Physically based deformations of implicit surfaces Michal Remiš

Implicit surfaces Implicit surface – group of points that form solution for equation: F(x,y,z)=0 Easy: Is the point on (inside/outside) the surface? Hard: Generate all points on the surface.

Simple algebraic primitives Sphere - x 2 +y 2 +z 2 -r 2 =0 Torus (R 2 − r 2 ) 2 + 2R 2 (z 2 − x 2 − y 2 ) − 2r 2 (x 2 + y 2 + z 2 ) + (x 2 + y 2 + z 2 ) 2 =0 etc.. plane, elipsiod,paraboloid, hyperboloid

Skeleton based surfaces Surface is defined by skeleton (points, lines, curves) and weight function(s). Each unit (e.g.point) of the skeleton contributes to function according to distance. General point skeleton based surface:

Convolution surfaces not point skeletons (lines, curves, polygons) weight function defines kernel for convolution skeleton abstracts final shape natural blending

Complex implicit models Complex implicit models may be defined by binary operations of primitives (union,intersection,…) blending may be involved for smooth transitions between objects deformations (warp, bend,…) may be applied to get desired shape Interpolation techniques Surface reconstruction

Geometrical Modelling of Living Cells

Deformation of implicit surfaces Implicit function defines volume Usually: –f(p) > 0 - outside of surface –f(p) = 0 - surface –f(p) < 0 - inside surface –gradient(f(p)) = normal of surface in p

Physically based deformations Based on physical laws Langrangian methods (mass spring systems, particles) Eulerian methods (fluid simulations) Set of differential equations must be calculated to determine correct responses

Nodal approach Langrangian method Skeleton connected by springs, hinges.. Skeleton of implicit surface is deformed according to external forces and deformation propagates through connections of nodes

Collisions of implicits Collision detection –test if node lies inside of other object (inaccurate) –sampling Response computation –deformation of skeleton by force computed from the amount of intersection –local implicit deformation

Layered model 1) Animate skeleton of each object integrating the forces 2) Detect object colisions, interpenetrations 3) Generate contact surface under collision, add deformation term 4) Compute forces that are to be applied next step  1

Deformation of surface under collision

Modeling contacts between objects -Interpenetration zone g 1 (p) = -f 2 (p), g 2 (p) = -f 1 (p) -Since we need to 0 = f 1 (p)+g 1 (p) = f 2 (p)+g 2 (p) whenever f 1 =f 2 f 1 =f 2

Deformation of propagation region In propagation region g i =h i (d) M i = -a i *g i,min g i =-f i  k = |gradient(f j,p 0 )|

Other issues -Volume preservation -Unwanted blending

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