Section 5 - Perspective

Question 1

define a projective plane

Answer 1

a projective plane is a:

• set P of objects called points
• set L of things called lines
• symmetric relation “on” between lines and points s.t.:
  1. Any two points are on a unique line
  2. Any two lines are on a unique point
  3. There exist four points, no three of which are on the same line

Question 2

“on” is sometimes called a …

Answer 2

incidence relation

Question 3

what is the simplest projective plane? what is its order?

Answer 3

the fano plane (order 7)

Question 4

how do we define the real projective plane (ℝℙ²)?

Answer 4

Take “points” to be lines through O in ℝ³, “lines” to be planes through O in ℝ³, and the “plane” to be the set of all lines through O in ℝ³. Then;

1. Any two “points” are contained in a unique “line because two given lines through O lie in a unique plane through O.
2. Any two “lines” contain a unique “point” because any two planes through O meet in a unique line through O.
3. There are four different “points,” no three of which are in a “line”: for example, the lines from O to the four points (1,0,0), (0,1,0), (0,0,1), and (1,1,1), because no three of these lines lie in the same plane through O.

Question 5

a line through O (a pt in ℝℙ²) is the set of all pts….

Answer 5

(ɑx,ɑy,ɑz), where (x,y,z)≠0 & ɑ∈ℝ

Question 6

a plane through O (a line in ℝℙ²) has equation….

Answer 6

ax+by+cz=0, where a,b,c∈ℝ and aren’t all 0

Question 7

if (x₁,y₁,z₁) & (x₂,y₂,z₂) determine different lines through O, how do we find the plane through O that contains them? (ℝℙ²)

Answer 7

solve:
ax₁+by₁+cz₁=0
ax₂+by₂+cz₂=0 , for a,b,c

Question 8

define a 3-dimensional projective space

Answer 8

a 3-dimensional projective space is a:

• set P of objects called points
• set L of things called lines
• set M of things called planes
• symmetric relation “on” between lines, points, and planes s.t.:
  1. Any two points are on a unique line
  2. Any three points not on a unique line are on a unique plane
  3. Any two lines are on a unique point
  4. Any three planes not on the same line are on a unique point

Question 9

does ℝ⁴ satisfy the projective space axioms?

Answer 9

yes, define:

• points as lines through O in ℝ⁴
• lines as planes through O in ℝ⁴
• planes as 3-d subspaces of ℝ⁴

Question 10

Suppose we have two lines l₁ & l₂ and a point P. How do we define a mapping f: l₁ → l₂∪{∞}?

Answer 10

by the point where the line through P & x (on l₁) meets l₂

Question 11

in terms of mappings what does k represent?

Answer 11

by Thales’ thm the distance between the images of equally spaced pts on l₁ is “magnified” by some constant k

Question 12

if l₁ & l₂ are // and P is at ∞, what is k?

Answer 12

k = 1 (lines through l₁ & l₂ are //)

Question 13

if l₁ & l₂ are // and l₂ is between P & l₁, what is k? (P≠∞)

Answer 13

k < 1

Question 14

if l₁ & l₂ are // and P is in the middle of l₁ & l₂, what is k?

Answer 14

k = 1

Question 15

if l₁ & l₂ aren’t //, is distance preserved during projection?

Answer 15

no

Question 16

when projecting if l₁//x-axis & l₂//y-axis, what is the slope of the line through P and the point on l₁ at x=n?

Answer 16

l₁/n

Question 17

when projecting if l₁//x-axis & l₂//y-axis, what is the equation of the line through P and the point on l₁ at x=n?

Answer 17

y = (l₁/n)x + k

Question 18

when projecting if l₁//x-axis & l₂//y-axis, the mapping P send x on l₁ to what pt on l₂?

Answer 18

l₁l₂/x

Question 19

what are the generating transformations?

Answer 19

any composition of reflections is a composition of the functions:
x + l
kx
1/x

Question 20

any composition of the functions x+l, kx (k≠0), 1/x is a function of what form?

Answer 20

f(x) = (ax+b)/(cx+d), where ad-bc ≠ 0

Question 21

what is a linear fractional?

Answer 21

a function f: ℝ∪{∞} → ℝ∪{∞} of the form f(x) = (ax+b)/(cx+d), where ad-bc ≠ 0

Question 22

why do we specify ad-bc ≠ 0 for a linear fractional?

Answer 22

it guarantees the function isn’t constant (i.e. all pts don’t map to the same pt)

Question 23

how do we realise f(x) = (ax+b)/(cx+d), where ad-bc ≠ 0, using generating transformations? (if c≠0)

Answer 23

• given x
• mult. by c → cx
• add d → cx + d
• mult by c → c(cx + d)
• reciprocate → 1/c(cx + d)
• mult. by bc-ad → (bc - ad)/c(cx + d)
• add a/c → a/c + (bc - ad)/c(cx + d)
• find common denominator and solve → (ax + b)/(cx + d)

Question 24

how do we realise f(x) = (ax+b)/(cx+d), where ad-bc ≠ 0, using generating transformations? (if c=0)

Answer 24

• given x
• mult. by a/d → (a/d)x
• add b/d → (a/d)x + b/d

Question 25

any composition of projections of ℝ∪{∞} realises a….

Answer 25

…linear fractional

Question 26

what is the cross ratio?

Answer 26

given 4 pts on a line (p,q,r,s), the cross ratio of the ordered 4-tuple is :
[p,q;r,s] = (r-p)(s-q)/(r-q)/s-p)

Question 27

do projections preserve the cross ratio?

Answer 27

yes, we can prove it using induction

Question 28

state the 4th pt determination thm

Answer 28

given any 3 pts p,q,r∈ℝℙ¹, any other pt x∈ℝℙ¹ is uniquely determined by its cross ratio

Question 29

state the existence thm related to the cross product

Answer 29

given 3 pts p,q,r∈ℝℙ¹ and 3 more p’,q’,r’, there exists a linear fractional f such that f(p)=p’, f(q)=q’, f(r)=r’

Question 30

state the uniqueness thm related to the cross product

Answer 30

there is exactly one linear fractional f that sends three pts p,q,r to three pts p’,q’,r’ respectively

Question 31

the linear fractional are precisely the functions on ℝℙ¹ that preserve the …

Answer 31

…cross ratio

Question 32

suppose 𝙸(p,q,r,s) (the invariant) is a function defined on quadruples of distinct pts so that 𝙸(p,q,r,s) = 𝙸(f(p),f(q),f(r),f(s)) for any linear fractional f. then we have that 𝙸(p,q,r,s) =

Answer 32

…. 𝙸(p’,q’,r’,s’)

we can think of 𝙸 as a function J where J(x) = 𝙸(p,q,r,s) for [p,q;r,s] = x