The Geometry of Relief
(11/19/2014: Draft not to be quoted without permission of the author)
E.M. Saniga*
University of Delaware
1. Introduction
Consider an object in space. A two dimensional model of that object as in painting can be
made using the well-known laws of perspective. A full three dimensional model as exists in
sculpture can be constructed without the laws of perspective.
A compressed three dimensional model of the object in space is known as a relief model. In one
direction the limit of relief is the compression of the space into two dimensions; in the opposite
direction the limit of relief is the uncompressed space. Any amount of compression can be made
in relief with the amount of compression usually expressed as 1/c where 1≤c.< ∞.
Relief models of an object in space, like a model in painting of an object in space, can and have
been governed by formal laws. Close inspection of ancient and renaissance relief sculpture as
well as some reliefs up to the present time indicates the plausibility of the existence of a formal
modelling technique based upon a structured compression of the object in space.
In this paper I address the concept of a “true relief” model as opposed to reliefs generated by
models such as illustrated by Rogers(1974, pp. 78-79). I define a true relief model as a model
where the order of depth of objects in space is preserved in relief and where the depth of
compression is formally proportional to the distance from the location of a viewer.
Published information on the laws of compression of relief is elusive, if it exists. Relatively
recent formal perspective based models of low or Bas relief have appeared. For example,
Cignoni et al.(1997) discusses a camera based algorithm where non linear scaling of the heights
provides less dynamic range to far objects than near. Using methods from research on tone
mapping Weyrich, et.al.(2007) has presented an algorithmic approach for generating very low
reliefs( Bas relief) with many options available to the sculptor such as selecting certain scene
elements for particular emphasis. Weyrich (2007, p.466) notes that “straightforward re-scaling
of the height field fades perceptually salient shape features in bas relief “: the height field is also
known as the depth map. Other similar approaches exist and differ mainly in the method of
depth compression. Kerber, et al. (2010) improves on one of the shortcomings of previous
*I wish to thank the American Academy in Rome for their support of this project and Yaoyin
Zhang for graphic and conceptual contributions to this paper.
work, the impact of depth compression on surface quality, with an approach that also features a
manageable number of control parameters.
Some evidence exists that the American artist Thomas Eakins lectured on the problem of
compression in relief. In the course of his research for his volume on Eakins, Homer(1992))
gained access to these lectures and provided copies to me (Homer, 2004). Some of this material
also appears in Eakins(2005). Eakins approach was that of true relief defined as above.
Unfortunately, the material available on this subject from the Eakins lectures is scant at best but
can be considered a starting point for the development of a geometric ( and algebraic) solution to
the problem of relief compression.
I wish to caution that this approach, while structured, allows the artist much leeway in relief
construction in the form of invention, something that is required should the resulting construction
be labeled as art. As Pope-Hennessy notes(1980, p. 160)) in reference to the Puzzi Madonna in
the Staatliche Museum, Berlin, “contrary to the claims of science” the “ relevance of the
diagonals established by the corners of the ceiling is to the relief as pattern, not to the relief as
space and .… cannot be regarded as other than empirical”
The purpose of this paper is to develop the relief laws geometrically and algebraically, not
algorithmically. While an algorithmic approach would be less time consuming it may suffer
from an over structuring of the problem from the point of view of an artist. Furthermore, an
algorithmic approach would not be of value to an art historian in the analysis of sculpture from
the past.
A solution to the problem I present in this paper allows rescaling of the depths of objects in
relief in a formal way, which I call true relief. In particular, I limit my development to the
horizontal and vertical projection of a point in relief viewed from that YZ plane as shown in
Figure 1. The projection of the third dimension is easily found in any text on two dimensional
perspective.
In the next section I develop these relief laws. In section 2 I address the problem of relief in
larger spaces and I draw some conclusions in the last section.
2. Mathematical development
Mathematically, the problem can be stated as follows:
Suppose we have a point in space with coordinates (Xp,Yp,Zp) such as the one depicted in Figure
1. We wish to find the horizontal (Zp’) and vertical projection (Yp’) of this point which, in relief,
now has the coordinates (Xp’,Yp’,Zp’).
Relief compression is expressed as a fraction of full relief, or sculpture. I assume without loss of
generality that in a relief of fraction 1/c, a finite space of depth z will be compressed such that
the new space in relief is exactly of depth ( 1/c) z. I also assume, as noted earlier, that the order
of objects in space is preserved in the new space defined by the relief and that the depth of relief
of the objects in space is proportional in a formal way to the distance from a viewer. .
Suppose we have a space bounded on the Z axis with limit Zm. This space is depicted in Figure 2.
We wish to place our eye at point E where the coordinates of this point are (Ye ,0). If our desired
relief depth is 1/c it follows that the relief boundary then becomes the line defined by Y= Zm/c.
Now, to ensure that no object or part of an object appears in relief past this boundary we
determine the ground plane of the relief as follows:
First, note that the equation of the line going from E to M in Figure 2 can be shown to be
Y=Ye-(Ye/Zm)Z .
(1)
Now, at the point where Z=Zm/c ,
Y=Ye(1-1/c).
(2)
Thus, the point where the lines A and B intersect is defined by the coordinates (Ye(1-1/c),Zm/c).
The equation of the line going from the origin O to this intersecting point is then
Y=Ye(1-1/c)/(Zm/c)Z.
(3)
And if Y=Ye,
Zr=(ZmYe/c)/Ye(1-1/c)).
(4)
Thus, the coordinates of point R become, since Yr=Ye, (Yr,Zr) where Zr is defined above.
Note that we have derived the ground plane of a relief of depth 1/c in a finite space of length Zm.
Four interesting points can now be made:
1. All points in relief in the first quadrant are bounded by this plane.
2. If we wish to model an object with no compression as in sculpture the ground plane is
horizontal.
3. If we wish to model an object as in a two dimensional painting, the ground plane
becomes vertical.
4. All depth orders in the initial space are preserved.
We now consider the relief projection in finite space of any point where the depth of relief is
1/c. Consider point P in Figure 3 with coordinates (Yp,Zp). The projection of this point is
(Yp’,Zp’). Note that the projected point occurs at the intersection of two lines. The first line is the
line from E to P. The equation of this line is
Y=Ye-(Ye-Yp)/Zp,
(5)
The second line is the line from a point on the vertical axis at coordinates (Yp ,0) to the point R
and the equation of this line is
Y=Yp-(Yp-Yr)/Zr.
(6)
These lines intersect at the point P’ with coordinates (Yp’,Zp’). Solving these equations
simultaneously we can show that
Zp’ =(Yp-Ye)/(ZpYp-ZpYr-ZrYe+ZrYp)/(ZrZp)
(7)
and
Yp’ =((Yr-Yp)/Zr)Zp’ +Yp.
(8)
These are the horizontal and vertical projections of the point P respectively.
When converting a space in relief the projections in the XY plane can be made with the usual
laws of perspective, giving thought to the fact that the vanishing points are at eye level in relief
rather than at the horizon.
A sculptor could assay a model by finding the coordinates of the model at a number of points,
say; the accuracy of the resulting relief model depends upon the number of points. The relief
projections of these points would then be given by equations 7 and 8 above calculated on a
calculator or with a computer program such as Microsoft Excel.
These projections may aid the sculptor in achieving a convincing relief. In very low relief where
1/c is very small, it is doubtful if it is necessary to compute too many projections as the scene is
dominated by planar perspective in the XY plane, and that is where care should be given.
Likely more convincing reliefs would accrue with the use of these equations in higher relief.
An example of how these relief projections can be used is provided in Figure 4 and Figure 5.
Figure 4 shows the projection in one quarter relief of three archways of a particular depth and
separation. (Note that our viewpoint is from the Y Z plane; thus the arches are not visible).
These projections were derived geometrically although they could have easily found using
equations (7) and (8).
Figure 5 shows three archways in a relief by Lorenzo Ghiberti ( Ghiberti, 2014) and is shown
from the X Y plane. Note that the steepness of the ground plane indicates that this relief is very
low; certainly not one quarter relief as in our example.
It is interesting to note that the depth ordering is preserved in Ghiberti’s relief, that there is a
reduction in the archways depth as the archways are further away and that there is a reduction in
the archways separation as the archways become further away. Using the definition earlier we
can label this as a “true relief”.
The actual model underlying this relief could be made with a few measurements should an art
historian desire to do so. Departures from the formal model made by the artist could also be
identified; this could be labeled “invention”..
3. Relief in large spaces
For very large spaces the idea of expressing relief compression as a fraction of the space loses
meaning. Here, it is likely that one could decide on the finished depth of the relief desired and
find a method of compression that preserves the depth orders of objects yet fits these objects into
the predetermined space. Various methods of achieving this goal are possible; one alluded to by
Eakins in his lectures, again only briefly, is to consider the relief plane as a non linear but convex
plane. I illustrate this in Figure 6 with an example of placing two rectangles in relief with an
arbitrary but predetermined depth of the relief space. Here, without loss of generality, I have
modeled the relief plane with a fourth power equation of the form
Y=a +bX4
It is a simple matter then to fit these equations to the two points R(Ye,Zr) and (Yp,0) for various
points P. In the example in Figure 6 we would only need two arcs : Arc R’ and Arc R’’ to
determine the relief projections of the rectangles given in figure 5. Note, for example, that the
projection of point j would be the intersection of the line from E to J with Arc R’’ as before .
In the construction of the relief one might find it more convincing to make the line from P’’ to C’’
linear.
Conclusion
I have provided a method to find the projections in the Y and Z plane of a relief model of depth
1/c. This method may be of value to the sculptor in producing a convincing relief, should that be
their desire. It might also be of value to an art historian to study how previous sculptors used the
laws of relief perspective or compression.
References
Cignoni, P., Montani,C. and Scopigno, R.,(1997), Automatic gereration of bas-and highreliefs, Journal of Graphics Tools, Vol2, No.3, pp 15-28
Eakins,T., with K.A. Foster and A.B. Werbel, A Drawing Manual by Thomas Eakins, Yale
University Press, 2005.
Ghiberti, L. (n.d.) In Wikipedia, retrieved November 6, 2914 from,
https://upload.wikimedia.org/wikipedia/commons/2/2a/Lorenzo_Ghiberti__Esa%C3%BA_e_Jac%C3%B3_-_Porta_do_Para%C3%ADso.jpg
Homer,W.I., (1999) “Eakins lectures”, personal correspondence with the author.
Homer(1992) Thomas Eakins: His life and Art, Abbeville Press
Kerber,J., Tevs,A, Belyaev,A., Sayer,R. and H Seidel, (2010), Real Time Generation of
Digital Bas Reliefs, Computer-Aided Design and Applications, Vol 7, No. 4, pp 465-478
Pope-Hennessy, L.L. ( 1980), The Study and Criticism of Italian Sculpture, Metropolitan
Museum of Art, NY, NY.
Rogers, L.R.,(1974), Relief Sculpture, Oxford University Press.
Weyrich,T.,Deng, J.,Basrnes,C., Rusinkiewicz, S., and A. Finkelstein, (2007) “Digital BasRelief from 3D scences, ACM Transactions on Graphics, Vol. 26 , No.3.
Figure 1. A Point in 3-Space
Figure 2. Determination of A Relief Scope
Figure 3. Horizontal And Vertical Projection of A Point P
Figure 4. Three Gates in Half Relief
Figure 5. Baptistry of S. Giovanni by Ghiberti
Figure 6. Relief Compression in a Very Deep Space