lie space, ongiven ashow plane, points howwhat to liefind on is athe plane value valuesWhy of ofthe a, do b, determinant you c, d think for theso
plane Doesthat thispasses give athrough useful test those to three tell whether points.
four
Question 6: The equation of a plane in space is ax + by + cz + d = 0. Given three points in
You
space, Question may givenfind points
show how
the 6:
lie
to
answer The
on
findequation a plane
the
to
values
question of
Why
of a a, plane 4do useful you
b, c, ind space here. think
for theis so
plane ax + that by + passes cz + d through = 0. Given those three three points points.
You space, Question may show findhow the 7: 6: toanswer Let The findp equation 0
the , to p 1
values question , p of 2 be of athree a, plane 4 useful b, c, distinct ind space here. for the points isplane ax in + that by space. + passes cz Now + d through = consider 0. Given those thethree cross three product points.
n You space, = (p may show 0
find phow 1 ) the ⇥ (p toanswer find 0 the 2 ). toWhat values question does of a, 4 this useful b, c, vector d here. for the mean plane geometrically that passes through Let p be those any three point points. on the
YouQuestion 7: Let p 0 , p 1 , p 2 be three distinct points in space. Now consider the cross product
n
plane
= Question
may formed find by the
(p
p 0 , p 1 and p 2 ; what is the geometric relationship between n and p p 0 (look at
0 p 1 ) ⇥ 7:
answer
(pLet 0
p 0 2
,
to
). p
question
What 1 , p 2 be does three
4 useful
thisdistinct here.
vector points mean geometrically in space. Now consider Let p bethe anycross pointproduct
on the
n the
plane = Question dot (pformed 0 product) p 1 ) by ⇥ 7: (p pLet Show that, if p = (x, y, z) the equation of the plane must be n.(p ) = 0.
0
0 , pp 0 1 and 2 ,). pWhat 1 p, p 2 ; 2 what be does three isthis the distinct vector geometric points meanrelationship geometrically in space. Now between consider Let np and bethe any p cross point p 0 (look product on the at
n
the plane = Question dot
(pformed 0 product)
p 1 ) by ⇥ 8: (p p Suppose Show 0 , p 1 pand 2 that,
). that What p 2 if ; what Ap does
= is(x, ais this square y, the z)
vector geometric the matrix, equation
meanand relationship geometrically
of itsthe inverse plane between and must
Let transpose np beand n.(p
any p exist, point p 0 )(look and =
on
0.
the are at
equal; the plane dot formed
Question
aproduct) matrix by
8:
with p 0 Show , p
Suppose
these 1 and that, properties p 2 if ; what p = (x,
that A is a
is is
square
called y, the z) geometric the an equation
matrix,
orthonormal relationship of the
and its inverse
matrix. plane between
and
Show mustnbe transpose
that, and n.(p for p
exist,
any p 0 )(look angle = 0. at
and are
,
the matrix M( ), defined by the array below, is orthonormal.
equal; Question
dot product)
a matrix8: with Suppose
Show that,
these properties that
if
A
p =
is
(x,
aissquare y, z) the
called an matrix,
equation
orthonormal and
of
its
the
inverse
plane
matrix. and
must
Show transpose
be n.(p
that, forexist, p 0 )
any angle and
= 0.
are,
the equal; Question
matrix a matrix M(
8:
), with defined
Suppose theseby properties that
the
A
array
is ais below,
square called is an matrix,
orthonormal.
and ⇥ its inverse matrix. and Show transpose that, forexist, any angle and are,
the equal; matrix a matrix M( ), with defined theseby properties the arrayisbelow, called cosis anorthonormal.
sin matrix. Show that, for any angle ,
sin cos
⇥
the matrix M( ), defined by the array below, cosis orthonormal. sin ⇥
cos
sin cos
⇥
Show also that M( 1 + 2 ) = M( 1 )M( 2 ); cos use sin this cos sinto argue that the inverse of M( ) is M( ).
Finally, for the point p = (0, 1), what geometrical figure is given by taking the points given by
Show also that M( 1 + 2 ) = M( 1 )M( 2 ); use sin this cos to argue that the inverse of M( ) is M( ).
M( )p for every possible value of
Finally,
Show also
for
that
the
M(
point 1 +
p = 2 )
(0,
= M(
1), what 1 )M(
geometrical 2 ); use this
figure
to argue
is given
that the
by taking
inverse
the
of M(
points
) is
given
M(
by
).
M(
Finally, Show Question also
)p for
forthat every
the9: M( point
possible You 1 + p are = 2 )
value given (0, = M( 1),
of four what 1 )M(
vectors geometrical 2 ); use in the thisplane, figure to argue xis 1
given and thatxthe by 2 , btaking inverse 1 and bthe of 2 , and M( points you ) isgiven M( are told by ).
that M( Finally, )p there foris every the a matrix point possible p M= value such (0, 1), that of what Mxgeometrical 1 = b 1 and Mx figure 2 = isb given 2 . Nowby iftaking you arethe given points a vector given xby
3 ,
Question 9: You are given four vectors in the plane, how M( )p do for youevery determine possible Mxvalue 3 (hint: of show how to find out what 1 and M is 2 from 1 and the data, 2 and you are told
and then apply
that
Question
there is matrix
9: You are
such
given
that
four
Mx
vectors
it to x 3 ).
1 in
1 and
the plane,
Mx 2 x 1 and
2 Now
x 2 ,
if
b 1
you
and
are
b 2
given
, and you vector
are told
3 how
thatQuestion do
there
you
is
determine
a matrix 9: You
Mx
Maresuch given
3 (hint:
that four
show
Mx vectors 1
how
= bin 1
to
and the plane,
find
Mx
out 2
what
= x 1 b 2 and . Now x
is 2 ,
from
if b 1 you and
the
are b
data, 2 given , and you
and
a
then
vector are told
apply
x 3 ,
it
how that
toQuestion do there 3 ).
youisdetermine a matrix 10: Write MxM such down 3 (hint: that the parametric show Mx 1 how = b 1 toequation and findMx out 2 of what = the b 2 . points MNow is from on if you athe line are data, that given and passes athen vector through apply x 3 ,
it how two topoints do x 3 ). youindetermine space, p Mx 1 and 3 p(hint: 2 ; nowshow writehow down to find the parametric out what Mequation is from the of the data, points and then on a apply plane
it Question 10: Write down the parametric equation of the points on line that passes through
that to contains x these two points and a third point, p 3 . Show how to use these parametric equations
two
Question 3 ).
points in space,
10: Write and some inequalities 1 and
down on the parameters 2 the
now
parametric
write down
equation
the parametric
of the points
equation
on a
of
line
the
that
points
passes
on
through
plane
twoQuestion to specify (a) all the line that lie between
that
points
contains these
space, 10: Write
two
p 1 and down
points
p 2
and
; the nowparametric write
third point,
downequation the p 1 and p 2 and (b) all the points on the triangle whose 3 parametric of the points
Show how to
equation a
use these
of line the that
parametric
points passes onthrough
equations
a plane
two thatpoints contains these space, two p points and a third point, p vertices are p 1 , p 2 and p 3 .
and some inequalities 1 and p
on the parameters 2 ; now write down the
to specify 3 . parametric Show how to equation use these of the parametric points on equations a plane
(a) all the points on the line that lie between
that and 1 and Question some contains inequalities these 2 and (b) 11: two
all Mon the ispoints athe points symmetric parameters and a third
on thereal topoint, triangle 2x2 specify matrix. p (a)
whose 3 . Show
vertices It allhas the howtwo points to
areeigenvalues, useon these parametric line 1 2 and ⇥ 0
that
3 and ⇥lieequations
1 . between
pand 1 and some p 2 inequalities and (b) all the on the points parameters on the triangle to specify whose (a) vertices all the points are p 1 , on p 2 the andline p 3 . that lie between
p 1 and Question 11: is symmetric real 2x2 matrix. It has two eigenvalues,
• If pboth eigenvalues are positive, show that, for any 2D vector x, 0 ⌅ x T Mx ⌅ 0 and (max(⇥ 1 Question 2 and (b) all the points on the triangle whose vertices are p
11: M is a symmetric real 2x2 matrix. It has two eigenvalues, 1 , p 2 and p
⇥ 3 . 0 , ⇥ 1 ))x T 0 and ⇥ 1 . x.
Question 11: M is a symmetric real 2x2 matrix. It has two eigenvalues, ⇥ 0 and ⇥ 1 .