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TRIA LAR ME HE
. .
Department of Computer Science and Engineering.
P R E S E N T E D B Y :P A R D E E P K U M A R
C I V I L A N D E N V I R O N M E N T A L E N G I N E E R I N G D E P A R T M E N T
,
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Curvature
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Parametric Representation of Surfaces
Curved sur ace described b a ol nomial in terms
of two parameters u and w,
=
Jacobian of curved surface
, , , , , ,
( )
( )
( )
( )
( )
( )
, , ,, ,
, , ,x y z
y z z x x yJ J J
u w u w u w
( ),y z
y z u uwhere
=,
w w
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Parametric Representation of Surfaces
Assum tion: Three Jacobian are not all zeros at
same time2 2 2
0J J J+ +
This assumption makes sure that curved surface: Do not degenerate to a point or a curve, and
Do not contain any singular points such a spikes.
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Parametric Representation of Surfaces
u-curve = P u w
w-curve = P(u0,w)
( ) ( ) ( )0 0 0, , ,x u w y u w z u w
( ) ( ) ( )0 0 0
, ,
, , ,
ou uu u u
x u w y u w z u w
=
u
Tangent Vectors
at pointP(u0,w0) , ,ow w
w w w=
w
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Parametric Representation of Surfaces( ) ( ) ( ) ( ) ( ) ( ), , , , , ,y u w z u w x u w z u w x u w y u w
( ) ( ) ( ) ( ) ( ) ( )
, ,
, , , , , ,o
o
u uw w
y u w z u w x u w z u w x u w y u ww w w w w w =
=
=
u wP P
( )
( )
( )
( )
( )
( )
, , ,, , , ,
, , , oooo
u ux y zw wu u
w w
y z z x x yJ J J
u w u w u w===
=
= =
u wP P
or non-zeroJacobian conditionto be true we must
have cross productabove to be nonzero.
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Parametric Representation of Surfaces
Curve in arbitrary direction on curved surface can be
d du dw= +
u w
PP P
expressed in terms of parameter t as P(u(t),w(t)). Tangent atP(u0 ,w0)
0 0 0t t t t t t = = =
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Parametric Representation of Surfaces
u w u wP P P P
( )2 2 2 2
x y zJ J J+ +u wP P
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Equation of Tangent Plane R be position vector on tangent plane
( )( ) ( )( ) ( )0 0 0 0, . 0, , . 0u w u w= =u wR - P e R - P P P
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Unit Tangent Vector
) ) ) ),Define t u t w t and curved surface t= u P u
[ ]Tangent vector u w u w = + = u
u w
w
PP P P uA
P
( ) ( ) ( )22
,T
agnitude s of tangent vector s t t t = =
2
P P P
[ ] [ ] [ ]2
2
s u w u ww w
= =
=
u u u w
u w 2
w w u w
P PP P P P
T =
( )1
, .
,t
Unit Tangent Vector = =
P uAt
2T uF u
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Curvature of a Surface
: ' . .d d dt
Lets now use length s as parameter = =
P P PP
, . ' 1.
'
s t s s
Length of tangent s
Thus Unit Tan ent Vector
= =
=
2P P
t P
( ) ( )2 0 02
0
' 'lim .
s
s s sd
ds s
+
P PPP" =
( )0 0
1lim lim
0s s
s
ss s
= = =
P"
( )0s
s + s
=
=
P" n = t'
P = t t2s + st t'
2s + s = P t n
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Curvature of a Surface
( ) ( ) ( ) ( )( ),Lets find fo r curve t u t w t on curved surface t=
P u P u
2 2
,u w
d du u w w u uw u w wu w
dt dt
+
+ + + = + + + + +
u w
u w
u w uu uw u w w w u w
=
P PP = P P P P P P P P
,Taking inner product of unit normal with both the expressions we get
=
e
e. P
2 2s + s s =
0
e. t e. n n.e
( ) ( ) ( )
( ) ( ) ( )2 2u uw u w wu w= + + + + +
uu uw u w w w u w
0 0
e. P e.P e.P e.P e.P e.P e.P
[ ]
2
T
T
u ww
Equating two we get s
= =
=
uu uw
w u w w
. .uG u
e.P e.P
n.e uG u
Second Fundamental Matrix of Curved S=G urfaces
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Normal Curvature
Space curve u on curved surface P(u,w). urve : n ersec on o curve sur ace w p ane con a n ng
tangent vector and unit normal at P. Curvature of C is called normal curvature relative to (du/dt)A
. Normal curvature is projected length of curvature vector of
curve u to e .
2
,n
T
Lets be normal curvature
2
n
T T
n T = =
u G u u G u
s u u
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Normal Curvature
is calledPr i n c ip l e D ir e ct io n o f N o r m a l Cu r v a t u r e .
, , , ,Let u w L M N = = = = = = uu uw w u w we.P e.P e.P e.P
2 2 2 2
, . , .T
n T
T T
E F G Then can be written as= = = =
2 2
u u w w
uG uP P P P
uF u
n n
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Normal Curvature
0 0,n nTo find extreme values set and to get
= =
( ) ( ) ( ) ( )0, 0:
n n n nL E M F M F N G
Elimin ating and from th ese equations gives
+ = + =
( ) ( ) ( )2 2 2
max min
2 0
, &
n n
n n
EG F EN GL FM LN M
Roots of this quadratic equations are always real an
+ + =
d
.
are maximum and minimum values of normal curvature called
the principle curvatures
max min
max min
, n n
n n
Total Curvature or Gaussian Curvature K
= =
+
F
,2
=
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Objective
urvature
EstimationMethods forTriangular Meshes
curvature
curvaturedirections
Set of Test Cases
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Curvature Estimation Methods
Curvature Calculation Methods for Triangular
Discrete Estimation
Fitting Methods
CurvatureDirections
Curvature tensor
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Goal of Test Cases
MeshRegularity
Noise in Mesh
Accuracy
and Stabilityof CurvatureEstimation
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Introduction
Set o test cases to model mesh variations was
presented. Accuracy of curvature calculation methods for
triangular meshes was assessed based on noise in
the data, mesh resolution, regularity and valence. tat st ca ana ys s was nc u e to a ress
different aspects of curvature estimation error.
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Curvature Estimation for Triangular Meshes
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Basic Approaches
Sur ace ittin usin anal tic unction that its the
mesh locally. Discrete estimation of curvature and curvature
directions.
Estimation of Curvature tensor from whichcurvature an curvature rect ons can ecalculated.
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Fitting Methods
Fittin methods var de endin on t e o anal tic
function chosen for the fitting. Function can be parametric or implicit.
F t unct on separate y at eac vertex.
Picking a local coordinate frame is useful.
.
Minimum number of vertices are picked depending onnumber of coefficients of the function being fit.
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Fitting Methods
Local Coordinates Origin at target vertex i.e. vertex where the curvature is being
calculated. Normal at target vertex is one of the axis of the system.
Normal can be calculates directly from the analytic function used.
Average of unit normals of triangular faces that surround target
vertex.
Parameterization Re resent sur ace as unction o two arameters u v).
F(u,v) = [x(u,v), y(u,v), z(u,v)] Simplest representation-Height function
F(u v) = [u v (u v)]
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Fitting Methods
Re resentin curved sur ace: F(u,v = [u, v, (u,v)]
Parametric coordinates can be found by projectingthe vertices onto the tangent plane.
Quadric Fitting
Cubic fitting with normal
( ) 2 2, ,i i i i i i i i iz f u v Au Bu v Cv Du Ev F= = + + + + +
Main focus is to calculate curvature directions.
Implicit conic functions
0i i i i i i i i i i i i
ax by cz dx y ey z fz x gx hy jz k + + + + + + + + + =
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Quadratic Fitting Methods
Representing curved surface: F(u,v) = [u, v, f(u,v)]
If origin is located at target vertex constant term can be( )
2 2
, ,i i i i i i i i iz f u v Au Bu v Cv Du Ev F= = + + + + +
roppe .
Dropping the linear term forces the normal to line up with
- .
( )
)
0, 01
, 0, 0
u
u vv
fF F
Unit normal vector at origin e f
= =
( ) ( ) ( )( ) ( ) ( )
2
22 2
0, 0 0, 0 1 1
0, 0 0, 0 0, 0,
u v u v
uu vv uv
F F f f
f f fGaussian Curvature K
+ +
= =
G
, ,u v+ +
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Quadratic Fitting Methods
Quadratic surface passing through the origin
Requirement that it passes through 5 points can be expressed bylinear system:
2 2z Au Buv Cv Du Ev= + + + +
( ), , , 1, 2, ..5i i iU V Z i =
1
2 2
1 1 1 1 1 1 2
3... ... ... ... ...
ZA
U U V V U V Z B
ZC
=
2 2
5 5 5 5 5 5 4
5
U U V V U V Z D
ZE
t s system as un que so ut on, t e un t norma an curvature atorigin are given by2
1D
AC BE and
( )2 2 2 21 11D E D E+ + + +
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Discrete Methods
Avoid computational cost associated with fittingalgorithms.
o no nvo ve so v ng e eas square pro ems.
Only provide subset of gaussian, mean and.
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Discrete Methods
Spherical Image Method
Angle Excess Method An le De icit Method
Integral of Absolute Mean Curvature Method
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Spherical Image Method
Non planar polygon P .
Spherical image isapproximated by spherical
polygon ni.
Curvature at O,( )1i o i
i
Ar n n n
+
=
1i i
iAr P OP +
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Angle Deficit Method
Non planar polygon Pi.
Spherical image isapproximated by spherical
i,i+1.
Join points on spherical
ima e with arcs. Area of spherical polygon is
angle deficit of polygon,
Area related to target point O, 1i i +
, 13
i i
i
S +
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Angle Deficit Method
, 12
i i +
Gaussian Curvature at O,, 1
1
i
i i
KS +
= i
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Discrete Methods
Integral Formulation
Per Face Tensor Calculation
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Evaluations
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Evaluations
Test cases to hi hli ht both detailed behavior o
curvature estimation methods and statistical erroranalysis.
Detailed behavior test case defines meshparameters to distinguish between noise and
.
Statistical analysis test case creates meshes,
regular and irregular mesh regions.
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Test Cases
2 2 2: 4Sphere x y z+ + = 2 2: 2Elliptical Paraboloid z x y= +
2 2 2
:
: 13 2 4
y n er x z
x y zEllipsoid
+ =
+ + =
2 2: 0.4 -Hyperboloid z x y=
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Test Cases
2 2
3 2
: 2Elliptical Paraboloid z x y= +
( )3 2 2
. -
: 0.15 2 - 2
on ey a e z x xy
Cubic Polynomial z x x y xy y
=
= + +
[ ]0.1 cos( ) cos( )
Trignometric Function
z x y = +
22 -
0.1x y y
Exponential Function
z e+=
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Detailed Mesh Parameters
To assess local curvature at a oint on sur ace.
Project the planar mesh onto surface of study. Center the mesh at tar et oint.
Mesh is regular N-ring neighborhood, N = {1,2,3}
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Detailed Mesh Parameters
1. Number of vertices in first ring, n.
2.The scale,
.
3. Target vertex displacement, dRT.
4. Adjacent vertex displacement, dRA.
5. Target vertex displacement, d
T,owar s a acen ver ex.
6. Adjacent vertex displacement, dA,.
7. Adjacent vertex displacement, d,towards a neighboring adjacent vertex.
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Statistical Analysis Case
Create a mesh1. 72 interior vertices (112 total),
2. Valence ranging from three to ten, and
3. onta n ng ot o tuse an non-o tuse tr ang es.
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Experimental Results
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Curvature Estimation Based on Fitting
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Curvature Estimation Based on Fitting
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Curvature Estimation Based on Fitting
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Curvature Estimation Using Discrete Methods
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Curvature Estimation Using Discrete Methods
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Curvature Estimation Using Discrete Methods
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Curvature Estimation Using Curvature Tensor
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Curvature Estimation Using Curvature Tensor
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Curvature Estimation Using Curvature Tensor
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Statistical Analysis Results
Considering both Gaussian and Mean Curvature,most accurate methods:1. The cubic fit with exact normal
-. ,
3. Two-ring conic fit,
4. Two-ring quadric natural fit, and
5. u c t w t ca cu ate norma .
-
curvature magnitude, where as two ring conic fitpredictions are larger.
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Discussion
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Discussion
Accuracy of conic fitting methods is very dependent ontype of surface being fit.
Two ring fitting methods are superior than one ringme o s.
One ring methods are highly noise sensitive.
,using weighted face average in quite sensitive.
Princi al curvature directions are more stable and less
sensitive to mesh regularity than curvature magnitude. Fitting methods based on two ring neighborhood is
recommen e t an t ree r ng, to avo ncrease cost.
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Thank You