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Smooth Two-Dimensional Interpolations: A Recipe for All Polygons

January 2005
Journal of Graphics GPU and Game Tools 10(2):27-39

January 2005
10(2):27-39

DOI:10.1080/2151237X.2005.10129192

Source
DBLP

Authors:

Elisabeth Anna Malsch

Thornton Tomasetti

Gautam Dasgupta

Columbia University

This paper presents a method for constructing smooth and bounded interpolations on any polygon, whether convex or concave in shape, or even one containing holes or isolated points in its interior. The resulting two-dimensional function distributes the value at any given vertex or internal node over the remaining portion of the domain. The representation depends only on simple geometrical properties such as lengths and areas. Accordingly, it is invariant with respect to any chosen coordinate system. The resulting set of interpolations is smooth within the domain. Within a triangle, the behavior is akin to a linear color gradient. If necessary, linear boundary behavior can also be assured. A Java implementation is available online.

Interpolants on a triangle, rectangle, skew quadrilateral, and pentagon.

…

Interior points.

…

Interpolants on concave polygons do not require tesselation.

…

Interpolants that are smooth and linear on the sides of a concave polygon.

…

Linearity on the sides.

…

Figures - uploaded by Gautam Dasgupta

Content may be subject to copyright.

Content uploaded by Gautam Dasgupta

Content may be subject to copyright.

The

Mathematica®

Journal

Algebraic Construction

of Smooth Interpolants

on Polygonal Domains

Elisabeth Anna Malsch

Gautam Dasgupta

A smooth and bounded interpolant can be constructed in explicit algebraic

form within any polygon, convex or concave. The resulting function is not

unique and accordingly can be adjusted to satisfy desired global conditions,

such as linear fields. The closed-form representation is obtained by com-

bining simple geometric descriptions, such as the side lengths and areas.

The interpolant distributes values given at discrete nodes smoothly over

the interior of the domain. On a convex polygon, the interpolant is a

rational function of the product of areas. On a concave or multiply con-

nected polygon, the interpolant is a function of areas and edge lengths,

which introduces a square root term.

‡

Introduction

There are many varied applications for a smooth and bounded interpolant. The

most straightforward is for computer graphics, for example in rendering and

coloring applications [1]. Smoothly distributing color defined at nodes over a

domain requires a bounded interpolating function. Interpolants can also be

employed as test functions for computational modeling [2]. For the finite ele-

ment method, they act as shape functions [3, 4]. Similar interpolants have been

applied to the analysis of biomedical growth and shape change [5].

Conventional methods for constructing interpolants do not distribute boundary

values smoothly over arbitrary polygonal shapes. Most either require a mesh or

apply only to very simple geometries [4, 6]. For example, the conventional finite

element development does not specifically address concavity; instead, concave

domains are tessellated into convex parts. An exception is the R-function construc-

tion that applies to any polygonal domain [7]. Unlike the proposed formulation,

it can only represent homogeneous boundary conditions.

In general, any function

uHx,yL

can be approximated as the sum of the products

of the value

at given nodes

Hxi,yiL

and an interpolant

NiHx,yL

(1)

uHx,yL>u

`Hx,yL=‚

i=1

uiNiHx,yL.

The derived interpolants

NiHx,yL

are linearly independent:

(2)

NiHxj,yjL= di,j.

Along any given boundary, only the values given at the end nodes prescribe the

behavior along the line segment:

(3)

`Ht xi+H1-tLxj,t yi+H1-tLyjL=fHui,uj,tL.

Within the domain, the interpolants are bounded, normalized between zero and

one, and smooth. The interpolant generated is not unique. Any function that

satisfies an elliptic operator and the boundary conditions satisfies the interpolant

requirements.

Using the same method to construct general bounded interpolants on any

polygon, linear interpolants are constructed. A linear interpolant satisfies the

additional requirements of boundary linearity:

(4)

`Ht xi+H1-tLxj,t yi+H1-tLyjL=t ui+H1-tLuj,

and exact reproduction of linear fields:

(5)

x=‚

i=1

xiNiHx,yL and y=‚

i=1

yiNiHx,yL.

The constant field is satisfied by construction:

(6)

1=‚

i=1

NiHx,yL.

In most cases, the linearity requirements do not prescribe a unique interpolant.

Therefore additional global conditions, such as higher-order fields, could also be

imposed.

‡

Input Data

The interpolant can be constructed on any two-dimensional slice of any poly-

tope, including polygons, polyhedra, or higher-dimensional objects. Accordingly,

the dimension of the data must be defined. The examples are in two dimensions.

In[1]:=

xy =8x, y<;

The list of vertices must also be defined; for example, here is a triangle with

three nodes.

In[2]:=

vertexConnect =881, 2, 3<<;

642 Elisabeth Anna Malsch and Gautam Dasgupta

Another example is a triangle enclosing a quadrilateral and an interior point.

In[3]:=

vertexConnect =881, 2, 3<,84<,88, 7, 6, 5<<;

For a simple interpolant, the quantity that is going to be distributed smoothly

over the domain must be defined at every nodal point. The nodal points not only

determine the structure of the domain, they are also the points at which the value

being distributed is given. The vertices of the polygon are a subset of the nodal

points. Nodal points and interpolation points are one and the same.

‡

Geometric Measures and Support Functions

It is useful to construct the interpolant with respect to geometric measures as

opposed to nodal locations. The benefit of such an approach is that it is invariant

of coordinate system and dimension.

The Distance Function

The distance between two points

and

with coordinates

Hx1,y1L

and

Hx2,y2L

is:

(7)

»p1 p2»=l12 ="##############################################

Hx1-x2L2+Hy1-y2L2.

The function can be written as the norm of the vector connecting the points,

norm[

p1-p2

]. The built-in Norm function would be appropriate if the points

were represented as complex numbers rather than as pairs of real numbers.

In[4]:=

norm@a_D:=Sqrt@a.aD

The Signed Area of a Triangle

The area of the triangle with vertices

with coordinates

Hx1,y1L

Hx2,y2L

Hx3,y3L

can be defined in terms of the determinant:

(8)

A= D@p1,p2,p3D=1ê2

ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ

x1y11

x2y21

x3y31

ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ

The area is signed:

D@p1,p2,p3D= -D@p2,p1,p3D

In[5]:=

area@p_, q_, r_D:=

1ê2 Det@Join@Transpose@8p, q, r<D,881, 1, 1<<DD ê;

Length@pDãLength@qDãLength@rDã2

Algebraic Construction of Smooth Interpolants on Polygonal Domains 643

The Unsigned Area Function

The unsigned area of a triangle can be written in terms of side lengths using

Heron’s formula:

(9)

A=è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

sHs-aLHs-bLHs-cL,

where

(10)

s=1

ÅÅÅÅÅÅ

2Ha+b+cL.

The resulting area is necessarily nonnegative. The area measure has no physical

meaning for a node located on the edge of the domain. In such a case, defining

the unsigned area to be one when the signed area is zero allows the concave

element shape function routines to apply to elements with side nodes.

In[6]:=

unsignedArea@p_, q_, r_D:=Module@8a, b, c, s<,

a=norm@p-qD; b =norm@q-rD; c =norm@r-pD; s =1ê2 Ha+b+cL;

Sqrt@sHs-aL Hs-bL Hs-cLDD

In[7]:=

unsignedArea@p_, q_, r_D:=1ê; area@p, q, rDã0

The Angle Functions

The cosine or sine of an angle defined by three vertices can be determined using

line lengths and the unsigned area. Care is taken to choose the proper sign for

the cosine and sine functions. For the angle b defined by the nodes

with

the angle at

(11)

cosHbL=†pq§2+†qr§2-†r p§2

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

2†pq§†qr§,

(12)

sinHbL=2†D@p,q,rD§

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

†pq§†qr§.

The angle b never has to be evaluated.

In[8]:=

cosAng@p_, q_, r_D:=Module@8pq, qr, rp<, pq =norm@p-qD;

qr =norm@q-rD; rp =norm@r-pD;Hpq^2 +qr^2 -rp^2LêH2 pq qrLD

In[9]:=

sinAng@p_, q_, r_D:=2 unsignedArea@p, q, rDêHnorm@p-qD norm@q-rDL

‡

Minimum Functions

The geometric definitions can be used to define minimum functions. For exam-

ple, the distance from node p to node q, |pq|, is zero and minimized when

p=q

and the area, defined by the three nodes p, q, r, is zero and minimized when

on the line

. The R-function formulation is similarly dependent on minimum

functions [7].

644 Elisabeth Anna Malsch and Gautam Dasgupta

Define a function of the variable point

zHx,yL=8x,y<

that is strictly zero along

the line qr:

(13)

†zHx,yL - q§+†zHx,yL-r§-†q-r§.

The equation applies in any dimension.

In[10]:=

perim@xy_, q_, r_D:=norm@xy -qD+norm@xy -rD-norm@q-rD

In[11]:=

Show@#, DisplayFunction Ø$DisplayFunction,

ImageSize Ø8200, Automatic<, PlotRange ØAllD&êü

Thread@8ContourPlot@#@8x, y<,8-1ê2, -1ê2<,81ê2, 1 ê2<D,

8x, -1, 1<,8y, -1, 1<, DisplayFunction ØIdentity,

PlotPoints Ø33, PlotRange Ø80, 1<D &êü

8norm@#1 -80, 0<D &, area, unsignedArea, perim<,

8Graphics@8PointSize@.02D, RGBColor@0, 0, 1D, Point@80, 0<D<D,

Graphics@8PointSize@.02D, RGBColor@0, 0, 1D,

Line@88-1, -1<,81, 1<<D<D, Graphics@

8PointSize@.02D, RGBColor@0, 0, 1D, Line@88-1, -1<,81, 1<<D<D,

Graphics@8PointSize@.02D, RGBColor@0, 0, 1D,

Line@88-1ê2, -1ê2<,81ê2, 1 ê2<<D<D<<D

From In[11]:=

-1-0.5 0 0.5 1

-1

-0.5

0.5

-1-0.5 0 0.5 1

-1

-0.5

0.5

-1-0.5 0 0.5 1

-1

-0.5

0.5

-1-0.5 0 0.5 1

-1

-0.5

0.5

Figure 1. Contours that are minimum at a point, along a line, and along a line segment.

Algebraic Construction of Smooth Interpolants on Polygonal Domains 645

Linearity

The perimeter function can be modified so that the contour lines are linear along

a broken line segment. Four nodal points are required: say,

. Construct a

function that is zero along the path

and linear along the paths

and

, with

angles

a = —opq

and

b = —pqr

. Again define the variable point

zHx,yL=8x,y<

(14)

AHx,yL=†pq§2-2 cosHbL†qzHx,yL§†pq§-†pzHx,yL§2+†qzHx,yL§2,

(15)

BHx,yL=†pq§2-2 cosHaL†pzHx,yL§†pq§+†pzHx,yL§2-†qzHx,yL§2.

The functions

c1Hx,yL

c2Hx,yL

, and

c3Hx,yL

are defined in terms of

AHx,yL

and

BHx,yL

(16)

c1Hx,yL=1

ÅÅÅÅÅÅ

2HAHx,yLsinHaL+BHx,yLsinHbLL†pq§,

(17)

c2Hx,yL=AHx,yL+BHx,yL,

(18)

c3Hx,yL=HAHx,yLcosHaL+BHx,yLcosHbLL †pq§.

Here we combine all the pieces:

(19)

c1Hx,yLH†pzHx,yL§ +†zHx,yL q§L2-†pq§2

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

c2Hx,yLH†pzHx,yL§ +†zHx,yL q§L-c3Hx,yL.

In[12]:=

perimLin@xy_, o_, p_, q_, r_D:=

Module@8A, B, pq, xyq, pxy, c1, c2, c3<,

pq =norm@p-qD;

xyq =norm@xy -qD;

pxy =norm@p-xyD;

A=pq^2 -2 cosAng@p, q, rD xyq pq -pxy ^2 +xyq^2;

B=pq^2 -2 cosAng@o, p, qD pxy pq +pxy ^2 -xyq^2;

c1 =1ê2HA sinAng@o, p, qD+B sinAng@p, q, rDL pq;

c2 =A+B;

c3 =HA cosAng@o, p, qD+B cosAng@p, q, rDL pq;

c1 HHxyq +pxyL^2 -pq ^2L ê HHxyq +pxyL c2 -c3LD

646 Elisabeth Anna Malsch and Gautam Dasgupta

In[13]:=

Show@8ContourPlot@Evaluate@

perimLin@8x, y<, #@@1DD,8-1ê2, 0<,81ê2, 0<, #@@2DDDD,

8x, -1, 1<,8y, -1, 1<, DisplayFunction ØIdentity,

PlotPoints Ø40D, Graphics@8RGBColor@1, 0, 0D,

Line@8#@@1DD,8-1ê2, 0<<D, Line@881ê2, 0<, #@@2DD<D,

RGBColor@0, 0, 1D, Line@88-1ê2, 0<,81ê2, 0<<D<D<,

DisplayFunction Ø$DisplayFunction, ImageSize Ø8200, Automatic<,

PlotRange Ø88-1, 1<,8-1, 1<<D &êü

888-1, -2<,81, 1 ê2<<,880, 1<,81ê10, -1ê10<<<

From In[13]:=

-0.75-0.5-0.25 0 0.25 0.5 0.75 1

-0.75

-0.5

-0.25

0.25

0.5

0.75

-0.75-0.5-0.25 0 0.25 0.5 0.75 1

-0.75

-0.5

-0.25

0.25

0.5

0.75

Figure 2. Contours of a function that are minimized along a line segment (in blue) and

linear along two adjacent broken line segments (in red).

‡

Functions for Displaying Interpolants

In[14]:=

outline@pts_D:=Line@Append@#, #@@1DDD &@ptsDD

name@pts_D:=

Table@Text@ToString@iD, pts@@iDD,8-2, 0<, Background ØHue@.2D,

TextStyle Ø8FontWeight Ø"Bold"<D,8i, Length@ptsD<D

nodes@pts_D:=Point êüpts

mask@pts__D:=

Polygon@Flatten@Append@#, #@@1DDD &êü888-.1, -.1<,8-.1, .1<,

8.1, .1<,8.1, -.1<< +H8#@@1DD, #@@2DD, #@@4DD, #@@3DD< &@

Flatten@Outer@List, ##D&üü

H8Min@#D, Max@#D< &êüThread@ptsDL, 1DDL, pts<, 1DD

output@i_, colorfunction_, f_, points_, cnct_, j_: 1D:=

Module@8fc, pContour, pts<,8

pts =Part@points, cnct@@1DDD;

fc =Compile@88x, _Real<,8y, _Real<<,

Evaluate@f@8x, y<, points, cnct, jDDD;

pContour =ContourPlot@Check@fc@x, yD@@iDD,

0., CompiledFunction::"cfn"D,

8x, Min@Thread@pointsD@@1DDD, Max@Thread@pointsD@@1DDD<,

8y, Min@Thread@pointsD@@2DDD, Max@Thread@pointsD@@2DDD<,

Algebraic Construction of Smooth Interpolants on Polygonal Domains 647

In[14]:=

DisplayFunction ØIdentity, H*ContourShadingØFalse*L

ColorFunction Øcolorfunction, Contours Ø20,

PlotPoints Ø60, PlotRange Ø80, 1<D; Show@pContour,

Graphics@Flatten@88RGBColor@1, 1, 1D<,8mask@ptsD<,

Polygon êüHpoints@@#DD &êüDrop@cnct, 1DL<DD,

Graphics@Flatten@88PointSize@.02D<,8outline êü

Hpoints@@#DD &êücnctL, name@pointsD, nodes@pointsD<<DD,

PlotRange ØAll, Frame ØFalse, AspectRatio Ø1D<D

‡

Interpolants for Convex Polygons

The interpolant is constructed as a combination of the geometric measures.

In[19]:=

convex@8x_, y_<, points_, cnct_, i_D:=

Module@8pts =Part@points, cnct@@iDDD,

areaTog, n, varArea, zeroArea<, n =Length@ptsD;

areaTog =Apply@area, Partition@pts, 3, 1, 82<D,81<D;

varArea =

Apply@area@##, 8x, y<D &, Partition@pts, 2, 1, 81<D,81<D;

zeroArea =Apply@Times, Partition@varArea,

Length@ptsD-2, 1, 8Length@ptsD<D,81<D;

HzeroArea *areaTogLêHzeroArea.areaTogLD

Interpolants are rational functions. For example, here is the set of functions for a

triangle, rectangle, skew quadrilateral, and pentagon.

In[20]:=

convex@8x, y<,880, 0<,81, 0<,81, 1<<,881, 2, 3<<, 1D êê Simplify

Out[20]=

81-x, x -y, y<

In[21]:=

convex@8x, y<,880, 0<,81, 0<,81, 1 ê2<,80, 1 ê2<<,

881, 2, 3, 4<<, 1D êê Simplify

Out[21]=

8H-1+xLH-1+2 yL, x -2 x y, 2 x y, -2H-1+xLy<

In[22]:=

convex@8x, y<,880, 0<,81, 0<,83ê2, 1<,8-1ê2, 1 ê4<<,

881, 2, 3, 4<<, 1D êê Simplify

Out[22]=

9-H7+6 x -16 yL H-2+2 x -yL

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

14 +12 x +59 y ,

2H7+6 x -16 yL Hx+2 yL

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

14 +12 x +59 y ,26 y Hx+2 yL

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

14 +12 x +59 y ,-28 H-2+2 x -yLy

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

14 +12 x +59 y =

In[23]:=

convex@8x, y<,880, 0<,81, 0<,83ê2, 1 ê2<,

8-1ê4, 1<,8-1ê2, 1 ê4<<,8Range@5D<, 1D êê Simplify

648 Elisabeth Anna Malsch and Gautam Dasgupta

Out[23]=

9H7+12 x -4 yL H-1+x-yLH-13 +4 x +14 yL

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

91 -48 x2+487 y +198 y2+2xH64 +33 yL,

-H7+12 x -4 yL Hx+2 yLH-13 +4 x +14 yL

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

91 -48 x2+487 y +198 y2+2xH64 +33 yL,

18 H7+12 x -4 yLyHx+2 yL

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

91 -48 x2+487 y +198 y2+2 x H64 +33 yL,

-184 H-1+x-yLyHx+2 yL

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

91 -48 x2+487 y +198 y2+2xH64 +33 yL,

28 H-1+x-yLyH-13 +4 x +14 yL

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

91 -48 x2+487 y +198 y2+2 x H64 +33 yL=

In[24]:=

Show@GraphicsArray@

Partition@Graphics@#@@1, 1DD, PlotRange ØAll, AspectRatio Ø1D&êü

8output@1, HRGBColor@1, H1-#L,H1-#LD &L,

convex, 880, 0<,81, 0<,81, 1<<,881, 2, 3<<D,

output@1, HRGBColor@H1-#L, 1, H1-#LD &L, convex,

880, 0<,81, 0<,81, 1 ê2<,80, 1 ê2<<,881, 2, 3, 4<<D,

output@1, HRGBColor@H1-#L,H1-#L, 1D&L, convex,

880, 0<,81, 0<,83ê2, 1<,8-1ê2, 1 ê4<<,881, 2, 3, 4<<D,

output@1, HRGBColor@1, H1-#L, 1D&L, convex, 880, 0<,81, 0<,

83ê2, 1 ê2<,8-1ê4, 1<,8-1ê2, 1 ê4<<,8Range@5D<D<,

2DD, ImageSize Ø8350, Automatic<D

From In[24]:=

1 2

Figure 3. Interpolants on a triangle, rectangle, skew quadrilateral, and pentagon.

Algebraic Construction of Smooth Interpolants on Polygonal Domains 649

‡

Interior Points

Points inside the domain are dealt with using the distance function.

In[25]:=

interior@8x_, y_<, points_, cnct_, 1D:=

Module@8pts =points@@Flatten@Select@cnct, HLength@#Dã1 &LDDDD<,

zeroDist =Apply@Times, Partition@Hnorm@8x, y<-#D&êüptsL,

Length@ptsD-1, 1, 8Length@ptsD<D,81<D;

zeroDist ê HPlus üü zeroDistLD

In[26]:=

points =88-.1, -.1<,81.1, -.1<,

81.1, 1.1<,8-.1, 1.1<,80, 0<,81, 1<,81, 0<<;

cnct =881, 2, 3, 4<,85<,86<,87<<;

interior@8x, y<, points, cnct, 1D@@1DD

Show@output@1, HRGBColor@1, 1, H1-#LD &L, interior, points, cnctD,

DisplayFunction Ø$DisplayFunctionD

Out[28]=

J"###########################################

H-1+xL2+H-1+yL2"#############################

H-1+xL2+y2Ní

J"###########################################

H-1+xL2+H-1+yL2"#############################

H-1+xL2+y2+

"###########################################

H-1+xL2+H-1+yL2è!!!!!!!!!!!!!!

x2+y2+"#############################

H-1+xL2+y2è!!!!!!!!!!!!!!

x2+y2N

From In[26]:=

1 2

Figure 4. Interior points.

‡

Concave Polygons

For concave polygons, the zeroArea[a,b,c] function is replaced with the

zeroPerim[a,b,c] function.

In[30]:=

varPerim@xy_, points_, cnct_, i_D:=

Module@8pts =Part@points, cnct@@iDDD<,

Apply@perim@xy, ##D&, Partition@pts, 2, 1, 81<D,81<DD;

650 Elisabeth Anna Malsch and Gautam Dasgupta

In[31]:=

concave@8x_, y_<, points_, cnct_, i_D:=

Module@8pts =Part@points, cnct@@iDDD, n, zeroPerim<,

n=Length@ptsD;

zeroPerim =Apply@Times,

Partition@varPerim@8x, y<, pts, cnct, iD, n -2, 1, 8n<D,81<D;

HzeroPerimL ê Apply@Plus, zeroPerimDD

The shape functions for a concave quadrilateral can easily be constructed. The

interpolant contains a square root. The branch points of the square root terms

add a discontinuity in the gradient of the interpolant. This discontinuity is

necessary for the modeling of re-entrant corners.

In[32]:=

points =88-1, -1<,80, 0<,81, -1<,80, 1<<;

cnct =881, 2, 3, 4<<;

concave@8x, y<, points, cnct, 1D êê Simplify

Out[34]=

9JJ-è!!!

5+"#############################

x2+H-1+yL2+"#########################################

H-1+xL2+H1+yL2N

J-è!!!

2+è!!!!!!!!!!!!!!

x2+y2+"#########################################

H-1+xL2+H1+yL2NN ì

J-è!!!

2-è!!!

5+"#############################

x2+H-1+yL2+è!!!!!!!!!!!!!!

x2+y2+

"#########################################

H-1+xL2+H1+yL2+"######################################

H1+xL2+H1+yL2N2

JJ-è!!!

5+"#############################

x2+H-1+yL2+"#########################################

H-1+xL2+H1+yL2N

J-è!!!

5+"#############################

x2+H-1+yL2+"######################################

H1+xL2+H1+yL2NN ì

J-è!!!

2-è!!!

5+"#############################

x2+H-1+yL2+è!!!!!!!!!!!!!!

x2+y2+

"#########################################

H-1+xL2+H1+yL2+"######################################

H1+xL2+H1+yL2N2

JJ-è!!!

5+"#############################

x2+H-1+yL2+"######################################

H1+xL2+H1+yL2N

J-è!!!

2+è!!!!!!!!!!!!!!

x2+y2+"######################################

H1+xL2+H1+yL2NN ì

J-è!!!

2-è!!!

5+"#############################

x2+H-1+yL2+è!!!!!!!!!!!!!!

x2+y2+

"#########################################

H-1+xL2+H1+yL2+"######################################

H1+xL2+H1+yL2N2

JJ-è!!!

2+è!!!!!!!!!!!!!!

x2+y2+"#########################################

H-1+xL2+H1+yL2N

J-è!!!

2+è!!!!!!!!!!!!!!

x2+y2+"######################################

H1+xL2+H1+yL2NN ì

J-è!!!

2-è!!!

5+"#############################

x2+H-1+yL2+è!!!!!!!!!!!!!!

x2+y2+

"#########################################

H-1+xL2+H1+yL2+"######################################

H1+xL2+H1+yL2N2=

Algebraic Construction of Smooth Interpolants on Polygonal Domains 651

In[35]:=

Show@GraphicsArray@

Graphics@#@@1, 1DD, PlotRange ØAll, AspectRatio Ø1D&êü

8output@1, HRGBColor@1, H1-#L,H1-#LD &L, concave,

88-1, -1<,80, 0<,81, -1<,80, 1<<,881, 2, 3, 4<<D,

output@1, HRGBColor@1, 1, H1-#LD &L, concave,

880, 1 ê2<,81, 0<,83ê2, 1 ê2<,8-1ê4, 1<,8-1ê2, 1 ê4<<,

881, 2, 3, 4, 5<<D<D, ImageSize Ø8350, Automatic<D

From In[35]:=

Figure 5. Interpolants on concave polygons do not require tesselation.

‡

Concave Polygons with Linearity on the Sides

The perim[a,b,c] function is replaced with perimLin[d,a,b,c,e]. Similarly the

varPerim[x,y] function is modified so that four vertices define the behavior of

the minimum along a line. The constructed interpolants on the quadrilateral and

pentagon are linear on the sides.

In[36]:=

concaveLin@8x_, y_<, points_, cnct_, i_D:=

Module@8pts =Part@points, cnct@@iDDD, zeroPLin, varPLin, areaTog<,

areaTog =Apply@unsignedArea, Partition@pts, 3, 1, 82<D,81<D;

n=Length@ptsD;

varPLin =

Apply@perimLin@8x, y<, ##D&, Partition@pts, 4, 1, 82<D,81<D;

zeroPLin =Apply@Times, Partition@varPLin,

Length@ptsD-2, 1, 8Length@ptsD<D,81<D;

HzeroPLin *areaTogL ê HareaTog.zeroPLinLD

The interpolants for a quadrilateral can be constructed as follows.

652 Elisabeth Anna Malsch and Gautam Dasgupta

In[37]:=

Show@GraphicsArray@

Graphics@#@@1, 1DD, PlotRange ØAll, AspectRatio Ø1D&êü

8output@1, HRGBColor@H1-#L,H1-#L, 1D&L, concaveLin,

88-1, -1<,80, 0<,81, -1<,80, 1<<,881, 2, 3, 4<<D,

output@1, HRGBColor@H1-#L, 1, H1-#LD &L, concaveLin,

880, 1 ê2<,81, 0<,83ê2, 1 ê2<,8-1ê4, 1<,8-1ê2, 1 ê4<<,

881, 2, 3, 4, 5<<D<D, ImageSize Ø8350, Automatic<D

From In[37]:=

Figure 6. Interpolants that are smooth and linear on the sides of a concave polygon.

The formulation for a concave element applies to any shape. For linearity on a

side the area function must be modified.

In[38]:=

unsignedArea@a_, b_, c_D:=1ê; Abs@area@a, b, cDD <10^ -6

Otherwise, the side node would cause singular behavior along the boundary.

In[39]:=

points =88-1, -1<,8-1ê3, -1<,

80, -1<,81ê5, -1<,81, -1<,81ê2, 0<,80, 1<<;

cnct =8Range@Length@pointsDD<;

Plot@Evaluate@

concaveLin@t points@@1DD +H1-tL points@@4DD, points, cnct, 1DD,

8t, 0, 1<, PlotRange ØAllD

From In[39]:=

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

Figure 7. Linearity on the sides.

This works even with multiple side nodes.

Algebraic Construction of Smooth Interpolants on Polygonal Domains 653

In[42]:=

Show@GraphicsArray@

Graphics@#@@1, 1DD, PlotRange ØAll, AspectRatio Ø1D&êü

8output@3, HRGBColor@H1-#L, 1, 1D&L, concaveLin,

88-1, -1<,80, -1<,81, -1<,80, 1<<,881, 2, 3, 4<<D,

output@5, HRGBColor@1, H1-#L, 1D&L, concaveLin,

88-1, -1<,8-1ê3, -1<,80, -1<,81ê5, -1<,81, -1<,81ê2, 0<,

80, 1<<,8Range@7D<D<D, ImageSize Ø8350, Automatic<D

From In[42]:=

123

1 2 3 4 5

Figure 8. Side node interpolants that are smooth and linear on the sides.

‡

Combining Parts

The boundary functions and interior point behaviors can be combined to

describe shapes with holes or interior points. Shape functions that apply to

separate boundaries and points can be combined.

In[43]:=

combPrt@8x_, y_<, points_, cnct_, convxH_D:=

Module@8funs, zero, xy =8x, y<<, funs =Table@Switch@cnct@@iDD,

a_?H#ãconvxH &L, convex@xy, points, cnct, iD,

a_?HLength@#D∫1 &L, concaveLin@xy, points, cnct, iD,

_, 81<D,8i, Length@cnctD<D;

zero =Apply@Times, Partition@Table@Apply@Times,

HIf@Length@cnct@@iDDD ∫1, varPerim@xy, points, cnct, iD,

norm@xy -points@@cnct@@iDDDD@@1DDDDLD,8i,

Length@cnctD<D, Length@cnctD-1, 1, 8Length@cnctD<D,81<D;

Flatten@Hfuns *zeroL ê Apply@Plus, Flatten@funs zeroDDDD

For example, construct a set of interpolants on a multiply connected domain that

contains a triangle and two interior points.

654 Elisabeth Anna Malsch and Gautam Dasgupta

In[44]:=

Show@GraphicsArray@Partition@

Graphics@#@@1, 1DD, PlotRange ØAll, AspectRatio Ø1D&êü8

output@1, HRGBColor@1, 2 ê3 H1-#L,H1-#LD &L, combPrt,

88-1, -1<,81, -1<,81, 1<,8-1, 1<,8-1ê2, -1ê2<,81ê2,

-1ê2<,80, 1 ê2<<,881, 2, 3, 4<,85, 6, 7<<,81, 2, 3, 4<D,

output@8, HRGBColor@7ê8 H1-#L, 1, H1-#LD &L, combPrt,

88-1, -1<,81, -1<,81, 1<,8-1, 1<,8-1ê2, -1ê2<,

81ê2, -1ê2<,80, 1 ê2<,81ê2, 1 ê2<,8-1ê2, 1 ê2<<,

881, 2, 3, 4<,85, 6, 7<,88<,89<<,81, 2, 3, 4<D,

output@8, HRGBColor@H1-#L, 1, H2ê3L H1-#LD &L,

combPrt, 88-1, -1<,80, -4ê5<,81, -1<,81, 1<,8-1, 1<,

8-1ê2, -1ê2<,80, 0<,81ê2, -1ê2<,80, 1 ê2<<,

881, 2, 3, 4, 5<,86, 7, 8, 9<<,81, 2, 3, 4<D,

output@7, HRGBColor@7ê8 H1-#L, 5 ê6 H1-#L, 1D&L, combPrt,

88-1, -1<,81, -1<,81, 1<,8-1, 1<,8-3ê4, -3ê4<,

8-1ê4, -3ê4<,8-1ê4, -1ê4<,8-3ê4, -1ê4<,

83ê4, 3 ê4<,81ê4, 3 ê4<,81ê4, 1 ê4<,83ê4, 1 ê4<<,

881, 2, 3, 4<,85, 6, 7, 8<,89, 10, 11, 12<<,81, 2, 3, 4<D<,

2DD, ImageSize Ø8350, Automatic<D

From In[44]:=

1 2

5 6

910

11 12

1 2

5 6

1 2

5 6

7 89

Figure 9. Interpolants that are smooth and linear on the sides of multiply connected

polygons.

Algebraic Construction of Smooth Interpolants on Polygonal Domains 655

‡

Enforcing Global Linearity

If some of the interpolating functions do not satisfy the same linearity conditions

that the convex shape functions satisfy, they can be altered. Given the entire set

of interpolating functions

NjHx,yL

corresponding to the

nodal points

Hxj,yjL

and the set of functions

RiHx,yL

that correspond to the points

Hxi,yiL

on the

convex hull, the following requirement can be imposed:

(20)

‚

j=1

aHxj,yjLNjHx,yL=aHx,yL.

When the function

aHx,yL

is either 1,

, or

, the convex interpolant satisfies the

requirement exactly over the

convex hull points:

(21)

‚

i=1

aHxi,yiLRiHx,yL=aHx,yL.

The functions

NiHx,yL

can be modified to ensure linearity:

(22)

NiHx,yL=RiHx,yL-‚

k=1

RiHxj,yjLNjHx,yL.

When the points

Hxj,yjL

are not the convex hull points and

p=n-m

, bounded-

ness is not necessarily preserved.

In[45]:=

makeLin@8x_, y_<, points_, cnct_, cnvxH_D:=Module@8nonLin, lin<,

8nonLin =combPrt@8x, y<, points, cnct, cnvxHD;

lin =convex@8x, y<, points, 8cnvxH<, 1D-

HReplacePart@combPrt@8x, y<, points, cnct, cnvxHD, 0, Map@List,

cnvxHDD.HMap@convex@#, points, 8cnvxH<, 1D&, pointsDLL;

Fold@ReplacePart@#1, lin@@#2DD, #2D&, nonLin,

Range@Length@cnvxHDDD<@@1DDD

For example, construct linearized interpolations on a quadrilateral.

656 Elisabeth Anna Malsch and Gautam Dasgupta

In[46]:=

Show@GraphicsArray@

Graphics@#@@1, 1DD, PlotRange ØAll, AspectRatio Ø1D&êü

8output@1, HRGBColor@1, H1-#L,H1-#LD &L, makeLin,

88-1, -1<,80, 0<,81, -1<,80, 1<<,881, 2, 3, 4<<D,

output@1, HRGBColor@H1-#L,H1-#L, 1D&L, makeLin,

880, 1 ê2<,81, 0<,83ê2, 1 ê2<,8-1ê4, 1<,8-1ê2, 1 ê4<<,

8Range@5D<D<D, ImageSize Ø8350, Automatic<D

From In[46]:=

Figure 10. Linearized interpolants.

‡

Conclusion

The resulting interpolants are smooth and bounded within their respective

domains and the sum of any set of interpolants is one. The functions are linearly

independent. Examples of the smooth and bounded behavior of the functions are

shown in the figures. The convex polygon shape functions automatically satisfy

constant and linear fields. This is not the case for the concave or multiply con-

nected domain representation. Nevertheless, the linearity constraint can be

imposed on the representation.

This type of interpolant is distinctly different from the available interpolant and

shape function formulations, since each interpolant is constructed to satisfy the

required conditions exactly: smoothness, boundedness, linearity on sides, and the

linear field conditions. Using this algebraic construction, closed-form interpo-

lants satisfying different field properties can also be derived.

‡

Acknowledgment

This research was supported by a National Science Foundation grant

(CMS:0202232). We are grateful to the reviewers for their careful reading and

many helpful comments.

Algebraic Construction of Smooth Interpolants on Polygonal Domains 657

‡

References

[1]

J. D. Foley, A. van Dam, S. K. Feiner, and J. F. Hughes, Computer Graphics Principles

and Practice, 2nd ed., New York: Addison-Wesley, 1996.

[2]

E. L. Wachspress, A Rational Finite Element Basis, New York: Academic Press, 1975.

[3]

R. Courant, “Variational Methods for the Solution of Problems of Equilibrium and

Vibrations,” Bulletin of the American Mathematical Society, 49, 1943 pp. 1–43.

[4]

O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method, 4th ed., New York:

McGraw-Hill, 1989.

[5]

G. Dasgupta and J. Treil, “Maxillo-Facial Frame: Finite Element Shapes,” The Mathemat-

ica Journal, 8(2), 2001 pp. 235–246.

[6]

P. J. Davis, Interpolation and Approximation, New York: Dover Publications, 1963.

[7]

V. L. Rvachev, T. I. Sheiko, V. Shapiro, and I. Tsukanov, “On Completeness of RFM

Solution Structures,” Computational Mechanics, 25, 2000 pp. 305–316.

About the Authors

Elisabeth Anna Malsch completed her B.S. (1999), M.S. (2000), and Ph.D. (2003)

from Columbia University, New York. Currently she is an Alexander von Humboldt

fellow, Bonn, Germany. She has been using symbolic computation in her undergradu-

ate, graduate and Ph.D. courses and research focused on engineering mechanics.

For details about her recent publications on concave finite elements and boundary

elements, see www.civil.columbia.edu/~malsch.

Gautam Dasgupta is a professor of civil engineering and engineering mechanics at

Columbia University, New York. His areas of research include computer mathematics,

waves in random media, earthquake engineering, and finite and boundary element

analysis with Mathematica. For details on his recent work, see

www.columbia.edu/~gd18.

Elisabeth Anna Malsch

Postdoctoral Researcher

Department of Civil Engineering and Engineering Mechanics

Columbia University

500 West 120th Street

New York, NY 10027

malsch@civil.columbia.edu

Gautam Dasgupta

Professor

Department of Civil Engineering and Engineering Mechanics

Columbia University

500 West 120th Street

New York, NY 10027

dasgupta@columbia.edu

658 Elisabeth Anna Malsch and Gautam Dasgupta

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