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Multiple-View Geometry

Chapter 1: Introduction

Projective Geometry

  • preserve straightness
  • Projective in homogeneous coordinates: both
    (x,y,1)(x, y, 1)
    (x,y,0)(x, y, 0)
    are defined

Affine Geometry

  • In Projective Space, we single out a particular line and call it the line at infinity.
  • With this line at infinity, we are able to define parallelism, and equal length of two intervals.

Euclidean Geometry

  • In Affine Geometry, we further single out two circular points in the line at infinity.
  • Two circular points:
    (1,±i,0)T(1, ±i, 0)^\mathrm{T}
    which satisfy a pair of real equations:
    x2+y2=0;w=0x^2 + y^2 = 0; w = 0
  • With two circular points, we are able to define angle and length ratios.
  • Representation:
    (x,y)(x, y)