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Ray Tracing Implicit SurfacesPowerPoint Presentation

Ray Tracing Implicit Surfaces

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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 (Kalra&Barr 89)
- Sphere Tracing (Hart 96)
- Interval Analysis (Mitchell 89)

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?

Lipschitz Functions

- Function f is Lipschitz ifL |f(x) – f(y)|/||x – y||for 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)

Lipschitz Arithmetic

- Let f(x) and g(x) be real functions over a compact subset of 3-D
Lip(f+g) = max(f+g)’ = max(f’+g’) max f’ + max g’ = Lip f + Lip g

Lip(fg) = max(fg)’ = max(f’g + fg’) max(f’ g) + max(fg’)

- Let h be a real function
Lip(f(h)) = max(f’(h) h’) = h’ Liph(x)f

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 can’t 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 > r then f 0 over the entire cell no implicit surface in cell
- Otherwise subdivide the cell

f(x)/L

r

Root Isolation

- Let G be a Lipschitz bound off’(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

- Directional derivative of f
- If f’(t)/G >t then derivative of f can’t get back to zero over interval
- Hence f is monotonic over interval
- Hence we can use Newton’s method to find root
- Otherwise subdivide interval

f(t)

f’(t)

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

Interval Analysis

- Replace values x with ranges of values [x0,x1]
- Interval arithmetic
[a,b] + [c,d] = [a+c,b+d]

[a,b] – [c,d] = [a-d,b-c]

[a,b] * [c,d] = [min(ac,ad,bc,bd), max(ac,ad,bc,bd)]

1 / [c,d] = [1/d,1/c]

[a,b]3 = [a3,b3]

- Implement function using interval arithmetic

f [a,b]

[a,b]

Interval Roots

- Let [y0,y1] = f[x0,x1]
- If y0 > 0 or y1 < 0 then no root over interval [x0,x1]
- Otherwise subdivide [x0,x1] and recurse
- Let [y’0,y’1] = f’[x0,x1]
- If y’0 > 0 or y’1 < 0 then f monotonic over interval [x0,x1]
- Then we can find root with Newton’s method
- Otherwise subdivide [x0,x1] and recurse

[y0,y1]

[x0, x1]

[y’0,y’1]

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