Computer Graphics Camera Projection Picking

1
Computer GraphicsCamera Projection / Picking

• CO2409
• Week 8 - Additional Material

2
Contents

• World / View Matrices Recap
• Projection Maths
• Pixel from World-Space Vertex
• World Space Ray from Pixel
• These notes are presented as additional material
  and are not examinable

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Model Space

• An models mesh is defined in its own local
  coordinate system - model space
• Each model is positioned with a matrix
• Transforming it from model space into world space
• This matrix is called the World Matrix

4
World to Camera Space

• Next consider how the models are positioned and
  oriented relative to the camera
• Convert the models from world space into camera
  space
• The scene as viewed from cameras position
• This transformation is done with the view matrix

5
Camera to Viewport Space

• Finally project the camera space models into 2D
• The 3D vertices are projected to camera position
• Assume the viewport is an actual rectangle in the
  scene
• Calculate where the rays hit the viewport 2D
  geometry
• This is done with the projection matrix

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Projection Details

• Cameras have two settings
• Field of View (FOV)
• Viewport distance (D)
• Viewport distance is same as the near clipping
  plane
• Where geometry slices through the viewport
• FOV works as a wide angle or zoom lens
• FOV can be different for width and height FOVX
  FOVY

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Projecting a Vertex

• Consider the projection of a single 3D vertex to
  2D
• Want 2D coordinates (XV, YV)
• YV not shown in diagram
• Calculate using similar triangles
• X / Z XV / D, so XV D X / Z
• In a similar way, YV D Y / Z
• This is the perspective divide
• Now have 2D coords, but still in camera space
  units
• Need to convert to pixels

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Converting to Pixels

• Calculate the actual viewport dimensions (camera
  space)
• tan(FOVX / 2) (WV / 2) / D
• so WV 2 D tan(FOVX / 2)
• similarly, HV 2 D tan(FOVY / 2)
• Then calculate
• XN 2 XV / WV
• YN 2 YV / HV
• This 2D coordinate(XN, YN) is in the range 1 to
  1
• Ready to convert to pixel position

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Converting to Pixels

• If the viewport width height (in pixels) are WP
  and HP
• then XP (XN 1) WP / 2
• and YP (1 - YN) HP / 2
• The second formula flips the Y axis (viewport Y
  is down)
• (XP, YP) are the coordinates of the final pixel
  we want

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Picking

• Sometimes we need to manually perform the
  projection process
• To find the pixel for a particular 3D point
• E.g. To draw text/sprites in same place as a 3D
  model
• Or perform the process in reverse
• Each 2D pixel corresponds to a ray in 3D space
  (refer to the projection diagram)
• Finding and working with this ray is called
  picking
• E.g. to find the 3D object under the mouse
• The algorithms for both follow they are derived
  from the previous slides

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Pixel from World-Space Vertex

• Start with world space vertex P
• Transform this vertex by combined view /
  projection matrix to give Q
• If Q.z lt 0 then the vertex is behind us, discard
• Otherwise do perspective divide
• Q.x / Q.z and Q.y / Q.z
• Finally, scale to pixel coordinates X,Y
• X (Q.x 1) (ViewportWidth / 2)
• Y (1 - Q.y) (ViewportHeight / 2)
• Use to draw text/sprites in same place as 3D
  entity

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World-Space Ray From Pixel 1

• Initial pixel (X,Y), first convert to point Q in
  the range -1 -gt 1
• Q.x (2 X / ViewportWidth) - 1
• Q.y 1 (2 Y / ViewportHeight)
• Set Q.z D (viewport / near clip distance)
• The result vertex will be exactly on the clip
  plane
• Calculate viewport size in camera space
• WV 2 D tan(FOVX / 2)
• HV 2 D tan(FOVY / 2)
• If FOVY not available
• HV WV ViewportHeight / ViewportWidth

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World-Space Ray From Pixel 2

• Convert Q into camera space
• Q.x WV / 2
• Q.y HV / 2
• Finally transform by the inverse of the view
  matrix to get a point in world space
• Then cast a ray from camera to this point
• Use this 3D ray to detect the entity at the pixel