Navigating and viewing the virtual world

   Camera properties and definition Perspective transformation Quaternion transforms for changing camera

 ◦ ◦ pinhole camera model tiny aperture finite size screen Light strikes screen aperture Light from world

  Recall that we used 3 matrices for transformation in the BasicEffect: ◦ ◦ ◦ world (for where the object is in the virtual world) viewing (describes transformation of world to canonical space in front of camera) projection (transform from 3D world to 2D screen) Separation allows us to deal with each independently

   operation to transform 3D world coordinates to 2D screen coordinates orthogonal projection: parallel rays perspective projection: rays pass through aperture (pinhole, human eye)

  Simple projection: ◦ (x,y,z,1)  (x,y) z value used in depth buffer

  only map objects within "canonical" viewing volume If the world is bigger (and not properly oriented) need coordinate transform to map into canonical volume

    ◦ ◦ ◦ Previously, assumed the camera was axis aligned Not typically the case!

Specify camera position, orientation Common mechanism: camera position viewing direction up direction

  ◦ ◦ ◦ Given gaze direction g, up direction h "Camera axes": say u, v, w w = -g/|g| (gaze direction is – z) u = h x w / | h x w | v = w x u

Orthographic Perspective

In classical perspective, have vanishing points Parallel lines appear to converge

  Orthographic projection works when we have a large aperture Our experience of the world is in distant objects look smaller

perspective

: aperture small object large object Objects have same apparent size

"frustum": a truncated cone or pyramid

near plane z = n far plane z = f

y s = y g (n/z g ) y s = y r (n/z r ) aperture y s y g n z g z r y r

y s = y r (n/z r )    Need to divide by z No division with matrices Again use homogeneous coordinates: "homogenize" operation

x y z 1

equiv

hx hy hz h for any nonzero h

1 0 0 0 0 1 0 0 0 0 (n+f)/n -f 1/n 0 0 0

n 0 0 0 0 n 0 0 0 0 n+f 1 0 0 -fn 0

n 0 0 0 0 n 0 0 0 0 0 n+f -fn 0 1 0 x y z 1

=

nx ny (n+f)z-fn z

n 0 0 0 0 n 0 n+f -fn 0 0 0 1 0 0 0 x y z 1 nx

=

ny (n+f)z-fn z

homogenize

nx/z ny/z (n+f) fn/z 1

n 0 0 0 0 n 0 n+f -fn 0 0 0 1 0 0 0 x y z 1 nx

=

ny (n+f)z-fn z

homogenize

nx/z ny/z (n+f) fn/z 1

compare: y s = y g (n/z g )

     We need to preserve Z ordering so that depth tests still work The perspective matrix preserves Z At z=f, preserves z=f At z=n, preserves z=n At intermediate values, preserves order

   Wide-angle perspective looks weird Human focal region has small solid angle – we do not experience severe perspective distortion Technically correct results may not be perceptually correct

  ◦ ◦ ◦ Recall that to specify camera position and orientation, need 6 quantities: 3 for 3D position “ forward direction” (unit axis, 2 scalars to specify) “up direction” (orthogonal to forward, can be specified just with an angle, one scalar) This info goes into the viewing transform

 Matrix.CreateLookAt( cameraPosition, // where the camera is targetPosition, // where camera looks at upVector // “up” direction (towards top) );

   ◦ Can apply transformations to viewing matrix translations, rotations… ◦ Can also recreate matrix at each frame inexpensive compared to everything else!

◦ ◦ Need to track camera orientation and position vector for position quaternion for orientation

 ◦ ◦ ◦ contains: position (vector) forward direction (vector) up direction (vector)  Want to be able to swing the camera sideways, up and down, spin (roll), plus move

 ◦ Change position according to current velocity x(t+dt) = x(t) + v(t)dt   Might have v(t) from player control, or velocity of body being followed Might have specific camera dynamics

 ◦ ◦ ◦ ◦ Often interpret player controls in terms of current heading move forward move backward strafe right, left change orientation (right, left, up, down, roll, all in current frame of reference)

 ◦ ◦ ◦ With forward and up known as part of camera, can change position easily move forward: x(t+dt) = x(t) + f(t)*s*dt  s = speed, f(t) = forward direction at time t can get sideways vector as u x f (cross product) moving sideways uses same principle as moving forward, just direction differs

   Store orientation as quaternion q Update forward, up directions using current q ◦ ◦ rotation of initial forward f 0 , initial up u 0 say p = (0,f 0 ), q’ = conjugate of q for q = (s,v), q' = (s,-v)  f = vector(qpq’)  In XNA, f = Vector.Transform(f0, q)

     Now, changing camera orientation is easy: Rotations about current forward, up, side axes ◦ available, or obtained from cross product Quaternion.CreateFromAxisAngle(axis, angle); ◦ describes rotation Compose with current quaternion: ◦ q = Quaternion.Multiply(q,qrot); Normalize q, and voila!