Stan Melax Graphics Software Engineer, Intel 3D Interaction Happens - - PowerPoint PPT Presentation

Interaction With 3D Geometry

Stan Melax Graphics Software Engineer, Intel

3D Interaction Happens with Geometric Objects

Rigid Body

Skinned Characters

Soft Body

{ Mesh geometry; Vec3 position; Quat orientation; }; { Mesh geometry; Vec3 position[]; Quat orientation[]; }; { Mesh geometry; Springs connectivity; };

Agenda – Interacting with 3D Geometry

Practical topics among:

• Core Geometry Concepts
• Convex Polyhedra
• Spatial and Mass Properties
• Soft Geometry and Springs

With examples and implementation issues.

Warning: Some Math Ahead

u v b a u v u v

𝑣 ∙ 𝑤 > 0

u v

𝑣 ∙ 𝑤 < 0

u v

𝑣 × 𝑤

u v 𝑤 × 𝑣

Vector Arithmetic Dot Product Cross Product

𝑁𝑤 =

𝑛00 𝑛01 𝑛02 𝑛10 𝑛11 𝑛12 𝑛20 𝑛21 𝑛22 𝑤𝑦 𝑤𝑧 𝑤𝑨

Matrix Stuff

𝑣𝑤𝑈 = 𝑣𝑦𝑤𝑦 𝑣𝑦𝑤𝑧 𝑣𝑦𝑤𝑨 𝑣𝑧𝑤𝑦 𝑣𝑧𝑤𝑧 𝑣𝑧𝑤𝑨 𝑣𝑨𝑤𝑦 𝑣𝑨𝑤𝑧 𝑣𝑨𝑤𝑨

𝑒𝑔 𝑒𝑤 = 𝜖𝑔

𝑦

𝜖𝑤𝑦 𝜖𝑔

𝑦

𝜖𝑤𝑧 𝜖𝑔

𝑦

𝜖𝑤𝑨 𝜖𝑔

𝑧

𝜖𝑤𝑦 𝜖𝑔

𝑧

𝜖𝑤𝑧 𝜖𝑔

𝑧

𝜖𝑤𝑨 𝜖𝑔

𝑨

𝜖𝑤𝑦 𝜖𝑔

𝑨

𝜖𝑤𝑧 𝜖𝑔

𝑨

𝜖𝑤𝑨

What’s an “outer product”?

𝑣 ∙ 𝑤 = 𝑣𝑈𝑤 = 𝑣𝑦 𝑣𝑧

𝑣𝑨

𝑤𝑦 𝑤𝑧 𝑤𝑨 = 𝑣𝑦𝑤𝑦 + 𝑣𝑧𝑤𝑧 + 𝑣𝑨𝑤𝑨

𝑣⨂𝑤 = 𝑣𝑤𝑈 = 𝑣𝑦 𝑣𝑧 𝑣𝑨 𝑤𝑦 𝑤𝑧 𝑤𝑨 = 𝑣𝑦𝑤𝑦 𝑣𝑦𝑤𝑧 𝑣𝑦𝑤𝑨 𝑣𝑧𝑤𝑦 𝑣𝑧𝑤𝑧 𝑣𝑧𝑤𝑨 𝑣𝑨𝑤𝑦 𝑣𝑨𝑤𝑧 𝑣𝑨𝑤𝑨

Familiar dot or inner product: Outer product:

Outer Product - Geometric View

Outer product 𝒗 ⊗ 𝒗 of a unit vector 𝒗 projects any vector along that direction.

u v u v

𝑣 = 1

𝑣 ∙ 𝑤

u v

𝑣(𝑣 ∙ 𝑤) 𝑣 𝑣 ∙ 𝑤 = 𝑣 ⊗ 𝑣 𝑤 𝑣 𝑣 ∙ 𝑤 ≠ 𝑣 ∙ 𝑣 𝑤

Distance of v along u (scalar) Projection of v along u (vector)

Outer Product for Plane Projection

𝑱 − 𝒗 ⊗ 𝒗 projects any vector onto plane with normal 𝒗.

u v

(𝑣 ⊗ 𝑣)𝑤

𝑤 − 𝑣 𝑣 ∙ 𝑤

u v u v

𝑤 − 𝑣 𝑣 ∙ 𝑤 = 𝐽𝑤 − 𝑣 ⊗ 𝑣 𝑤 = (𝐽 − 𝑣 ⊗ 𝑣 )𝑤

𝑣 𝑣 ∙ 𝑤

• r

−𝑣 𝑣 ∙ 𝑤 −𝑣 𝑣 ∙ 𝑤

Remove portion of v that runs along u

Numerical Precision Issues

3 “collinear” points

Triangles and Planes

• Specify Front and Back

Could use: 𝐵𝑦 + 𝐶𝑧 + 𝐷𝑨 == 𝐸 Prefer convention: 𝐵𝑦 + 𝐶𝑧 + 𝐷𝑨 + 𝐸 == 0 Given a point [xyz] its distance above plane is: 𝑞𝑦 𝑦 + 𝑞𝑧 𝑧 + 𝑞𝑨 𝑨 + 𝑞𝑥

• <0 below
• =0 on plane
• >0 above

𝑞𝑦 𝑞𝑧 𝑞𝑨 Plane 𝑞 = 𝑞𝑦 𝑞𝑧 𝑞𝑨 𝑞𝑥 0 0 0 𝑞𝑥 Triangle (𝑏 𝑐 𝑑) CCW winding

a b c

Intersection of 3 planes

𝑄 = 𝑞0𝑦 𝑞0𝑧 𝑞0𝑨 𝑞1𝑦 𝑞1𝑧 𝑞1𝑨 𝑞2𝑦 𝑞2𝑧 𝑞2𝑨 , 𝑐 = 𝑞0𝑥 𝑞1𝑥 𝑞2𝑥 , 𝑤 = 𝑤𝑦 𝑤𝑧 𝑤𝑨 p0 p1 p2 p0 p1 p2 v

𝑄𝑤 = −𝑐 𝑤 = −𝑄−1𝑐

𝑞0𝑦 𝑤𝑦 + 𝑞0𝑧 𝑤𝑧 + 𝑞0𝑨 𝑤𝑨 + 𝑞0𝑥 == 0 𝑞1𝑦 𝑤𝑦 + 𝑞1𝑧 𝑤𝑧 + 𝑞1𝑨 𝑤𝑨 + 𝑞1𝑥 == 0 𝑞2𝑦 𝑤𝑦 + 𝑞2𝑧 𝑤𝑧 + 𝑞2𝑨 𝑤𝑨 + 𝑞2𝑥 == 0

What if we have 4 planes?

As house grows bottom up, how will the 4 roof planes come together at the top of this house?

Floor Only Walls Roof Planes Roof Planes

• verhead view

Example:

Determining How 4 Planes Meet

? 𝑞0𝑦 𝑞0𝑧 𝑞1𝑦 𝑞1𝑧 𝑞0𝑨 𝑞0𝑥 𝑞1𝑨 𝑞1𝑥 𝑞2𝑦 𝑞2𝑧 𝑞3𝑦 𝑞3𝑧 𝑞2𝑨 𝑞2𝑥 𝑞3𝑨 𝑞3𝑥 𝑦 𝑧 𝑨 1 = p0 p1 p2 p3

Note: these are top down views

• f a convex region

If 4 planes meet at a point xyz then we’ve found a solution to: Only possible if matrix singular.

Determining How 4 Planes Meet

? d==0 d<0 d>0 d = 𝑞0 𝑞1 𝑞2 𝑞3 = 𝑞0𝑦 𝑞1𝑦 𝑞0𝑧 𝑞1𝑧 𝑞2𝑦 𝑞3𝑦 𝑞2𝑧 𝑞3𝑧 𝑞0𝑨 𝑞1𝑨 𝑞0𝑥 𝑞1𝑥 𝑞2𝑨 𝑞3𝑨 𝑞2𝑥 𝑞3𝑥 p0 p1 p2 p3

Notes: these are top down views of a convex region. Planes well “behaved” all point same way. Planes in CCW order. Det of any 3x3 subblock from first 3 xyz rows is >0.

Intersect Line Plane

p v1 v0

time

distance above plane

1

𝑞𝑦𝑧𝑨 ∙ 𝑤𝑢 + 𝑞𝑥

𝒆𝟏

t=? 𝑢 = −𝑒0 𝑒1 − 𝑒0

𝑞𝑦𝑧𝑨 = 𝑞𝑦 𝑞𝑧 𝑞𝑨

𝑗𝑛𝑞𝑏𝑑𝑢 = 𝑤0 + 𝑤1 − 𝑤0 𝑢

𝑒𝑗𝑡𝑢 = 𝑞𝑥 𝑒1 = 𝑞𝑦𝑧𝑨 ∙ 𝑤1 +𝑞𝑥 𝑞𝑦𝑧𝑨 𝑒0 = 𝑞𝑦𝑧𝑨 ∙ 𝑤0 +𝑞𝑥

𝒆𝟐

= − 𝑞𝑦𝑧𝑨 ∙ 𝑤0 + 𝑞𝑥 (𝑞𝑦𝑧𝑨 ∙ 𝑤1 + 𝑞𝑥) − (𝑞𝑦𝑧𝑨 ∙ 𝑤0 + 𝑞𝑥)

Simple Ray Triangle Intersection Test

• Intersect ray with plane and

get a point p.

• For each edge va to vb
• Cross product with p-va
• Dot result with tri normal n
• <0 means outside
• Backside hit if dot(n,ray)>0

p

Ray Mesh Intersection

• Check Against Every Polygon
• (spatial structures can rule out many

quickly)

• Detecting a “hit” - might not be closest
• Numerical Robustness – don’t slip

between two adjacent triangles.

• Mesh “intact” – t-intersections, holes

caused by missing triangles.

Multiple impact points – take closest. Numeric precision can result in plane intersection point landing just

• utside each time, thus missing

both neighbors.

Solid Geometry

• “Water-Tight” borderless manifold mesh:
• Every edge has 1 adjacency edge going
• ther way
• Consistent winding. Polygon normals all

face to exterior.

• The mesh is the boundary representation
• the infinitely thin surface that separates

solid matter from empty space.

Inside/Outside Test of a point

• Cast a long ray from point
• point is inside if it first hits

the backside of a polygon.

• point is outside the object if

it first hits a front side or nothing at all. ray p

Convex Polyhedra

Convex Mesh

• Neighboring face normals tilt away.
• Volume bounded by a number of planes.
• Every vertex lies at or below every face.

Many algorithms much easier when using convex shapes instead of general meshes. 𝐿 𝑑𝑝𝑜𝑤𝑓𝑦 ↔ ∀𝑏, 𝑐 ∈ 𝐿 → 𝑏𝑐 ⊆ 𝐿

𝒃 𝒄 𝑳

Math textbook:

Convex In/Out test

A point is inside a convex volume if it lies under every plane boundary. No testing with vertices or edges. p q w

Convex Ray Intersection

• Crop Ray with all front facing planes: 𝑜 ∙ 𝑤1 − 𝑤0 < 0
• Impact at v0 if under all backside planes

v1 v0 v0 v0 v0

Convex Line-Segment Intersection

• Trim v0 for Front facing planes 𝑜 ∙ 𝑤1 − 𝑤0 < 0
• Trim v1 for Back facing planes 𝑜 ∙ 𝑤1 − 𝑤0 > 0

v0 v1 v0 v1 v0 v1

Convex-Convex Contact

• Separating Axis Theorem –

Two non touching convex polyhedra are separated by a plane.

• The contact between two touching

convex polyhedra will be either

• A point
• A line segment
• A convex polygon

Physics engines like convex objects

Convex Hull from points

Convex Hull - smallest convex polyhedron that contains all the points.

• Convex hull of a mesh will contain the mesh.
• Often a sufficient proxy for interaction and

collision

Techniques

• Expanding outward: QuickHull [Eddy ’77]
• Reducing inward: Gift Wrapping [Chand ’70]

Compute 3D Convex Hull

• Start with 2 back to back
• triangles. (or a tetrahedron)
• Find a vertex outside current

volume (above faces).

• Find edge loop silhouette (around

all faces below point)

• Replace with new fan of polys
• Remove Folds (if any)
• Rinse and Repeat

Hull Numerical Robustness

Generated skinny triangle with bad normal. Flip edge to fix Grow Hull

Hull Tri-Tet Implementation

• Simple Triangle-mesh based approach
• When point d above any [abc] add tetrahedron

[abcd] (triangles [acb][dab][dbc][dca]) to mesh.

• Prune any back-to-back tris such as [abc] and [bac]

a c b

d

a c b d a c b d

Hull Tri-Tet Numeric Robustness

b a c d e

• 5 points {a,b,c,d,e} are coplanar-ish at one end of the point cloud
• But next point e tests above triangle [abc] but below [adb].
• Silhouette is {abc} for which we extrude a new tetrahedron up to e
• This produces triangles [eca],[ebc] (blue) facing the right way and

[eab] (red) facing the wrong way – based on known interior point.

• This provides the hindsight to see that [adb] should also be extruded

b c e b c e a a d d b c e a d

Simplified Convex Hull

• Off-the-shelf solutions typically

generate the complete hull.

• All hull vertices may not be needed

and may be inefficient for usage.

• Instead use greedy incremental

algorithm picking next vertex that increases hull size the most.

• Stop when vertex count or error

threshold reached Full Hull Simplified

Minimize Number of Planes vs Points

Minimize Planes:

• Compute full hull
• Dual points

𝑞𝑦 𝑞𝑥 𝑞𝑧 𝑞𝑥 𝑞𝑨 𝑞𝑥

• Compute simplified hull
• Take Dual

12 Vertices 20 Planes 20 Vertices 12 Planes

Convex Decomposition

Manual Creation of Collision Proxy Automatic Mesh Decomposition [Ratcliff], [Mamou] Use skinned mesh bone weights to create collision hulls

3DS Max Screenshot

Example: Convex Bones For Collisions

Ragdolls on Stairs Hand catching coins

Constructive Solid Geometry Boolean Operations

Solid Geometry Boolean Operations

Applications for Booleans

Art tools (already have CSG)

• Modeling and level design

Game Usages (less fidelity required):

• Destruction
• Geomod (tunneling)

A convex based approach may suffice.

Convex Cropping and Intersection

• Like general mesh cropping,

but only 1 open loop that is itself a convex polygon.

• Intersecting two convexes

can be done by cropping one with all the planes of the other

• Non-convex operands can be

implemented as a union

Convex Cropping Robustness

• Floating point issues can occur even in this most

basic of mesh operations.

• One solution is to utilize plane equations to

determine face/plane connectivity.

Correct Slice Bad Slice Pre Slice

Robustness with Quantum Planes

Let all planes p of CSG Boolean operands satisfy:

𝑁 = 𝑡0𝑞0𝑦 𝑡0𝑞0𝑧 𝑡0𝑞0𝑨 𝑡1𝑞1𝑦 𝑡1𝑞1𝑧 𝑡1𝑞1𝑨 𝑡2𝑞2𝑦 𝑡2𝑞2𝑧 𝑡2𝑞2𝑨 𝑤 = −𝑁−1 𝑡0𝑞0𝑥 𝑡1𝑞1𝑥 𝑡2𝑞2𝑥

∀𝑞∃𝑡 ∶ 𝑡𝑞 = 𝑡 𝑞𝑦 𝑡 𝑞𝑧 𝑡 𝑞𝑨 𝑡 𝑞𝑥 , 𝑡 𝑞{𝑦,𝑧,𝑨,𝑥} ∈ ℤ, 𝑡𝑞{𝑦,𝑧,𝑨} ≤ 𝑙

∃𝑏, 𝑐: 𝑤{𝑦,𝑧,𝑨}= 𝑏 𝑐 , 𝑏, 𝑐 ∈ ℤ , 𝑐 ≤ det 𝑁 ≤ 6𝑙3

∴ ∃𝜁∀𝑣, 𝑤: 𝑣 − 𝑤 < 𝜁 ↔ 𝑣 = 𝑤

Recall how vertices are the intersection of 3 or more planes Generated points have Finite denominator A fixed epsilon can now be used to indicate if two points are the same.

Destruction – geometry modification

Texturing break

Objects in Motion Spatial Properties

Spatial and Mass Properties

• With a solid mesh we can correctly

derive properties that affect motion:

• Volume
• Center of mass
• Covariance (and 3x3 Inertia Tensor)

Triangle Center and Area

center = 𝑏 + 𝑐 + 𝑑 3

a b c

abs area = (𝑐 − 𝑏) × (𝑑 − 𝑐) 2 a b

c

a b

c

area = 𝑐 − 𝑏 × 𝑑 − 𝑐 ∙ 𝑜 2

n

Given a pre set normal n, result is signed. So area could be negative. i.e. Center of mass. There are other ways to define “center”.

Cross Product - Edge Choice Irrelevant

a b

c

𝑑 − 𝑐 × 𝑏 − 𝑑 a b

c

𝑏 − 𝑑 × 𝑐 − 𝑏 a b

c

𝑐 − 𝑏 × 𝑑 − 𝑐

All the same

= =

Area of Polygon (2D)

? Triangle Summation:

• Pick a reference point p. Normal n known.
• For each edge (a,b) sum area of triangle

(p,a,b): 1

2 𝑏 − 𝑞 × 𝑐 − 𝑏 ∙ 𝑜

• Signed Area - upside down triangles cancel
• ut extra area if p outside.

p p

• =

p

Solid’s Area Weighted Normals Cancel

Sum of cross products for each tetrahedron face is zero.

u v w

𝑣 × 𝑥 + 𝑥 × 𝑤 + 𝑤 × 𝑣 + 𝑤 − 𝑣 × 𝑥 − 𝑣 = 0

w-u

Polygon Normal

• Pick reference point p (origin)
• For each edge (ab) in polygon,

sum cross product:

p p

+ =

p

𝑏 − 𝑞 × (𝑐 − 𝑏)

𝐹𝑒𝑕𝑓 (𝑏,𝑐)

?

For a trapezoid explanation search for “Newell Normal”

Tetrahedron Volume and Center

center

• f mass = 𝑏 + 𝑐 + 𝑑 + 𝑒

4

a c b d

volume = (𝑑 − 𝑏) × 𝑐 − 𝑏 ∙ (𝑒 − 𝑏) 6

a c b d

= 𝑑 − 𝑏 𝑐 − 𝑏 𝑒 − 𝑏 ÷ 6

Triple product in determinant form

Tetrahedron Integration

a c b d Edges: u,v,w a=[0 0 0] (origin)

𝑔 𝛽 + 𝛾 + 𝛿 𝑒𝛿 𝑒𝛾 𝑒𝛽

𝑥(1− 𝛽 𝑣 − 𝛾 𝑤) 𝑤(1 − 𝛽 𝑣) 𝑣

u v w

Letting f(X)=1, we get our volume:

1 𝑒𝛿 𝑒𝛾 𝑒𝛽

𝑥(1− 𝛽 𝑣 − 𝛾 𝑤) 𝑤(1 − 𝛽 𝑣) 𝑣

= 𝑣𝑤𝑥 6 𝛽 + 𝛾 + 𝛿 𝑣𝑤𝑥 6 𝑒𝛿 𝑒𝛾 𝑒𝛽

𝑥(1− 𝛽 𝑣 − 𝛾 𝑤) 𝑤(1 − 𝛽 𝑣) 𝑣

= 𝑣 + 𝑤 + 𝑥 4

General Form: Letting f(X)=X/vol for center of mass (relative to vertex a):

Tetrahedral Summation (3D)

𝑤𝑝𝑚𝑣𝑛𝑓 = 𝑣 𝑤 𝑥 /6

(𝑣,𝑤,𝑥)∈𝑛𝑓𝑡ℎ

𝑑𝑓𝑜𝑢𝑓𝑠 𝑝𝑔 𝑛𝑏𝑡𝑡 = 𝑣 + 𝑤 + 𝑥 4 𝑣 𝑤 𝑥 6 /𝑤𝑝𝑚(𝑛𝑓𝑡ℎ)

(𝑣,𝑤,𝑥)∈𝑛𝑓𝑡ℎ

• rigin

Center of Mass Affects Gameplay

Catapult geometry:

Inertia Calculation

2D:

𝑛𝑝𝑛𝑓𝑜𝑢 𝑝𝑔 𝑗𝑜𝑓𝑠𝑢𝑗𝑏 = 𝑠2 = 𝑦2 + 𝑧2 = 𝑦2 + 𝑧2

Variance in x Variance in y

3D: 3x3 Inertia Tensor Related to covariance

𝑧𝑧 + 𝑨𝑨 −𝑦𝑧 −𝑦𝑨 −𝑦𝑧 𝑦𝑦 + 𝑨𝑨 −𝑧𝑨 −𝑦𝑨 −𝑧𝑨 𝑦𝑦 + 𝑧𝑧 Fixed Axis

Covariance (origin = center of mass)

• If object was a collection of point masses v

𝑤𝑤𝑈 = 𝑤𝑦2 𝑤𝑦𝑤𝑧 𝑤𝑦𝑤𝑨 𝑤𝑧𝑤𝑦 𝑤𝑧2 𝑤𝑧𝑤𝑨 𝑤𝑨𝑤𝑦 𝑤𝑨𝑤𝑧 𝑤𝑨2

• Single Tetrahedron (0,u,v,w)…

𝑏𝑣 + 𝑐𝑤 + 𝑑𝑥 ⨂(𝑏𝑣 + 𝑐𝑤 + 𝑑𝑥) 𝑒𝑑 𝑒𝑐 𝑒𝑏

1−𝑏−𝑐 1−𝑏

∗ 𝑤𝑝𝑚_𝑏𝑒𝑘

1

Tetrahedron (0,u,v,w) Covariance

u,v,w are vectors from center of mass to triangle on mesh

Inertia Tetrahedral Summation

𝑑𝑗,𝑘 = 2𝑣𝑗𝑣𝑘 + 2𝑤𝑗𝑤𝑘 + 2𝑥𝑗𝑥

𝑘 + 𝑣𝑗𝑤𝑘 + 𝑤𝑗𝑥 𝑘 + 𝑥𝑗𝑣𝑘 + 𝑣𝑗𝑥 𝑘 + 𝑤𝑗𝑣𝑘 + 𝑥𝑗𝑤𝑘

120

𝑣,𝑤,𝑥 ∈𝑛𝑓𝑡ℎ−𝑑𝑝𝑛

𝑣 𝑤 𝑥 6

𝑗𝑜𝑓𝑠𝑢𝑗𝑏: 𝑈 = 𝐽𝑒𝑓𝑜𝑢𝑗𝑢𝑧 (𝑑0,0 + 𝑑1,1 + 𝑑2,2) − 𝐷

Center Of Mass 𝑗𝑜𝑢𝑓𝑠𝑢𝑗𝑏: 𝑈 = 𝑧𝑧 + 𝑨𝑨 −𝑦𝑧 −𝑦𝑨 −𝑦𝑧 𝑦𝑦 + 𝑨𝑨 −𝑧𝑨 −𝑦𝑨 −𝑧𝑨 𝑦𝑦 + 𝑧𝑧

Sum for each triangle uvw, Covariance matrix C:

Inertia Tensor and Object Motion

𝜕 = 𝑟 𝑈−1 𝑟−1 ℎ 𝜕 → 𝑡𝑞𝑗𝑜 ℎ → 𝑛𝑝𝑛𝑓𝑜𝑣𝑛 𝑟 → 𝑝𝑠𝑗𝑓𝑜𝑢𝑏𝑢𝑗𝑝𝑜 𝑈 → 𝐽𝑜𝑓𝑠𝑢𝑗𝑏

Time Integration

Updating state to the next time step.

• Position: 𝑞𝑢+𝑒𝑢 = 𝑞𝑢 + 𝑤𝑓𝑚𝑝𝑑𝑗𝑢𝑧 ∗ 𝑒𝑢
• Orientation:

𝑟𝑢+𝑒𝑢 = 𝑡 ∗ 𝑟𝑢 𝑡 = 𝜕 𝜕 sin ( 𝜕 𝑒𝑢 2 ), cos ( 𝜕 𝑒𝑢 2 ) 𝑟𝑢+𝑒𝑢 = 𝑟𝑢 + 𝜕 2 𝑟𝑢 𝑒𝑢 𝑟𝑢+𝑒𝑢 = 𝑟𝑢 + 𝑒𝑟 𝑒𝑢 𝑒𝑢 Build a Quat for Multiplication Add Derivative

lim

( 𝜕 𝑒𝑢)→0 𝑡 → 𝜕

2 𝑒𝑢, 1

𝑡 ∗ 𝑟𝑢 = 0001 ∗ 𝑟𝑢 + 𝜕 2 𝑒𝑢, 0 ∗ 𝑟𝑢 𝑡 ∗ 𝑟𝑢 = 𝑟𝑢 + 𝜕 2 , 0 ∗ 𝑟𝑢 𝑒𝑢

Proof it’s the same:

Time Integration without Numerical Drift

𝑟𝑢+𝑒𝑢 = 𝑟𝑢 + 𝜕(𝑟𝑢) 2 ∗ 𝑟𝑢 ∗ 𝑒𝑢

𝑙1 = 𝜕(𝑟𝑢) 2 ∗ 𝑟𝑢 𝑙2 = 𝜕(𝑟𝑢 + 𝑙1 ∗ 𝑒𝑢/2) 2 ∗ (𝑟𝑢 + 𝑙1 ∗ 𝑒𝑢 2 ) 𝑙3 = 𝜕(𝑟𝑢 + 𝑙2 ∗ 𝑒𝑢/2) 2 ∗ (𝑟𝑢 + 𝑙2 ∗ 𝑒𝑢 2 ) 𝑙4 = 𝜕(𝑟𝑢 + 𝑙3 ∗ 𝑒𝑢) 2 ∗ (𝑟𝑢 + 𝑙3 ∗ 𝑒𝑢) 𝑟𝑢+𝑒𝑢 = 𝑟𝑢 + 𝑙1 ∗ 𝑒𝑢 6 + 𝑙2 ∗ 𝑒𝑢 3 + 𝑙3 ∗ 𝑒𝑢 3 + 𝑙4 ∗ 𝑒𝑢 6

Forward Euler Runge Kutta

Time Integration Euler vs RK4

Forward Euler:

• Spin drifts

toward principle axis

• Energy gained

Runge Kutta

• Spin orbits as

expected

• Energy stays

constant

Soft Body Objects

Soft Body Meshes

• Every vertex has its own

position and velocity

• Vertices are connected via

springs or constraints.

Object Construction

Cloth Cube Lattice Table

• Connection topology can differ from visual mesh
• Stiffness can vary to simulate different objects and materials

Time Integration – Simulating Soft Body

Kinematic

• Connections as constraints
• Iterative position projection
• Easy to Implement
• Numerically Robust (will

never explode)

• Most common system used
• May not converge under

stress (compression or stretch)

Dynamic

• Connections as springs
• Forward Euler can only handle

weak “springy” forces

• Implicit Integration required

for stiff springs

• Damped behavior
• Converges to force-correct

state

• Harder to Implement

Kinematic Solver

Realistic Behavior Easy To Implement Algorithm: repeat a few times for each constraint move endpoints toward rest-length

Implicit Integration Spring Network

• Forward Euler

𝑤𝑢+𝑒𝑢 = 𝑤𝑢 + 𝑛−1𝑔𝑝𝑠𝑑𝑓𝑢 𝑒𝑢

• Implicit Euler

𝑤𝑢+𝑒𝑢 = 𝑤𝑢 + 𝑛−1𝑔𝑝𝑠𝑑𝑓𝑢+𝑒𝑢 𝑒𝑢

𝑔𝑝𝑠𝑑𝑓𝑢+𝑒𝑢 = 𝑔

𝑢+𝑒𝑢 = 𝑔 𝑢 + 𝜖𝑔

𝜖𝑞 Δ𝑞 + 𝜖𝑔 𝜖𝑤 Δ𝑤 , 𝑞𝑝𝑡𝑗𝑢𝑗𝑝𝑜 𝑞 𝑤𝑓𝑚𝑝𝑑𝑗𝑢𝑧 𝑤

Derivatives of Force

𝑒𝑔 𝑒𝑤 = 𝜖𝑔

𝑦

𝜖𝑤𝑦 𝜖𝑔

𝑦

𝜖𝑤𝑧 𝜖𝑔

𝑦

𝜖𝑤𝑨 𝜖𝑔

𝑧

𝜖𝑤𝑦 𝜖𝑔

𝑧

𝜖𝑤𝑧 𝜖𝑔

𝑧

𝜖𝑤𝑨 𝜖𝑔

𝑨

𝜖𝑤𝑦 𝜖𝑔

𝑨

𝜖𝑤𝑧 𝜖𝑔

𝑨

𝜖𝑤𝑨 Jacobian

𝑔 = −𝑙 𝑞 − 𝑠 𝑞 𝑞

p p r f f

Δ𝑔

start end

• =

=

Δ𝑞

=

Force at spring endpoint Compressed Spring: Stretched Spring: How force changes as endpoint moves along spring direction:

Δ𝑔 Δ𝑞 = −𝑙

Derivatives of Force – Endpoint moves orthogonal

p p f f

Δ𝑞 Δ𝑔

• =

= f f

Δ𝑞 Δ𝑔

• =

=

Stretched: Compressed:

𝑞𝑡𝑢𝑏𝑠𝑢 𝑞𝑓𝑜𝑒 Δ𝑔

=

Δ𝑞

=

How force changes as endpoint moves lateral to spring dir:

Δ𝑔 Δ𝑞 = −𝑙 (1 − 𝑠 𝑞 )

• r

Derivatives of Force – 3x3 Jacobian

𝜖𝑔 𝜖𝑞 = −𝑙 𝑞 ⊗ 𝑞 𝑞 ∙ 𝑞 + 𝐽 − 𝑞 ⊗ 𝑞 𝑞 ∙ 𝑞 1 − 𝑠 𝑞

Avoid compression singularity by clamping r/||p|| .

General change in Force for a given change in position:

𝐵 Δ𝑤 = 𝑐 , 𝐵 = 𝐽 − 𝜖𝑔 𝜖𝑤 𝑒𝑢 − 𝜖𝑔 𝜖𝑞 𝑒𝑢2, 𝑐 = 𝑔 𝑒𝑢 + 𝜖𝑔 𝜖𝑞 𝑤 𝑒𝑢2

Solve for Δ𝑤 in linear system:

(note: slightly oversimplified)

Implicit Integration

Force Based Stiff Springs Realistic Behavior Responsive Convergence (Jitter Free)

Summary

Moving beyond triangles, dot and cross product:

• Objects as 3D Solids
• Convex parts
• Spatial props
• “2nd Year” Maths
• Object Motion
• Stiff systems

𝜈𝑛𝑛𝛽𝑠𝛿 Variety of topics:

Interacting with 3D Geometry

Volume Integration Time Integration

Q&A

Stan Melax stan@melax.com

Hand Tracking – if someone asks….

Tracking model Depth data Fit model