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161

PERFORMATIVE SURFACES:GENERATING COMPLEX GEOMETRIES USING PLANAR FLOW PATTERNS

Masoud Akbarzadeh Institute of Technology in Architecture, ITA / ETH

Drainage Patterns, Arthur David Howard

ABSTRACT

This research explains the development process of a design tool that can construct complex

surface geometries using only two-dimensional plan drawings. The intention behind this tool is to

address certain complex behavior of surface geometries such as hydrological characteristics. This

paper briefly explains the historic and mathematic description of surface data structures, accord-

ing to Cayley, Maxwell and Morse. This is followed by a brief introduction of the surface network/

critical graph extraction technique in GIS. Additionally, the algorithm of contour extraction from a

simple critical graph to reconstruct a surface is explained. In the final section the lessons learned

from the previous sections are used to develop algorithms for a tool which uses only plan draw-

ings to construct complex surfaces. Three algorithms are explained in the final section among

which the third one is considered to be the most complete and promising approach. Therefore,

some design examples are presented to show the flexibility of the tool. At the end, this paper pro-

vides suggestions and discussions to reflect further ideas in order to improve the tool in future.

1

INFORMATION ACADIA 2013 ADAPTIVE ARCHITECTURE 162

1. INTRODUCTION

1.1 MOTIVATION“Geometry is understood to be a constitutive part of architecture… Architects do not produce geometry, they consume it.” —Robin Evans,

“The Projective cast” (Evans 1995)

Architectural design tools are becoming more adaptable and

interactive, every day, in informing architects with the different

consequences of their design. However, the design intentions are

becoming so specific that architects find it easier to build their

own special tools to directly address their design intentions.

This paper is addressed to architects who would like to build their

own tools based on hydrological features of a surface. Obviously,

this is not a tool, which analyzes a certain concept, but dictates

the design to have certain hydrological characteristics.

In spring 2011, while I was working on my thesis towards an archi-

tectural design degree, I came across an interesting design prob-

lem—a parametric river. The geometry of the river is a function of

the terrain geometry. In geomorphology, the complex geometry of

the different terrains can be described by surprisingly simple draw-

ings called drainage patterns (Figure 1) (Howard 1967: 2246-2259).

1.2 ON PROJECTION AND ARCHITECTUREWhat drainage patterns represent in geomorphology is very sim-

ilar to projections in architectural drawings. In Architecture, we

simplify the description of three-dimensional objects by using

(Normal Orthographic) two-dimensional projections of them. Such

projections are called plans, sections and elevation. Never the

less, there are still some practically important shapes which can-

not be represented and manipulated easily. Therefore, the repre-

sentation of the terrain geometry (topography) is one instance that

is highly interested among geometers, geographers, computer

scientists and even architects. Let us have a look at the various

methods of surface (topography) construction and representation

in different design related disciplines to realize what makes this

difficult to manipulate such geometries.

1.3 SURFACE GENERATION IN COMPUTER GRAPHICS AND ARCHITECTURE

1.3.1 INTERPOLATION OF SECTIONAL CURVES

A surface can be constructed as an interpolation of multiple sec-

tional curves in two main directions of u, v or s, t. Although this

surface will have precise sectional properties, to reach to reason-

able resolutions in plan, it is necessary to provide a large number

of sectional curves, in closer intervals and in both directions, a

task which is difficult and time consuming (Figure 2).

1.3.2. CONTOUR MANIPULATIONS

This method gives the designers control in plan, but highly

restricts them in section (Figure 3).

1.3.3. DIGITAL ELEVATION MODEL

DEM is a technique in surface reconstruction, which gives each

pixel a height with respect to its color range in an image. In this

method, it is very hard to control the precision of the result.

Therefore, it is extremely difficult to use as a design tool (Figure 4).

1.3.4. TRIANGULATION

This method can represent highly complex geometries using

simple triangles. However, it needs an enormous amount of time

in design in order to reach certain levels of resolution in plan and

section. In addition, it has poor global manipulation possibilities

for complex geometries (Figure 5).

1.4 PROBLEM STATEMENT

Considering the stated facts in surface design and representa-

tion, I restate the problem here: is it possible to design complex

surface geometries simply by using only plan drawings? To

answer this question properly, this research introduces the read-

ers with the historic and mathematical definition of surfaces in

the following sections and uses them as a foundation to develop

a tool in the final section.

2. OBJECTIVES

The main outcome of this research is a tool and its relevant algorithms

to design and control the properties of the surface which was not

possible easily using the mentioned techniques in previous section.

Therefore, the final outcome is presented though the following steps:

2.1. SURFACE DATA STRUCTURE

Explained in section 3, Surface data structure describes how to

reduce the whole information of a surface into a simple connec-

tivity graph of maximum, minimum, saddle points, and pathways

that connects them to each other. Surface data structure and its

extraction technique will be described historically and mathemati-

cally in section 3.1, and 3.2.

2.2. REGENERATING SURFACESIn section 3.3 the contour extraction technique from the surface net-

work graph will be explained. Consequently, an example will be used

to explain the process of surface generation from a network graph.

2.3. PROPERTIES OF SURFACE NETWORK MODULE The lessons learnt from previous steps are used as the basis to

develop an algorithm, which will be the final outcome of genera-

tion algorithms in this paper.

2.4. SURFACE GENERATION ALGORITHMSFinally, the development process of a tool will be presented in

section 3.4. This section will reveal the chronological steps toward

the completion of the final algorithm for the design tool.

163

Relationship between Maximum, Minimum, and Saddle points

a. b.

c. d.

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PERFORMATIVE SURFACESAKBARZADEH

a.

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a. Saddle point; b. local maximum; c. local minimum; d. coarse line; and e. ridge line8

a. b.

1. Maximum2. Minimum3. Saddle

3

2

1

2

2

a-b. projecting a point grid onto the existing geometry of a surface to construct square surface patches

9

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a. Surface patch; b. saddle point; c. minimum point; and d. maximum point10

Interpolation of sectional curves: a. sections; b. plan; and c. Interpolated result/ surface

04

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Contour manipulations: a. elements and intervals; b. plan; and c. surface result

a. b. c.

3

a. b. c. DEM representation: a. pixels; b. image representation of a surface; c. surface result4

a. b. c.

Triangulated surface: a. Vertices and their connectivity; b. plan; and c. surface representation5

a. Slope and contour lines; b. ridge and coarse lines; c. drainage paths; and d. plan view of the paths

contour Lines

Slope Lines

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c. d.

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3. METHODOLOGY

3.1. SURFACE DATA STRUCTURE

3.1.1. HISTORY

Prior to inventing a tool, it is necessary to obtain a good under-

standing of the geometrical properties of a topographic surface.

There are some information on the surface that are necessary

to keep and there are some that are not. The important infor-

mation can be called the topological information of a surface or

Surface data structure.

In 1859, Cayley, introduced an intuitive definition of surface data

structure (Cayley 1859: 108-111). He defines different areas of a

surface based on the shape of the contours (indicatrix). According

to him, there are three types of contours in the whole area of

a continuous topography: circular, hyperbolic and parabolic.

Respectively, there are two types of points on the surface; max-

imum/ minimum points, inside the circular contour and saddle

points on the hyperbolic contours.

He also mentions that moving from the saddle point upward with

the steepest slope will take us to the local maxima; whereas,

moving downward will take us to the local minima. This path is

always perpendicular to the parabolic contour lines (Figure 6).

Maxwell completed the intuitive description of surface data struc-

tures (Maxwell 1870). According to him, there is a mathematical

relationship between the number of maximums, minimums and

saddle points (Figure 7): Number of maximums minus number of

passes equals to one, and number of minimums minus number of

saddles equals one:

Maximums - Saddles = 1

Minimums - Saddles = 1

Maximums + Minimums - Saddles = 2

3.1.2 SURFACE DATA STRUCTURE: MATHEMATICAL REPRESENTATION

After Cayley and Maxwell, Morse presented the mathematical

definition for important points on surfaces and their connecting

graph. According to Morse, the second derivative of the surface in

two major directions of its curvature is responsible for establish-

ing the critical points on the surface (Morse 1965). If traveling from

saddle point to each local extremum point, we should always

travel along a path on which one of the primary curvatures of the

surface is always zero (Figure 8). This Graph in mathematics is

called critical graph or surface network.

3.2. SURFACE NETWORK EXTRACTION METHODSSurface data structures are used in geo-computation and GIS to

reduce the amount of unnecessary information of the terrains,

while keeping the most important data, which can represent

Surface with its surface network graph

a. A general surface network diagram; and b. Plan view of different types of surface networks

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Subdivision of the graph into three main parts: upper, middle and lower parts13


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a. Top, middle, and bottom part of the graph; and b. adding all parts together14


3 2

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the main features of that terrain. There are different methods to

extract these networks from a given terrain, but in this study the

bilinear surface patch techniques will be mentioned (Morse 1965;

Pfaltz 1976). Accordingly the following assumptions are necessary:

Assumptions

In surface network extraction, each surface or topography is a

continuous function, z= f(x,y). There is no whole or under cut

in the geometry of the assumed surface—extraction of surface

network graph is based on bilinear surface patch technique

(Schneider 2003) (Figure 9).

The properties of critical points of a surface can be easily discov-

ered using surface patches (Figure 10). Following the steepest path

from saddle point downward and upward will take us to local min-

imum and local maximum. These paths make a connectivity graph

which is the surface network graph or critical graph (Figure 11).

3.3. REGENERATING SURFACE USING SURFACE NETWORK GRAPHIn Figure 12, the general topology of surface networks are rep-

resented (Schneider et al. 2004). Only the first type of surface net-

works will be visited. Let us review the process of adding infor-

mation to the surface in order to regenerate it.

To extract the contours and find their connectivity information,

we may need to divide the whole graph into mainly three

parts (Figure 13): upper part—from the highest maximum to the

highest saddle points, lower part—from the lowest minimum

and the lowest saddle point, and the area in between. In the

top and bottom parts, contour curves are closed and circular

(Cayley 1859). The connectivity information for the contours in

the middle part is extractable from the height information of

the contours and their connectivity information in the surface

network graph.

a. Simple surface network module; b. aggregation; c. contour extraction; and d. final topography

16

a.

b. c.

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a.

b. c. d.a.

b. c. d.

a.

b. c. d.

a. Contour extractions and smoothening; b. smoothening level I; c. level II; and d. level III

a. Side by side aggregation of the Surface network units; and b. descending aggregation of the surface networks

Step by step transformation algorithm

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1. Maximum 2. Minimum3. Saddle

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Topologically similar network15



After connecting the contours, we will be able to have continuous

closed curves to reconstruct the surface (Figure 14).

The contour extraction section shows that any given graph, which

follows the rules of surface networks, can be translated to a con-

tour representation of topography. Figure 15 shows design possi-

bilities of a simple network. For instance, a designer can start with

a simple module of a surface network graph and aggregate them

to create a network of topography. This aggregation can be turned

into a contour representation, which in turn can be described with

a surface (Figure 16).

Secondary articulation algorithms can be used to change the

geometry of a contour, which in turn will be resulted in more elab-

orate surfaces (Figure 17).

3.4. GENERATING PERFORMATIVE SURFACES

3.4.1. INTRODUCTION

In this section I will introduce three main algorithms based on the

foundation of the surface network graphs.

Assumptions

1. In all algorithms the surface is a result of transformation of

two-dimensional point grids into three-dimensional point grids.

2. Boundary conditions of the point grid are rectangular for the

simplicity in representation of the comparable elements.

3.4.2. PERFORMATIVE AGGREGATION OF CRITICAL GRAPH

Let us take a look at the properties of a single module of a critical

graph mentioned in section 3. Ridge lines are the ones which connect

saddle points to minimum points. Coarse lines, on the contrary, con-

nect saddle points to maximum points. Change in the height proper-

ties of each of the graph’s points will change the whole properties of

the graph as we can no longer call that a complete network (Figure 18).

Surface transformations algorithm Completion process; a. propagation process; b. cell activation process; and c. transformation process

Physical manifest of the process

20 21

22

Step n

Step 4

Step 3

Step 2

Step 1

Step n

Step 4

Step 3

Step 2

Step 1

Step n

Step 4

Step 3

Step 2

Step 1

Step n

Step 4

Step 3

Step 2

Step 1

Step n

Step 4

Step 3

Step 2

Step 1

Step n

Step 4

Step 3

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Step 1

Step n

Step 4

Step 3

Step 2

Step 1

Step 1 Step 2 Step 3 Step n

a.

b.

c.

Step nStep3Step2Step 1

Step by step process of activation of cells

Step by step process of activation of cells

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Step 1 Step 2 Step 3 Step 4

a.

b.

c.

167

a. Algorithm II; b. algorithm I26a. b.

Grid Size : 50 x 50a. b.

a. Point grid; and b. input Flow pattern on the point grid

a. Point grid; and b. input Flow pattern on the point grid

Each cell is compared to eight primary directions of the flow to rationalize the unitized direction vector

a. the area covered by downward only; b. covered by both; and c. covered with either / or

a. Rationalized connected network; and b. shortest distance drawn from each point

Slope-finder algorithm; a. upward direction; b. downward direction; c. superimpo-sition; and d. surface network graph

Physical model of the rationalized surface based on flow direction (left)Physical model of the discrete flow pattern for another surface geometry (right)

27

28

29

30

31

32

33+34

a. b.

j+1, j-1j, j-1j-1, j-1

j -1, j+1

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b. c.

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a. b.

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c. d.

Inverse Slope

Slope2

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Surface transformation algorithm II25

Step 1

Step 2

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3.4.3. SURFACE GENERATIVE ALGORITHM I

The whole process starts with an input line drawing as the flow

diagram on the point grid. At the first step all the points of the

point grid are transported to higher elevation except the points

which are connected to the input lines (Figure 19). The cell, which

has all the points up and one point down can make a ridge line.

In the next step all the transported points, except the ones that

are connected to the both sides of the ridge lines, are transported

again to higher elevation. Each cell in this transformation can

create only one ridge line which will be connected to the previous

ridges. This technique recursively generates connected ridge lines

as well as coarse lines on the surface. The process continues,

until there is no points that is not connected to the network of

ridge and coarse lines (Figure 19- 22).

FACE GENERATIVE ALGORITHM II

This algorithm is in continuation of the previous algorithm with a

major modification. In the new version, after activating the cells in

the first step, only those points are activated that are not shared

with the activated cells in the previous step (Figure 23-25).

Comparing the results from the two algorithms shows that the

second algorithm generates smoother and cleaner surface, but

this should not overshadow the interesting features of the first al-

gorithm (Figure 26). The first algorithm, can transfer flow in multiple

directions and create more dynamic flow of water. The second

algorithm, on the other hand, provides with a faster and shorter

paths for the flow. The most important disadvantage of both is the

low resolution of the input drawings and also the results.

4.5. SURFACE GENRATIVE ALGORITHM IIIThe last algorithm provided in the research is developed to

overcome the problems and disadvantages of the previous algo-

rithms. Flow of water always follows the shortest paths on the

surface or topography (Figure 27). In this method, the shortest

distance of each point from the flow patterns are calculated

(Figure 28). Unitizing the shortest direction for each point on the

grid will result in a discontinuous field of lines and points. In

order to create a continuous field of lines and points, a simple

rationalization algorithm is used. This algorithm is explained in

the following section.

4.5.1. DISCRETE DRAINAGE NETWORK

Let us have a closer look at the unitized field of points and flow

directions (Figure 29). For each point there exists eight surrounding

neighbors and eight primary directions. In order to make a con-

nected network of points and lines, one must rationalize the direc-

tion of the flow. The angle of the unitized vector is compared to

the primary direction for each point. The minimums of all angles

are chosen and the corresponding primary directions are drawn to

create a connected network of flow (Figure 29).

slope = % 0.0

slope = % 0.1

slope = % 0.3

slope = % 0.6

a.

b.

c.

a.

b.

c.

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e.

c. d.

pl

Point

k

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a. Area of influence; b. input polyline; c. distance from point grid; and d. distance translation to height; e. transformed point

35

Different quantity of z creates different slopes for the surfaces36

Different quantity of z creates different slopes for the surfaces37

169

In order to cover the whole area of the surface, this technique

needs to be applied in two opposite directions: steepest path

upward and steepest path downward.

This rationalization technique is also applied to an existing surface

geometry. Figure 32 to 34 represent physical models of the rational-

ized surfaces based on discrete flow direction for different surface

geometries. The surface network graph of each surface is clear in

these models. Where all these lines converge to a point is a local

minimum or maximum, and where all the points are diverging

from is a saddle point.

4.5.2. ALGORITHM DESCRIPTION

Let us call the points closest to each segments of input flow, the

influence area for the very segment (Figure 35). Now that we are

dealing with each segment separately, we can simply measure

the shortest distance from each point on the plan and translate it

into the height component for the same point on the grid (Figure

35). Changing the coefficient of the height can result in different

surface edge heights and slopes (Figure 36, 37).

4.5.3. GLOBAL DRAINAGE

So far we constructed a transformed point grid, which sits on the

planar flow line (Figure 35). If the global drainage of the surface is

intended we need to construct the surface with respect to a spatial

curve. For this reason we need to construct the curve in three dimen-

sions, as if a drop of water starts at the highest point of the curve,

then, follows the slope of the curve. Without changing its path along

the curve, it finally arrives to the lowest height of the curve.

Two types of curve can be used in this algorithm: polylines/ con-

trol point curves, and branching polylines/ control point curves

(Figure 38). The process of constructing spatial curve is quite sim-

ple. Based on the input slope parameter and length of the curve,

the height of each control point is calculated. The result is a spa-

= end domain of the line

= Start domain of the line

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Spatial polyline generation: a. polyline; b. curve; and c. branching polylines and control polylines; d. branching sequence

38

Break in the geometry resulted from direct translation in plan and height caused by spatial curve

39

Point grid pre-transformation based on the spatial curves40

Superimposition of linear height change and re-transformation of point grid due to spatial curve

41



tial curve, which cannot fit on a single plane in three-dimensional

space. For branching geometry, the method is still the same.

However, in branching poly lines/ curves, the algorithm, each time,

calculates the shortest path from the root or the lowest point of

the flow to each branch (Akbarzadeh 2012).

4.5.4. POINT GRID PRE-TRANSFORMATION

Applying the height transformation of the grid with respect to the

spatial curve might result into the unwanted breaks in the con-

tinuity of the resulted geometry (Figure 39). In order to overcome

this problem we need to transform the original point grid prior to

this transformation. This problem can be resolved by adjusting the

point grid with respect to the spatial curve before calculating the

shortest distance algorithm.

First we need to move the point grid to the highest point of the

spatial curve and then relocate the points on the z axis based on

their distance to the projected curve on the plane (Figure 40).

After this step, the transformed grid is ready to be used for shortest

distance Algorithm. The superimposition of these two algorithms

will result in a continuous surface, which has a global direction of

drainage (Figure 41). Some design examples are also included to

show the possibilities of the invented tool (Figure 42, 43).

4.5.5. NON- LINEAR TRANSFORMATION OF THE HEIGHT

The inherent potential of this algorithm is in its linear transfor-

mation of point grids from two dimensions to three dimensions.

This transformation can be easily changed into other non-linear,

trigonometric, or logarithmic functions (Figure 44). This adds to the

variety of different complex geometries that can be constructed

using these algorithms (Figure 45).

5. CONCLUSIONS

5.1. SUMMARYThis research presents the process of developing a tool for

designing complex surfaces using only plan drawings, which

reflects the hydrological properties of a surface. The outcome has

the following features:

1. Using simple and adjustable plan drawings to start:

This tool uses only plan drawings to create three-dimensional

geometry. These drawings are easily adjustable and the result is

quickly accessible.

2. Easy tool for design and sketch:

This tool provides designers with a powerful tool to explore

unique geometries using plan drawings. It works with different

groups of curves and the result can be seen very fast.

3. Unique Geometries not achievable using existing tools:

The geometries resulted from this tool are unique and existing

tools and methods such as interpolation curves, contour methods

Design Sample Using branching polylines42a

Design Sample Using branching polylines42b

Design sample using only plan drawings of curves 43a

Design sample using only plan drawings of curves43b

171

and triangulation are not capable of producing such geometries in

a fast and intuitive way as this tool is.

4. Constructible elements in result:

This tool uses discrete flow pattern in representation of a sur-

face, which can easily be translated into constructible element in

practice. In other words, it fills the gap from surface generation to

surface rationalization for practical use.

5. Water collection performance:

Since the whole idea of the surface generation is based on the

flow patterns, the first and foremost exquisite results of this tool

are types of geometries, which can collect and direct water.

5.2. DISCUSSION AND FUTURE DEVELOPMENTSThere are definitely multiple directions to complete and utilize this

tool in the future. This tool requires some criteria to evaluate its

performance and capabilities. The evaluations in design and analy-

sis results in the following discussions:

1. Edge control

At the moment, the designer can only control the clearance or the

maximum height of the roof or landscape. This property can be

advanced with further improvement of the algorithm to provide

the designers with more control over the edge geometry.

2. Holes and Undercuts

Some topological improvements are necessary in the algorithm to

handle holes and openings in the geometry.

3. Structural Properties

Finally, there are some inherent structural potential in the result of

this algorithm. This tool results in a nonlinearly folded geometry,

which requires further structural explorations. In the case that

structural properties of these geometries are improved, this tool

can translate simple two-dimensional curves to three dimensional

constructible structures.

pl

Point

f (x) = axf (x) = sqrt (ax)

f (x) = sin (ax)

k i

ij

Pointij

Pointij

Pointijpl

k i

pl k i

pl k i

pl k i

pl k i

a.

b.

c.

Linear versus non-linear transformation of point into 3D space44

Use of non-linear transformation in generating surface geometry45



WORKS CITED Akbarzadeh, M. 2012. Designing Performative Surfaces: computational interpretation of flow pattern drawings, Thesis (S.M.), Massachusetts Institute of Technology, Dept. of Architecture

Cayley, A. 1859. 246. On Contour and Slope Lines, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, XVIII, 108-111.

Evans, Robin. 1995. The Projective Cast: architecture and its three geometries. Cambridge: MIT press, 366-370.

Howard, Arthur David. 1967. Drainage analysis in geologic interpretation: a summation, American Association of Petroleum Geologists Bulletin, v. 51, 2246-2259.

Maxwell, J.C. 1870. On Contour Lines and Measurements of Heights, The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 40, 421-427.

Morse, S. P. 1965. A mathematical model for the analysis of contour line data, Technical Report, Dept. Electrical Engineering, New York University, New York, 400-124.

Pfaltz, J. L. 1976. Surface networks, Geographical Analysis 8(1), 77-93.

Schneider, Bernard and Jo Wood. 2004. Construction of Metric Surface Networks from Raster-Based DEMs, Topological Data Structures for Surfaces, ed. Sanjay Rana, John Wiley and Sons, Ltd, 53-70.

Schneider, B. 2003. Surface Networks: extension of the topology and extraction from bilinear surface patches, 7th International Conference on Geocomputation.

Rinaldo, Andrea, and Ignacio Rodrguez-Iturbe. 2001. Fractal River Basins: Chance and Self-Organization, Cambridge: Cambridge University Press.

Nelson, Stephen A. 2012. Physical Geology, EENS 111, Tulane University, http://www.tulane.edu/~sanelson/eens1110/streams.htm

MASOUD AKBARZADEH is currently a PhD student at Institute of Technology in Architecture, ETH. He researches

on 3D Graphic Statics using 3D reciprocal diagrams. He has

received a Master of Science degree in Architecture—(SMArchS

Compuation) from Massachusetts Institute of Technology in

2012 as well as a Master of Architectural Design degree (MArch)

also from MIT in 2011. He has received the prestigious SOM

award for research and travel in architecture and urban design

in 2011. Prior to MIT, he received a Master of Science Degree in

Earthquake Engineering and Dynamics of Structures from Iran

university of Science and technology.

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