differential geometric methods in geometric modeling differential geometric methods in geometric...

Differential Geometric Methods in Geometric Modeling

Prof. Dr. F.-E. Wolter

July 13, 2011

Prof. Dr. F.-E. Wolter Differential Geometric Methods in Geometric Modeling July 13, 2011 1 / 52

Surface Contact Criteria

Outline

1 Surface ContactCriteriaHigher Order ContactApplications

2 Distance ComputationsConceptsApplications



Criteria for Second Order Contact

Problem:Develop Criteria, which employ minimal 1-dimensional Contact- orCurvature Conditions and control all Curvatures in a Surface Point.

1-dimensional Curvature- or Contact-Condition means e.g. Prescribing:

1 Normal Curvature respective a direction in a point

2 Tangential Contact with another surface along a curve



Contact-Curve-Theorem

Theorem on Curvature Equality of two surfaces along a tangentialcontact curve:Asssume that both regular C 2-surfaces F ,G have tangential contact alonga regular C 1-curve β(s). If at any point p of β(s) both surfaces F ,G havethe same normal curvature for a direction ~t transversal to β(s) then F ,Ghave indentical I- and II fundamental tensors at p.

p

t

1Pegna, J.; Wolter, F.-E., ”A Simple Practical Criterion to Guarantee Second Order Smoothness of Blend Surfaces”,

Proceedings of the 15th ASME Design Automation Conference: Advances in Design Automation., vol. 1, ASME, 1989, p.93-105

2J. Pegna, F.-E. Wolter, ”Geometrical Criteria to Guarantee Curvature Continuity of Blend Surfaces”, J MECH DESIGN

114 (1992) no. 1, 201-210



Point-Contact-Theorem (Three-Tangents-Theorem)

Theorem on Curvature-Equality of two surfaces in a tangentialcontact point:If two surfaces with tangential contact in a point p have in p commonnormal curvatures for 3 pairwise linearly independent directions ~ti , thenboth surfaces have identical I - and II fundamental tensors in p.

p

t

t

t1

2

3

1Pegna, J.; Wolter, F.-E., ”A Simple Practical Criterion to Guarantee Second Order Smoothness of Blend Surfaces”,

Proceedings of the 15th ASME Design Automation Conference: Advances in Design Automation., vol. 1, ASME, 1989, p.93-105

2J. Pegna, F.-E. Wolter, ”Geometrical Criteria to Guarantee Curvature Continuity of Blend Surfaces”, J MECH DESIGN

114 (1992) no. 1, 201-210



Proof: Notations

Take local representations of two surfaces by two height functions

z = αf (x , y) , α ∈ {1, 2}

with corresponding parametrizations:

αf (x , y) =[x , y ,α f (x , y)

]TPoint:

p =[xp, yp, zp

]T; p =

[xp, yp

]TTangent directions:

~ti = αf ′(p)

[xi

yi

]1 ≤ i ≤ 3



Proof (cont.)

Assuming for both surfaces αf common normal curvatures for 3 directions

~ti = αf ′(p)

[xi

yi

]=⇒

1k(xi ,yi )︷︸︸︷x2i 1fxx +2xiyi 1fxy +y2

i 1fyy√1+ 1f 2

x + 1f 2y |~ti |2

=

2k(xi ,yi )︷︸︸︷x2i 2fxx +2xiyi 2fxy +y2

i 2fyy√1+ 2f 2

x + 2f 2y |~ti |2

(1)

1f (p), 2f (p) tangential in p ⇒

(1fx(p), 1fy (p)) = (2fx(p), 2fy (p))

(2) and (1) =⇒

x2i 1fxx +2xiyi 1fxy +y2

i 1fyy =x2i 2fxx +2xiyi 2fxy +y2

i 2fyy



Proof (cont.)

L

1fxx2 1fxy

1fyy

= L

2fxx2 2fxy

2fyy

∀1 ≤ i ≤ 3 with L :=

x21 x1y1 y2

1

x22 x2y2 y2

2

x23 x3y3 y2

3

det(L) = det

[x1 x2

y1 y2

]det

[x2 x3

y2 y3

]det

[x1 x3

y1 y3

]6= 0,

as ~ti , 1 ≤ i ≤ 3 pairwise linearly independent. As

=⇒ (1fxx , 1fxy , 1fyy ) = (2fxx , 2fxy , 2fyy )

in p.


Surface Contact Higher Order Contact

Outline





Definition of Higher Order Contact

Definition: Let f (x , y) and g(x , y) be two Cn-smooth height functionsrepresenting two surfaces f , g which share the point

p = (0, 0, f (0, 0)) = (0, 0, g(0, 0)) .

Contact of n-th order at p means that the Taylor expansions off (x , y) and g(x , y) agree up to order n at the point (0, 0).

Directional contact of n-th order for a direction vector (x1, y1)means that the Taylor expansions for the two univariatefunctions d(s) := f (sx1, sy1) and e(s) := g(sx1, sy1) agreefor s = 0 up to order n.



General Point Contact Theorem

Theorem: 1 Two surfaces f , g have contact of n-th order at p if andonly if they have directional contact of n-th order for (n + 1) pairwiselinerly independent tangent directions at the point p.

1Wolter, F.-E.; Tuohy, S.-T., ”Curvature Computations for degenerate surface patches”. Computer Aided Geometric

Design 4 (1992), no. 9, 241-270



General Curve Contact Theorem

Theorem: Let f (x , y) and g(x , y) be two Cn-smooth height functionsrepresenting two surfaces f , g which have tangential contact along a curveα(s). Let V (s) be a vector field tangential to f along α(s) which istransversal to α(s). Assume that f and g have directional contact of ordern in direction V (s) for all points α(s). Then f and g have contact oforder n in all points α(s).

1Hermann, T.; Lukacs, G.; Wolter, F.-E., ”Geometrical criteria on the higher order smoothness of composite surfaces”,

Computer Aided Geometric Design 9 (1999), no. 16,907-911


Surface Contact Applications

Outline





Curvature Computations for Degenerate SurfaceRepresentations

Problem:Determine curvatures of a surface piece F whose representation f (1u, 2u)is degenerate but where the point set F has geometrically meaningfulcurvature.f () degenerate means there are points where rank f ′ < 2.

u2

1u

f( , )u21u



Curvature Computations for Degenerate SurfaceRepresentations

Solution:

Determine in a point p ∈ F the second derivatives of anon-degenreate parametrization of F .

Use those derivatives for curvature computations.

Choose for a parametrization of F the height function defined overthe tangent plane at F in p.

1Wolter, F.-E.; Tuohy, S.-T., ”Curvature Computations for degenerate surface patches”. Computer Aided Geometric Design



Curvature Computations

Let f (1u, 2u) : R2 → R3 surface

gij = ∂iuf ◦ ∂juf I-Fundamental Tensor

bij = N ◦ (∂iu∂juf ) II-Fundamental Tensor

with N Surface Normal and i , j ∈ {1, 2}.

Gauss-Curvature K =det(bij)det(gij)



Degeneration to a triangle Surface Patch

p

1

1

z

N

z=h(x,y)

y

x

¹

u²

f( , ) ²uu¹u

f (1u, 2u) = p for 0 ≤ 2u ≤ 1=⇒ ∂2uf (1, 2u) = 0=⇒ det(gij(1, 2u)) = 0=⇒Classical Curvature Computations Impossible!



Positive Gauss Curvature at the Degenerate Vertex

Octant of Sphere of Radius r=2, Represented by a Bezier-Patch

Elliptic Bezier-Patch Examples and Close Up



Non-Positive Gauss Curvature at the Degenerate Vertex

Parabolic Bezier-Patch Examples and Close Up

Hyperbolic Bezier-Patch Examples and Close Up



Example without well-defined Curvature at the DegenerateVertex

Here the parametric coordinate functions x(u, v) , y(u, v) , z(u, v) arepolynomials in u, v !



An equivalent Curve Example

x

y

y=h(x)

x(u) = u2 , y(u) = u3 , 0 ≤ u ≤ 1y(x(u)) = (x(u))3/2

y = h(x) = x3/2 ist C 1-smooth,but has no second order Taylor approximation in x = 0.



Problem

Problem: Under what Conditions are the Surface Curvatures at theDegenerate Vertex well defined?Is the computed surface curvature or are the computed second derivativesof the height function independent of the chosen 3 curves emating fromthe degenerate vertex?

Yes, if the Height function z = h(x , y)has a second order Taylor-Approximation.

However, even for degenerate parametric polynomial surfaces a C 1-smoothHeight function need not have a second order Taylor-Approximation at thedegenerate vertex.



Solution

Determine in a point p ∈ F the second derivatives of anon-degenerate parametrization of F .

Use those derivatives for curvature computations.

Choose for a parametrization of F the height function defined overthe tangent plane at F in p.


Surface Contact Conclusions

Conclusion

1 The 3-Tangent-Theorem gives necessary and sufficient conditions toensure 2-order-Point-Contact for surfaces.

2 This theorem can be used to compute candidates for 2-order partialsof a (non-degenerate) local height function representation of adegenerate surface. The height function is defined over the tangentplane of the surface point set at a degenerate surface vertex.

3 Those candidates give well-defined 2-order-partials and yield welldefined surface curvatures, provided the surface’s height functionrepresentation has a 2-order-Taylor-approximation at the degeneratevertex.

Contact Order 2 case of Curve-Contact-Theorem andPoint-Contact-Theorem are joined work with Joe Pegna (RPJ)


Distance Computations Concepts

Outline





Notations

Let M,M1,M2 ⊂ Rk be submanifolds with or without boundary.M,M1,M2,⊂ Rk may be given parametrically e.g. by splines orimplicitely as solutions of (e.g. algebraic) equations.

dist (.,.) will refer to Euclidean distance.



Problems

1 Let p ∈ Rk . Compute points nearest on M to p.

2 Compute:min{dist(a, b) | (a, b) ∈ M1 ×M2} ormax{dist(a, b) | (a, b) ∈ M1 ×M2}

3 Let c(t) be a curve in Rk . Trace a curve of nearest foot pointsπ(c(t)) on M. i.e.dist(π(c(t)), c(t)) =dist(M, c(t))

4 Compute all pairs of points in M1 ×M2 with distance 0.(Intersection-Problem)



Differential Equations for orthogonal Projectionsparametric case

f(u(t))u(t)

2u1u

1u

f( , )

(t)α2u

with: t ∈ [0, 1]α(t) = f (u(t)) + r(t)N(u(t))α′ = f ′(u(t))u′ + r(t)N ′(u(t))u′ + r ′(t)N(u(t))fi = ∂i f , Ni = ∂iNα′ = (f1f2)u′ + r(t)(N1N2)u′ + r ′(t)N(u(t))



Differential Equations for orthogonal Projectionsparametric case

[α′f1α′f2

]=

[w11 w12

w21 w22

]︸︷︷︸

W

[1u′

2u′

]

wij = fi fj + rNi fj

u′ =

[1u′

2u′

]= W−1

[α′f1α′f2

]

1Pegna, J.; Wolter, F.-E., ”Designing and Mapping Trimming Curves on Surfaces Using Orthogonal Projection”,

Proceedings of the 15th ASME Design Automation Conference: Advances in Design Automation, (B. Ravani, ed.), vol. 1, NY:ASME, 1990, p. 235-245

2Pegna, J.; Wolter, F.-E., ”Surface curve design by orthogonal projection of space curves onto free-form surfaces”, (1996),

45-52, J. MECH DESIGN 118 (1): 45-52 MAR 1996



Differential Equations for orthogonal projectionsimplicit case

α (t)

N( )S

(t)β

(t)βS = {(x , y , z)|g(x , y , z) = 0}N = 5gα(t) = β(t) + r(t)N(β(t))α′(t) = β′(t) + r(t)N ′(β(t))β′(t) + r ′(t)N(β(t))α′(t)− (r ′)tN(β(t)) = (ID + r(t)N ′(β(t)))︸︷︷︸

M(t)

β′(t)



Differential Equations for orthogonal projectionsimplicit case

α′(t)− r ′(t)N(β(t)) = M(t)β′(t)α′ − (Nα′)N = β′ + r(∂1Nβ

′1 + ∂2Nβ

′2 + ∂3Nβ

′3)

α′ − (Nα′)N = (ID + r(t)N ′)︸︷︷︸M(t)

β′

β′(t) = M−1(t)(α′ − (Nα′))

1Pegna, J.; Wolter, F.-E., ”Designing and Mapping Trimming Curves on Surfaces Using Orthogonal Projection”,

Proceedings of the 15th ASME Design Automation Conference: Advances in Design Automation, (B. Ravani, ed.), vol. 1, NY:ASME, 1990, p. 235-245

2Pegna, J.; Wolter, F.-E., ”Surface curve design by orthogonal projection of space curves onto free-form surfaces”, (1996),

45-52, J. MECH DESIGN 118 (1): 45-52 MAR 1996



Approximation of the Orthogonal Projection

Non-linear Inversion:

r 00 0r 0( )α t0tt f( , )

0

N( )r

-1

1f(t,r)

f (x,y)

r r

t

t

0

A = ([0,1])α

0 ( )α t0

0

( , )

f (t, r) := α(t) + rN(α(t))

Compute (approximate) the inverse map f −1(x , y): Ifdet(f ′(t0, r0)) 6= 0 then f (t, r) is locally around f (t0, r0) = (x0, y0)invertible.

Compute the Taylor-approximation for f −1(x , y) at f (t0, r0)



Localize Extrema of the Distance Function by TopologicalMethods (Gradient field-Rotation-Index)

Let p ∈ Rk and let Q(u, v) a parametric surface in Rk .

ϕ(u, v) := dist(p,Q(u, v))

Lokalize Extrema of ϕ by searching zeros of 5ϕ with (Rotation-) Index +1

1Kriezis, G.A.; Patrikalakis, N. M.; Wolter, F.-E., ”Topological and Differential-Equation Methods for Surface

Intersections”, Computer Aided Design 24 (1992), no. 1, 41-55



Rotation Index

The Rotation (Index) counts the number of rotations of 5ϕ along aclosed curve.

Rotation Index = -1Rotation Index = 1

If the rot-index of 5ϕ along a closed curve γ is non-zero then there existsal least one critical point of ϕ in the domain bounded by γ.

Rotation Index for an isolated Extrema is +1



Rotation Index



Rotation Index

V (u, v) = {χ, ψ}

W (V , γ) =1

2π

∫γ

χdψ − ψdχ

χ2 + ψ2

Θ = arctan

(ψ

χ

)for − π < Θ < π

W =1

2π

n−1∑i=0

4Θi with 4Θi = Θ(i+1) mod n −Θi and − π < 4Θi < π



Elimination of Search Areas with Taylor Estmiation

ϕ(S0 + ε) ≥ ϕ(S0) + ϕ′(S0)ε− maxS0≤S≤S0+ε

ϕ′′(S)ε2︸︷︷︸≥ ϕ(S)

Leopold Kronecker (1869):Vector Field Index of a Vector Field Z on Rn (at 0)

1Area(Sn−1)

RSn−1ε

1|Z |n det(Z ,∂u1Z ,...,∂un−1Z)du1...dun−1

with (u1 . . . un−1) coordinates for the Sphere Sn−1ε



Minimal Distance between Submanifolds

Let a(s), c(t) be two closed curves in Rk .

c(t)

a(s)

Search min{ϕ(s, t)| ϕ(s, t) :=dist(c(s), a(t))}

Find local Extrema of ϕ(s, t) by locating zeros (with Index 1) of 5ϕ byVectorfield Index method.



The Cut Locus CM

b(t)M

MC

MC

Cut Locus CM := {q ∈ Rk | q has atleast 2 minimal joins to M}Trace nearest foot points on M of thecurve b(t). A normal ray is distanceminimal to M until it reaches the cutlocus CM of M !

Background for all this is contained in:

1Wolter, F.-E., ”Distance Function and Cut Loci on a Complete Riemannian Manifold”, Archiv d. Math. 1(1979), no. 32,

92-962

Wolter, F.-E., ”Cut Loci in Bordered and Unbordered Riemannian Manifolds”, Ph.D. thesis, Technical Univ. Berlin 19853

Wolter, F.-E., ”Cut Locus and Medial Axis in Global Shape Interrogation and Representation”,(1993), MIT, Dec. 1993(revised version) Published as: MIT Design Laboratory Memorandum 92-2 and MIT Sea Grant Report, 1992.

4Abrams, S.; Bardis, L.; Chryssostomidis, C.; Clement, A.; Jinkerson, R.; Patrikalakis, N. M.; Wolter, F.-E., ”Inspection

and Feature Extraction of Marine Propellers”, Journal of Ship Production 9 (1993), no. 2, 88-1065

Rausch, T.; Wolter, F.-E.; Sniehotta, O. , ”Computation of Medial Curves in Surfaces”,(1996), 43-68, Hannover, August1996. Published in: Conference on the Mathematics of Surfaces VII (September 1996), Institute of Mathematics and itsApplications, IMA Conference Series, pp. 43-68, 1997



The Cut Locus

Define the path length to the cut locus for all unit normals N(x) of M bya function

s(N(x)) := sup{λ|Segment(x + [0, λ]N(x)) is distance minimal from(x + λN(x)) to M}

If we find for q ∈ Rk an orthogonal foot pointO(q) ∈ M, with |q − O(q)| ≤ s(N(O(q))) and

N =q − O(q)|q − O(q)| then O(q) is a point nearest on M

to q.

M

O(q) q

N



Computing dist (M , CM)

Determine dist(M,CM) to support the computation of dist(q,M) for apoint q close to M:

If O(q) is an orthogonal projection of a point p ∈ Rk on a submanifold Mand if,

dist(q,O(q)) ≤dist(M,CM)then O(q) is a point nearest on M to q.



dist (M , CM) in a Special Case

Let c : [0, 1]→ Rk be a smooth closed curve yielding the submanifoldM = c[0, 1] .

η

M r m

δ

t

u

α=min{dist(c(u),c(t)) | u≤t−δ} δ=πrm

max0≤t≤1

|c′(t)|

dist (M,CM) = min{rm, η}rm = minimal curvature radius of cη = 1

2 lenghth(shortest binormal segment joining 2 points of c)


Distance Computations Applications

Outline





Application Examples for Distance Computations

Quality Control in Manufactoring 1

Medial Curves and Surfaces 2

Cut Locus in Shape Classification 3

Automatic Mesh Generation 4

Surface Intersection (Small Loops) 5



Rausch, T.; Wolter, F.-E.; Sniehotta, O. , ”Computation of Medial Curves in Surfaces”,(1996), 43-68, Hannover, August1996. Published in: Conference on the Mathematics of Surfaces VII (September 1996), Institute of Mathematics and itsApplications, IMA Conference Series, pp. 43-68, 1997.

3Wolter, F.-E., ”Cut Locus and Medial Axis in Global Shape Interrogation and Representation”,(1993), MIT, Dec. 1993

(revised version) Published as: MIT Design Laboratory Memorandum 92-2 and US National Seagrant Library, MIT Sea GrantReport, 1992


Intersections”, Computer Aided Design 24 (1992), no. 1, 41-555

Abrams, S.; Bardis, L.; Chryssostomidis, C.; Clement, A.; Jinkerson, R.; Patrikalakis, N. M.; Wolter, F.-E., ”Inspectionand Feature Extraction of Marine Propellers”, Journal of Ship Production 9 (1993), no. 2, 88-106



Quality Control in Manufacturing

Check, if measured points of manufactured object are close enough to theDesign object.(More thee than 3 dimensions may be relevant if beyond 3 spacecoordinates physical parameters for material properties enter.)





Propeller Blade



Medial Curves

The differential equation concept for orthogonal projections yields alsodifferential equations for medial curves between 2 curves.

γ

α

β γ ⊂ {w ∈ R2| dist(α,w) = dist(β,w)}

In Experiments with Splines: Position accuracy of γ 10−11, after Splineinterpolation 10−8

1F.-E. Wolter. Distance Computations for Curves and Surfaces. Mini-Symposium on Engineering Geometry at SIAM

Conference on Geometric Design, Tempe, Arizona, 19892

F.-E. Wolter. Geodesic Offsets, Computations and Applications. International Symposium on Freeform Curves andSurfaces 95, Mathematical Research Institute Oberwolfach, Germany, 1995.

3Rausch, T.; Wolter, F.-E.; Sniehotta, O. , ”Computation of Medial Curves in Surfaces”,(1996), 43-68, Hannover, August

1996. Published in: Conference on the Mathematics of Surfaces VII (September 1996), Institute of Mathematics and itsApplications, IMA Conference Series, pp. 43-68, 1997



Cut Locus and Shape Classification



Automatic Mesh Generation

Medial Axis of a Multiply-Connected Shape



Mesh

Finite Element Mesh of a Multiply Connected Domain



Surface Intersection Curves

Intersection of Two Biquartic Bezier Patches

Localization of small loops

Binormal


Intersections”, Computer Aided Design 24 (1992), no. 1,41-55


Distance Computations Conclusions

Conclusions

Methods use local and global Differential Geometric Concepts!1 Local

1 Tensorial differential equations for orthogonal projection2 Approximation (non-linear) inverse (of Normal map)3 Elimination of search areas e.g. by Taylor estimates

2 (Semi-)GlobalVectorfield Index-Method

3 GlobalGlobal Differential Geometric tools employing the Cut Locus


differential geometric methods in geometric modeling differential geometric methods in geometric...

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