Physically based deformations of implicit surfaces

Michal Remiš

Implicit surfaces

Implicit surface – group of points that formsolution 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 - x2+y2+z2-r2=0• Torus

(R2 − r2)2 + 2R2(z2 − x2 − y2) − 2r2(x2+ y2 + z2) + (x2 + y2 + z2)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 defineskernel for convolution

• skeleton abstractsfinal 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, interpenetrations3) Generate contact surface under collision,

add deformation term4) Compute forces that are to be applied

next step 1

Deformation of surface under collision

Modeling contacts between objects

-Interpenetration zone g1(p) = -f2(p), g2(p) = -f1(p)

-Since we need to 0 = f1(p)+g1(p) = f2(p)+g2(p) whenever f1=f2

f1 =f2

f1 =f2

Deformation of propagation region

• In propagation region gi=hi(d)

Mi = -ai*gi,min

gi=-fi k = |gradient(fj,p0)|

Other issues

- Volume preservation

- Unwanted blending

Thank you for your attention.