Prologue - Archimedes A Roman soldier came up to him unexpectedly and commanded him to follow; which Archimedes declined to do before he had worked out his problem at hand. The soldier, enraged, drew his sword and ran him through. -Plutarch (version 1)
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Prologue - Archimedes A Roman soldier came up to him unexpectedly and commanded him to follow; which Archimedes declined to do before he had worked out.
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Prologue - Archimedes
A Roman soldier came up to him unexpectedly and commanded him to follow; which Archimedes declined to do before he had worked out his problem at hand. The soldier, enraged, drew his sword and ran him through.
-Plutarch (version 1)
Prologue - Sophie Germain Paris, 1776-1831 Self-educated to age 18 Monsieur Le Blanc attends
Ecole Polytechnique Lagrange, Legendre, Gauss Elasticity and Fermat’s
Last Theorem
Alyn RockwoodMitsubishi Electric Research Labs
Le savant n’ étudie pas la nature parce que cela est utile; il l’étudie parce qu’il y prend plaisir et il y prend plaisir parce qu’elle belle. Si la nature n’ était pas belle, elle ne vaudrait pas la peine d’être connue.
-Poincare.
Implicit Fields
What is an implicit Field?
Generalization of a Distance Field
Shape Field in Gray Height field
A field that “specifies a minimum distance to a shape.” [Frisken, Perry, Rockwood, Jones, Siggraph2000]
Distance Field
The Zero Set of the field gives the shape - no polygons
Distance Field
Offsets of zero set yield volumes
Discrete Sampling -
The Data Size Problem - volume data The Solution - sample and interpolate Discrete Structure - octree, for example Adaptively Sampled Distance Fields (ADFs)
Store distance (implicit) values at vertices of an octree
Reconstruction
A single trilinear field can represent highly curved surfaces
Rendering
Surface rendering– ray casting with analytic– surface intersection
Volume rendering– back-to-front sampled– ray casting
Cloud of Points Rendering
Sample, bin and sort into z-buffer
Dense
Samples
Screen
View
What is an implicit Field?
Formally: An implicit field is a displayable, scalar field d: RnR where the zero set d-1(0) is non-empty.
Object oriented definition: structure + procedure Includes distance fields, but no “minimum” calculation Includes implicits, interpolated, procedural, others Implicitizes; thus the term implicit Examples follow
Example - Weierstrass Egg Carton
d(x) = z - k kcos(kx) cos(ky), k=1,…, Continuous, but
not differentiable! Rendering by
cloud-of-points,
z-buffering, depth
based intensity
Parameter space – unit cube Gamut space – deformed cube
w- axis
green
. (u,v,w) . d(x) = ||(u,v,w)||
blue v-axis
red
u-axis
implicit field at x is metric of its parameter (u,v,w).