Ray Tracing Implicit Surfaces

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Ray Tracing Implicit Surfaces

• John C. Hart
• CS 319
• Advanced Topics in Computer Graphics

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Problem

• Intersection of ray r(t) with implicit surface
  f(x) 0
• Easy to solve f(r(t)) when f algebraic
• Implicit surfaces can be defined by an arbitrary
  function
• More general techniques
• LG Surfaces (KalraBarr 89)
• Sphere Tracing (Hart 96)
• Interval Analysis (Mitchell 89)

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Problem Statement

• Given a cell (cube), can one guarantee that the
  implicit surface
• passes through it?
• does not pass through it?
• Sign of function values at corners
• Different signs guarantees surface intersects
  cell
• Same signs surface may or may not intersect cell
• How can we determine definitively if a surface
  intersects any given cell?

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Lipschitz Functions

• Function f is Lipschitz if L?? f(x) f(y)/x
  yfor some value L
• Smallest such L is the Lipschitz constant Lip f
• Lipschitz constant is the maximum slope of a
  continuous function
• L max f(x)
• Function might by Lipschitz only over a given
  region (e.g. x2)
• Function might not be Lipschitz (e.g. sin 1/x)

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Lipschitz Arithmetic

• Let f(x) and g(x) be real functions over a
  compact subset of 3-D
• Lip(fg) max(fg) max(fg) ? max f
  max g Lip f Lip g
• Lip(f g) max(f g) max(fg f g) ?
  max(f g) max(f g)
• Let h be a real function
• Lip(f(h)) max(f(h) h) h Liph(x) f

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Lipschitz Guarantee
f(x)

• Let L?? Lip f be a Lipschitz bound of f, and let
  f(x)?? 0
• Then f ? 0 over the region(x f(x)/L, x
  f(x)/L)
• Why? Because f cant get back to zero fast enough
• Let x be the center of a cell, and let r be the
  radius of the cell (distance to farthest corner)
• If f(x)/L gt r then f ? 0 over the entire cell ?
  no implicit surface in cell
• Otherwise subdivide the cell

f(x)/L
r
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Root Isolation

• Let G be a Lipschitz bound of f(t)
  df(r(t))/dt
• Derivative of f(r(t))?
• Directional derivative of f
• ? df(r(t))/dt ?f ?? rd
• Change in f in ray direction
• If f(t)/G gt??t then derivative of f cant get
  back to zero over interval
• Hence f is monotonic over interval
• Hence we can use Newtons method to find root
• Otherwise subdivide interval

f(t)
f(t)
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Sphere Tracing

• Let d(x,A) be a signed distance bound
• d(x,A)?? min x y for all y in A
• If f is Lipschitz with bound L then f(x)/L is a
  signed distance bound
• d(x,A) is radius of a ball guaranteed not to
  intersect implicit surface A
• March along ray by distance steps
• x r0
• x d(x,A) rd
• Intersection when steps converge

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Interval Analysis

• Replace values x with ranges of values x0,x1
• Interval arithmetic
• a,b c,d ac,bd
• a,b c,d a-d,b-c
• a,b c,d min(ac,ad,bc,bd), max(ac,ad,bc,b
  d)
• 1 / c,d 1/d,1/c
• a,b3 a3,b3
• Implement function using interval arithmetic

f a,b
a,b
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Interval Roots

• Let y0,y1 fx0,x1
• If y0 gt 0 or y1 lt 0 then no root over interval
  x0,x1
• Otherwise subdivide x0,x1 and recurse
• Let y0,y1 fx0,x1
• If y0 gt 0 or y1 lt 0 then f monotonic over
  interval x0,x1
• Then we can find root with Newtons method
• Otherwise subdivide x0,x1 and recurse

y0,y1
x0, x1
y0,y1