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Mathematics of the Simple Camera. ©Alexandre Meyer 2002, Anthony Steed 2001-2003. Overview. Simple Camera Scenes with spheres COP on +z COP = Centre Of Projection. Simple Camera (Cross Section). COP = Centre Of Projection. Y. d. y max. Z. -Z. COP. y min. View From the Camera.

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mathematics of the simple camera

Mathematics of the Simple Camera

©Alexandre Meyer 2002, Anthony Steed 2001-2003

overview
Overview
  • Simple Camera
    • Scenes with spheres
    • COP on +z

COP = Centre Of Projection

simple camera cross section
Simple Camera (Cross Section)

COP = Centre Of Projection

Y

d

ymax

Z

-Z

COP

ymin

view from the camera
View From the Camera

(xmax, ymax)

(xmin, ymin)

forming the rays
Forming the Rays
  • Map screen pixels (M by N window) to points in camera view plane

(xmax, ymax)

(M-1, N-1)

(0,0)

(xmin, ymin)

forming the rays1
Forming the Rays
  • Consider pixel i,j
  • It corresponds to a rectangle

width = (xmax-xmin)/M

height = (ymax-ymin)/N

  • Our ray goes through the centre of the pixel
  • Thus the ray goes through the point

(xmin + width*(i+0.5), ymin + height*(j+0.5), 0.0)

forming the rays2
Forming the Rays
  • Thus the ray from the COP through pixel i,j is defined by

p(t) = (x(t), y(t), z(t)) =

(t*(xmin + width*(i+0.5)),

t*(ymin + height*(j+0.5)),

d - t*d)

ray casting
Ray Casting
  • Intersection of Sphere and line (sphere at origin)
  • Substitute the ray equation in the sphere equation and solve!
  • Get an equation in t of the form

At2 + 2Bt + C = 0

ray casting1
Ray Casting

If B2 – AC < 0 then the ray doesn’t intersect the sphere.

If B2 -AC = 0 the ray is tangent to the sphere

If B2 – AC > 0 then there are two roots given by

t = (-B  (B2 – AC))/A

chose the lowest value one (the one closest to the COP)

ray casting2
Ray Casting
  • Intersection of Sphere and line (general case)
    • Sphere is centred at (a,b,c)
    • Translate the start of the ray by (-a,-b,-c)
    • Proceed as before
conclusions
Conclusions
  • We can now draw images
    • Forming rays from the camera
    • Intersecting those rays with objects (spheres) in the scene
  • But
    • No colour – merely binary detection operation
    • Camera is static - at the moment we must move the objects in front of the camera to be able to see them
    • Need more interesting scenes!