Consider an ellipsoid in 3D space that is represented by the quadric A b w ˜ T b T c w ˜ = 0 , where A is a 3 × 3 matrix, b is a 3 × 1 vector, and c is a scalar. For a normalized camera we can write...

1. 
   

   
   Consider an ellipsoid in 3D space that is represented by the quadric
   

   
   

_{A b}

w

˜


^T

b



^T



c

w

˜ = 0

,

where

A



is a 3

×



3 matrix,

b



is a 3

×



1 vector, and

c



is a scalar.

For a normalized camera we can write the world point

w

˜ in terms of the image point

x

˜ as

w

˜ = [

x

˜


^T







,




s

]


^T






where

s



is a scaling factor that determines the distance along the ray

x

˜.

1. 
   

   
   Combine these conditions to produce a criterion that must be true for an image point
   
   x
   
   ˜ to lie within the projection of the conic.
   

   
   


2. 
   

   
   The edge of the image of the conic is the locus of points for which there is a single solution for the distance
   
   s
   
   . Outside the conic there is no real solution for
   
   s
   
   
   
   and inside it there are two possible solutions corresponding to the front and back face of the quadric. Use this intuition to derive an expression for the conic in terms of
   
   A
   
   
   ,
   
   
   
   
   b
   
   
   
   and
   
   c
   
   . If the camera has intrinsic matrix
   
   Λ
   
   , what would the new expression for the conic be?