Section 5 - Perspective Flashcards
define a projective plane
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
“on” is sometimes called a …
incidence relation
what is the simplest projective plane? what is its order?
the fano plane (order 7)
how do we define the real projective plane (ℝℙ²)?
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;
- Any two “points” are contained in a unique “line because two given lines through O lie in a unique plane through O.
- Any two “lines” contain a unique “point” because any two planes through O meet in a unique line through O.
- 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.
a line through O (a pt in ℝℙ²) is the set of all pts….
(ɑx,ɑy,ɑz), where (x,y,z)≠0 & ɑ∈ℝ
a plane through O (a line in ℝℙ²) has equation….
ax+by+cz=0, where a,b,c∈ℝ and aren’t all 0
if (x₁,y₁,z₁) & (x₂,y₂,z₂) determine different lines through O, how do we find the plane through O that contains them? (ℝℙ²)
solve:
ax₁+by₁+cz₁=0
ax₂+by₂+cz₂=0 , for a,b,c
define a 3-dimensional projective space
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
does ℝ⁴ satisfy the projective space axioms?
yes, define:
- points as lines through O in ℝ⁴
- lines as planes through O in ℝ⁴
- planes as 3-d subspaces of ℝ⁴
Suppose we have two lines l₁ & l₂ and a point P. How do we define a mapping f: l₁ → l₂∪{∞}?
by the point where the line through P & x (on l₁) meets l₂
in terms of mappings what does k represent?
by Thales’ thm the distance between the images of equally spaced pts on l₁ is “magnified” by some constant k
if l₁ & l₂ are // and P is at ∞, what is k?
k = 1 (lines through l₁ & l₂ are //)
if l₁ & l₂ are // and l₂ is between P & l₁, what is k? (P≠∞)
k < 1
if l₁ & l₂ are // and P is in the middle of l₁ & l₂, what is k?
k = 1
if l₁ & l₂ aren’t //, is distance preserved during projection?
no
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?
l₁/n
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?
y = (l₁/n)x + k
when projecting if l₁//x-axis & l₂//y-axis, the mapping P send x on l₁ to what pt on l₂?
l₁l₂/x
what are the generating transformations?
any composition of reflections is a composition of the functions:
x + l
kx
1/x
any composition of the functions x+l, kx (k≠0), 1/x is a function of what form?
f(x) = (ax+b)/(cx+d), where ad-bc ≠ 0
what is a linear fractional?
a function f: ℝ∪{∞} → ℝ∪{∞} of the form f(x) = (ax+b)/(cx+d), where ad-bc ≠ 0
why do we specify ad-bc ≠ 0 for a linear fractional?
it guarantees the function isn’t constant (i.e. all pts don’t map to the same pt)
how do we realise f(x) = (ax+b)/(cx+d), where ad-bc ≠ 0, using generating transformations? (if c≠0)
- 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)
how do we realise f(x) = (ax+b)/(cx+d), where ad-bc ≠ 0, using generating transformations? (if c=0)
- given x
- mult. by a/d → (a/d)x
- add b/d → (a/d)x + b/d
any composition of projections of ℝ∪{∞} realises a….
…linear fractional
what is the cross ratio?
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)
do projections preserve the cross ratio?
yes, we can prove it using induction
state the 4th pt determination thm
given any 3 pts p,q,r∈ℝℙ¹, any other pt x∈ℝℙ¹ is uniquely determined by its cross ratio
state the existence thm related to the cross product
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’
state the uniqueness thm related to the cross product
there is exactly one linear fractional f that sends three pts p,q,r to three pts p’,q’,r’ respectively
the linear fractional are precisely the functions on ℝℙ¹ that preserve the …
…cross ratio
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) =
…. 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
