Affine and Projective Geometry

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Affine and Projective Geometry

• Chand T. John

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R2

• Imagine a bucket filled with vectors.
• A vector is an ordered pair of real numbers.
• These real numbers are called coordinates of the
  vector.
• A vector is usually written by listing its
  coordinates in parentheses.
• For example, a vector consisting of 3 and then 5
  is written as (3, 5).

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R2

• Pretend our bucket contains all possible vectors
  that is, it contains all possible ordered pairs
  of real numbers.
• This bucket is called R2, because each vector in
  R2 has 2 real numbers.
• This bucket has an infinite number of vectors,
  since there are an infinite number of real
  numbers.

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R2

• A vector with coordinates (x, y) can be drawn as
  an arrow that goes horizontally by x units and
  vertically by y units.

R2
y
x
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Operations in R2

• There are some operations we can do with vectors
  in R2addition, subtraction, and scalar
  multiplicationto produce another vector in R2.
• Pretend we have two vectors u (ux, uy) and v
  (vx,vy). We can add these vectors by adding
  their coordinates. Adding these vectors results
  in another vectorr u v (ux uy, vx vy)

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Operations in R2

• We can also subtract these vectors by sutracting
  their coordinates. This produces another
  vectors u v (ux uy, vx vy)
• Finally, we can multiply a vector by any real
  number c (called a scalar) to get another
  vectort cu (cux, cvx)

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Pictures of Operations in R2
Subtraction s u v
Addition r u v
v
u
r
v
s
u
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Pictures of Operations in R2
Scalar multiplication t cu
t
u
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E2

• Imagine we also have an infinitely large square,
  which we will call E2.
• Each exact location on E2 is called a point.

Points
E2
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Moving around on E2

• Suppose we are at point p on E2.
• Suppose we pick a vector v out of the bucket R2.
• We can add v to p to move to another point q on
  E2. To do this, simply place the arrow v on E2
  so that the tail end of v is at p. Then the head
  of v is at q

q
R2
v
p
E2
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Subtraction on E2

• We can also subtract point p from point q to get
  the vector v as a result v q p

q
R2
v
p
E2
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Center of Mass

• Seesaw

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Center of Mass on E2
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Calculating Center of Mass
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Barycentric Combination

• Change units so sum of masses 1
• Barycenter center of mass
• Mobius and der barycentrische calcul
• Possible on E2 even though addition of points
  is not allowed

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Linear and Affine Space

• Explain one-to-one correspondence between from R2
  to E2 and back pick a point called origin and
  two nonparallel vectors called basis vectorsthis
  origin basis vectors is called a coordinate
  system
• Key difference between E2 and R2 is that E2 has a
  notion of position, whereas R2 is just a bucket
  full of vectors with no other structure to it
• R2 is called a vector space or linear space, and
  E2 is called an affine space

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Linear versus Affine

• Difference between degrees Celsius (oC) vs.
  Celsius degrees (Co)
• Affine relationship oF 1.8oC 32
• Linear relationship Fo 1.8Co
• Again linear difference between affine points

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Linear versus Affine

• mechanical design align coord systems gt vectors
  have same coordinates in all of those coordinate
  systems, so vectors instead of points is
  convenient representation for forces/velocities/et
  c for computation

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Standard E2 Coordinate System

• Show standard coordinate system on E2 and some
  example points. Related to R2-E2 one-to-one
  correspondence. Can show how points are actually
  represented as origin basis vectors with scalar
  multiplication.

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Linear Transformations

• Emphasize that linear transformations move us
  from one vector to another.
• But remember, vectors in R2 are just floating
  around all the time, they have no fixed position.

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Affine Transformations

• Affine transformations move us from one point to
  another, or another way of looking at it is one
  coordinate system to another.
• Points in E2 have a fixed position, so affine
  maps move us from one specific point to another,
  unlike linear maps which move us from one vector
  to another, with no fixed position at any time.
• Affine transformation is linear transformation
  translation so just like linear transformation
  but with notion of position (hence translation).

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The Big Picture So Far

• Linear spaces have no notion of position.
• Linear maps transform one vector into another.
• Affine spaces have a notion of position.
• Coordinate systems can be defined on affine
  spaces to give special numbers representing each
  point.
• Any coordinate system is transformed into another
  using an affine map.

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E3

• Just to introduce real projective plane show
  standard coordinate system with basis vectors
  pulled out of R3 and z 1 plane, which we call
  RP2

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RP2

• Show point coordinates and vector coordinates on
  the real projective plane (in homogeneous
  coordinates)

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Projective

• Projective maps, line-point duality, homogeneous
  coordinates, line at infinity
• Weak perspective projection, perspective
  projection, orthogonal projection, parallel
  projection, cameras
• Affine transformations in E2 as 3 x 3 matrices in
  homogeneous coordinates