OLLING, PROJECTING, 
STRETCHING OF CURVED 
SURF ACES INTO A PLANE 
Sandi Berk 
Institute of Geodesy, Cartography and Photogrammetry 
FGG, Ljubljana 
Received f or publication: September 2 1999 
Prepared for publication: December 8 1999 
Abstract 
Problems of representing curved surf aces on the plane are 
discussed. Fram a set of captured points, a wire frame of the 
swface can be created using the Delaunay triangulation. In 
this way, the curved surf ace is approximated by small planar 
facets. 
A method of stretching of the wire frame of the surface onto 
a plane is presented. The stretching is perf ormed in such a 
manner that the sum of squares of linear distortions 
throughout the frame, as a whole, reaches a minimum value. 
The idea comes from the Airy's criterion far the projection of 
minimal distortions. 
Keywonls: adjustment, Airy's criterion, curved surface, 
Delaunay triangulation, linear distortion, stretching, 
wireframe 
1 INTRODUCTION 
The first cartographic representations of the Earth were made in good faith that it 
is a fiat and disk-shaped form. The beginning of the second step in the 
development of cartography dates back in the antiquity, the Pythagorean era (6th 
century BC) when the notion of Earth as a curved surface had begun to ripen. 
Evidence arose proving the Earth is a sphere. The consequence of this was the 
appearance of first cartographic projections (Jovanovič, 1983). However, problems 
arising from the projection of surfaces do not occur in cartography only. Such needs 
also exist in the conservation of cultural heritage ( e.g. restoration of frescos and 
ornaments), medicine and elsewhere. It has become obvious that the mapping of 
surfaces, which cannot be predefined, represents a serious problem. The purpose of 
this paper is to illustrate the problems arising in projecting curved surfaces onto a 
plane. Its aim is also to present one of the applicable solutions used in 
non-cartographic purposes. 
2 REPRESENTING CURVED SURFACES ONA PLANE 
Lei us recall some of the basic characteristics of smooth curved surfaces ( e.g. 
Jovanovič, 1983). The sections with planes including a normal onto the surface at 
Geodetski vestnik 43 (1999) 4 
the given point are called normal sections. At the given points, the normal sections 
may have different radii of curvature. The two normal sections, the one with the 
highest and the one with the lowest radius of curvature, are always mutually 
perpendicular. They are called principal normal sections. The corresponding radii of · 
curvature are designated as R 1 and R 2 and are called principal radii of curvature at a 
given point. By using the two radii we may define the full or the Gaussian curvature 
of a surface at a given point: 
K=(R,-R,f. 
A specific group o surfaces consists of those surfaces that can be generated though 
moving a straight line in space. These are called ruled surfaces; at least one straight 
line lying wholly in the plane can be placed through each point of such a surface. 
However, this is nota sufficient condition allowing us to unroll the surface onto a 
plane. For example, a onesheet hyperboloid is also a ruled surface (Figure 1, left). A 
surface that can be unrolled onto a plane without any distortions is called a 
developable surface. For each point lying on such a surface, the Gaussian curvature 
equals O. 
K=O 
Figure 1: Onesheet hyperboloid (left) and conical surface (right); 
both surf aces are ruled surfaces, the conical surf ace is also developable one 
Developable surfaces are generated by moving a straight line along a curve 
(directrix). Provided the straight line is moved parallel to itself in the given direction, 
a cylindrical surface is obtained. However, provided the straight line has a fixed point 
or vertex, a conical surface is obtained (Figure 1, right). Only with such surfaces it is 
possible to speak about unrolling or unfolding onto a plane. All other surfaces are 
impossible to project onto a plane without deforming the content (Jovanovič, 1983). 
Now, we are facing the basic problem in mathematical cartography: how to project a 
curved surface and its elements onto a plane by reducing the deformation of the 
content to a minimum level. A projection of a surface onto a plane without 
deformation would mean the lengths of all linear elements as well as the angles and 
areas have been preserved. 
procedure bringing us to the desired result is called projecting or mapping of 
curved surface onto o plane. Both terms are usually taken to be synonymous, 
although in a narrow sense, projecting denotes a geometrical procedure where 
relations between points in a given surface and their corresponding points in an 
Geodetski vestnik 43 (1999) 4 
auxiliary ruled surface ( e.g. curved surface of a cylinder or cone) or directly in a 
plane are set up with a central or parallel bundle of straight lines. The term mapping 
denotes the mathematical connection between points on a surface ancl the 
corresponding points (image) in a plane. There are few cartographic projections that 
can actually be treated in terms of geometric projecting; names of individual 
projections ( e.g. conical, cylindrical, horizontal) indicating the type serve merely in 
didactic purposes (Jovanovič, 1983). In literature we may encounter the division of 
cartographic projections into geometrical and rnathematical. 
3 DEFORMATIONS IN CURVED SURFACE MAPPING rrhe extent of the deforrnation in a mapped point (in a plane) is denoted with the 
linear or particular scale. This is a ration between an infinitesimal linear element 
( a straight line) in the projection plane ancl the corresponding linear element in the 
surface. Therefore, 
A'B' 
c 1 = lim--, 
• ' 13->A AB , 
3.1 
where AB stand for the are length in a plane and A'B' for the lcngth of the image of 
the former in the plane (Maling, 1973, Jovanovič, 1983). The linear scale should not 
be mistaken for the principal scalc or the stated scale of the map. This is a reduction 
of the map content which can be performed prior to or after the mapping into a 
plane. The recluction has no effect on the cleformation of the content. Technically 
speaking, the linear scale is related to a specifically clefined direction in a given point, 
for it depends on the clirection from which the point B in the equation 3.1 is neared 
to the point A. This rclation is shown in the following equation:c A = c A ( a) . 
The ratios between the linear elements on t~1e surface and th~ir respective_images m the plane neecl to be preserved to the h1ghest extent poss1ble. The cleviat1ons 
of the linear scale from the principal unit 
is called lincar clistortion. Linear distortion may be either positive or negative, 
depending on whether the linear element is contractecl or extendecl. 
4 SELECTING OPTIMAL CURVED SURFACE JPROJECTIONS 
3.2 
The question imposing itself is how to select the optimal projection. A ranclom 
number of clifferent conformal as well as equal-area projections can be 
constructed for each surfacc. However, this is not possihle for surfaces not being 
developable for the projection into a plane cannot be conformal and equal-area at 
the same tirne. The selection of the optimal projection is clictated by the shape and 
sizc of thc mapped region, and by the shapc of the smface that is being mapped or 
from which it is being mapped, as well as by our intentions. For the purposes of 
various cartomctric tasks ( angle or clirection measurement, measmement of area, 
rneasurement of length), projections are selectecl according to the type and thc 
nature of distortions (Maling, 1989). Whcn thc mappcd region is dcfined or properly 
outlined as well, beside the surface being mapped into the plane, we may speak about 
an optimal projection. On the whole, this is a projection containing minimal 
distortion. What we have here is the Airy's criterion requiring the sum of squared 
Geodetski vestnik 43 ( 1999) 4 
linear distortions along the entire mapped area to be minimal (Maling, 1973). The 
mathematical form of the criterion is as follows: 
JI(2f e2 (a). da). dS = min~ 
S a=O 
where e( a) denotes linear distortion in a given point and direction, and S denotes the 
mapped area. 
The projection conforming to the abovementioned criterion is called the 
projection of minimal distortions. . 
Beside the projec~ion_ of (abs~ll~tely) ~ini~al distortions, we may als.o speak about a conformal pro3ect1on of mm1mal d1stortions as well as about a equal-area 
projection of minimal distortions. The condition stands for the two projections as 
well. However, they also need to comply with the conformity and the equal-area 
condition. The derivation of such optimal projections is usually too demanding and a 
task impossible to perform analytically even for relatively simple surfaces and 
mapped regions with simple definitions. The conformal projection of minimal 
distortions for the mapping of an area of an ellipsoid bound with a closed polygon 
was introduced only recently (Nestorov, 1997). In practice, the projection can be 
performed only through numerical methods. 
S NON-CARTOGRAPHIC PR.OJECUONS OF CUR.VED SURFACES 
rrhe procedures of mapping the earth's surface into a plane are a subject of 
mathematical cartography. The procedures are based upon the fact that we have 
to deal with a relatively simple surface: sphere or a ellipsoid of revolution (spheroid). 
Notwithstanding, the derivation of cartographic projections - as we have seen - can 
be often a rather demanding task. If we do not limit our deliberations to a sphere or 
ellipsoid but start dealing with surfaces in general, we encounter complex problems. 
Each even slightly more complex surface becomes in practice analytically insuperable 
for the point of view of mapping into a plane. This means that the effort needed for 
the derivation of corresponding projection equations, probably would not pay off. 
However, if we desire to solve the problem analytically, we need to approximate the 
surface in question to such a surface for which corresponding projection equations 
have already been derived. One of the solutions to the problem in which we 
renounce the analytical approach shall be described in the text that follows. 
6 FORMAUON OF WIRE FRAMES OF CURVED SURi<ACE 
The metric analysis of a surface first requires the determination of a sufficient 
number of appropriately distributed points on the surface. The densiry of points 
depends on the cur,ature of the surface in the given region and on the required 
accuracy. Usually, well-definable points of details on the surface are selected. The 
coordinates of these points (x, y, z) can be determined, for example, with procedures 
found in stereophotogrammetry. An arbitrary (local) rectangular coordinate system is 
applied in the procedure. Now, the surface can be defined analytically (as an 
approximation surface of a specific 1:-ype) or presented in the form of a triangular 
irregular network - TIN. The letter is defined as a mesh of adjacent planar triangles 
Geodetski vestnik 43 (1999) 4 
differing size, shape and indination 
( clepencling 011 thc gcomci:ry of the 
to the distribr~tion of 
1-ihc triangular network is created ·with a set r)f qnrinm wrth the 
_ Dclaunay tnangulation. The is thc most eftcctive 
triangular approximation of tbe the obtainccl ti·iangles 2n-c aln,ost 
equilateral (Clarke, 199S). 'The principal of thi:: m 
the three-dirnensional space is vcry sirr1plc. Threc poin.ts z,rc sclecteci :frorn 
a set of acquirecl points. A minimal is r:reated si:ill incladi.ng the thrce selected 
points. A check is run to vcrify vvhethei- there arr.: ,my othcr points frorn the set fitting 
into a similar sphere. Provided that no other points are thtc: three selectecl 
points represent the Delaunay trianglr::. 'The creaticm of a 
uniform ancl autornatcd with the application of the H thc 
eclge of thc network (periphernl polygon) is r,ot clefinf,d 
triangulation results in a convex lrnll. 'iVith an the elimi1wtion 
of edge acute-anglecl triangles ir-: corners, the of th,:; 
region may be left to a computer. The obtai1c1ecl ""'·"··h"·•,.V- nct'No:ck is called the 
surfacc wire frame. 
fter the procedure describecl above has ber::n carriecl out, thc surface is 
_L_ approximated to small plana1 facets, the sir:i.cs of these facr:ts of triangle.::;) 
forn] a truss co11struction. The rnain ide'.:l is to transforrn thc ol)tained construction 
into planar form and to cause as littlc distortion č\S hereinafter 
the projection of the surface wire frarne into a plane shdl be reforrecl to as thc 
stretching of the smface wirc frame into a plane. In practice, the indicated solution is 
simple to translate into the adjustment of the trilateration network. 
Actual (spatial) lengths of sides oi triangles acqnire the rnle of distance 
measurement. The rcsult of thc adjustrncnt are re:.;idu2,ls causing the wire frarne to 
acquire planar form and the sum of wcightecl residuais sqttarecl to bc rninimaL 
Therefore, 
= 111111, 7 1 
where N denotes thc number of alI network vi denot,:;s the resv:lual of the i-th 
sicle and Pi dcnotes the weight of the i-th side. The choicc of adequate vreights bears 
great importancc. The linear scales of sides after are Pvnn~coc as 
C = 1 
cl + v 
whcre di denotcs the actual length of the sidF:: Let ES i1".seri: t11e 
equation 3.2 ancl thc residual of thc i-fr, side is 0 ·,,· . .-,-"•c 0 r1 č,1; 
e 
v, 
d 
The A.iry's criterion Js applied for 
sum of sguarecl !inear disto:·ticns 
to 
Geodetski vestnik 43 (1999) 4 
7.2 
7.2 into the 
7.3 
f e~ = mm. 
i = 1 
The equation 7A is mercly a different form of the equation 7,L By taking into 
consideration the equation 7.3, the following is obtained: 
7A 
The weight of the length of the i-th sicle must be inversely proportional to the square 
of the length itsclf. Such a choice of weights ensures that the sum of squarecl linear 
distortions along the entire construction is minimal during stretching. The procedure 
results in an optirnally stretched surface wirc frc1_me which is a very favorable 
approximation of the optirnal prnjection (projccbon of rninimal distortions) of the 
given surface into a plane; the densification of points in a surface woulcl near the 
result obtained through such an optimal projection. 
A trilateration network set i.n such a manner is adjustecl as a simple planar 1etwork. The adjustment procedure is an iterative one. The coordinates of the 
acquired points (vertices of triangles) -without the z-coordinate are taken as initial 
approximation coordination. The first approximation is a rectangular projection of a 
wire frame into a ;,.')'-plane. The wire frarne bas to be shifted in the local coordinatc 
system in order to make the regression plane of v1ire frame vertices (vertices of 
triangles) as horizontal as possible. It needs to be pointed out that in the adjustment 
of trilateration networks insignificant adjustment residues are expected to be done. 
Therefore, the eguation systems of these networks rnay be linearized. In our case, the 
linearization may be performed only vvhen the surface is approximated enough to a 
developable surface. Otherwise, meeting the minimal distortion condition poses a 
rather difficult taslc 
8 EXAMPLE OF STRETCHING INTO A PLANE 
The procedure clescribecl above of stretching into a plane bas becn developed and 
Jl used for the planar presentation of frescos in the church of St. Mary of Aliete in 
Izola. Beside other products, the commissioner, the Municipality of Izola, the 
Intermunicipial Institute for the Preservation of the Environment and Cultural 
Heritage Piran, ordered the production of eight photographs of frescoes stretched 
into a plane for the purposes of restoration works. The contractor implementing the 
project was the Institute of Geodesy, Cartography ancl Photogramr11etry FGG (Oven, 
Berk, 1998). Nowadays, metric clocumenting of cultural heritage objects is bascd 
upon three-dimensional models of objects forming wire frames of individual Iateral 
sides (Kosmatin Fras et al., 1998). Therefore, all phases of acquisition ancl 
presentation of such surfaces are a matter of routine. The results of acquisition and 
proccssing one of the frescoes are prescnted in this paper. The presented fresco is 
located on a semicircular arch on the church ceiling. 
he frcsco was acquirecl with 92 detail points. The Dclaunay triangulation 
produced the wire f;:ame of the fresco (Figure 3), formecl by 126 triangles ancl 
217 triangle sides. In this particular case, stretching the surface rneant to solve an 
overdeterminecl systern of 217 equations with 184 92) nnknowns. The 
306 __ ,, - .... ;.-l!l -4""'""'""·~""' 
"'-'ls!,ar<CVS:¾'0'd Geodetski vestnik 43 ( 1999) 4 
procedure requirecl 5 adjustmcnt iterations. With thc stretching of the fresco into a 
plane, thc distance between the far left and the far right vertex of the fresco 
increased from 4.792 m to 6 . 113 m, i.c. for 1.321 mor 27,6 %. The highest lincar 
clistortion of a wire frame side reached the value of 0.0094, i.e. 94 %o. After the 
stretching 01' the wirc frame lrnd been performed, a plarrnr frame was obtained 
(Figure 4 ), serving as a basis for a planar presentation of thc given content. The 
object dcalt 1,vith was a scanned photograph of the fresco, i.e. ornament (Figure 2). 
The edgc matching was performed througb resampling. The object was resampled 
region by regi on vvith regard to the selected dctail points (wire frarn.e vertices ). 
Bilinear transformation was applied. The final result was a pbotograph of a fresco 
stretched into a plane (Figure 5) and printed ona scale of 1:10. 
Additional data on the used equipment and software: The stereopairs were taken Yvith the R.olleiflex 6006 metric camera. The 
stereorestitution vvith the acquisition of characteristic points of the frescoes was 
. performecl with the Adam Promap System analytical instrm11ent. The Delaunay 
software package (produced for this specific purpose) was applied in the creation of 
thc wire frame. The stretching of the frame into a plane ,,;vas carriec: out with the 
Trim software package (Berk, Janežič, 1995). The resampling of scanned 
photographs with respect to the stretched frame -.vas pcrformed with the Adobe 
Photoshop software package. 
Figure 2: Photograph of" the fresco 
Geodetski vestnik 43 ( 1999) 4 
the 
Geodetski vestnik 43 (1999) 4 
9 CONCLUSION 
Gcnerally, t~e presrentation of curved surfacres in a plan~ is not foasible without any distornons or content A w1de array or cartograph1c proJect10ns, enablmg us 
to control the distortions, is availablc for clealing with simple surfaces (the curvecl 
surface of the sphere or the ellipsoid of revolution). Hmvever, vvhen the complexity 
of the surface increases, it can be approximated to a higher-order surface fitting it 
closely. To derive the corresponding projection for the surface in question is all but 
an easy task. The paper laid out the possibility in which YNe renounce the analytical 
solution. Each surface can be approximated to plamir triangular facets on the basis of 
a set of random points on these surfaces. The optimal approximation is achieved 
through Delaunay triangulation. The sides of obtained triangles form a surface wirc 
frame which is stretched into a plane keeping the distortions as minimal as possible. 
Such stretching represents a very favorable approximation of the optimal projection 
of a surfacc into a plane. The described procedure can be translated into the 
adjustment of a corresponding trilateral network by changing the roles of individual 
values. Existing software packages for the adjustment of geodetic networks can be 
applied for stretching. The surfacc wire frame stretched into a plane than serves as a 
basis for the planar presentation of the surface content 
Sources and bibliogrnphy: 
Berk, S., Janežič, M., TRIM - program za izravnavo triangulacijskih mrež. Geodetski vestnik, 
Ljubljana, 1995, Vol. 39, No. 4, pp. 271-279 
Clarke, K. C., Analytical and Computer Cartography. 2nd Edition. Prentice-Hall, Englewood Cliffs 
(New Jersey), 1995 
Jovanovic, V, Matematička kartografija. Vojnogeografski institut, Beograd, 1983 
Kosmatin Fras, M. et al., Sodobne oblike metričnega dokumentiranja objektov kuliume dediščine. 
Geodetski vestnik, Ljubljana, 1998, Vol. 42, No. 4, pp. 383-390 
Maling, D. H, Coordinate Systems and Map Projeciions. George Philip and Son Limitecl, London, 
1973 
Maling, D. H., Mcasurements from Maps. Principles and methods of cariomel,y. Pergamon Press, 
Oxford, 1989 
Nestorov, I. G., Camprel, A New Adaptive Conformal Cartographic Projection. Cartography and 
Geographic Infonnation Systems, American Congress on Sun1eying and Mapping, Bethesda 
(ivlmyland), 1997, Vol. 24, No. 4, pp. 221-227 
Oven, K., Berk, S., Izdelava fotogrametričnega posnetka stropa cerkve sv. Marije Alietske v Izoli. 
Tehnično poročilo. Inštitut za geodezijo in fotogrametrijo FGG, Ljubljana, 1998 
Review: Sergej Čapelnik 
Dr. Bojan Stopar 
Geodetski vestnik 43 ( l 999) 4