International audienceWe propose a framework based on integral geometry that enables to express and to solve all sunlighting problems: the direct problems for assessing sunlighting in a scene, and the inverse ones for achieving sunlighting constraints in a design context. The heart of our method is the sunlighting volume that we denote ∏(P, T). We describe this notion and we show how to implement it in an architectural and urban design process

Siret, Daniel

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Inverse simulation of sunlighting: a geometrical
framework for architectural and urban design
Daniel Siret
To cite this version:
Daniel Siret. Inverse simulation of sunlighting: a geometrical framework for architectural and urban
design. CISBAT’01, Scartezzini J.-L., Sep 2001, Lausanne, Switzerland. pp. 231-236. ￿halshs-
02472154￿
Daniel Siret
« Inverse simulation of sunlighting : a geometrical framework
for architectural and urban design »
Published in Proceedings of CISBAT’01
EPFL, Lausanne, Sept. 2001
pp. 231-236
Contact
 siret@cerma.archi.fr
INVERSE SIMULATION OF SUNLIGHTING : A GEOMETRICAL
FRAMEWORK FOR ARCHITECTURAL AND URBAN DESIGN
D. Siret
CERMA, School of Architecture of Nantes, rue Massenet, 44300 Nantes, France
ABSTRACT
We propose a framework based on integral geometry that enables to express and to solve all
sunlighting problems : the direct problems for assessing sunlighting in a scene, and the
inverse ones for achieving sunlighting constraints in a design context. The heart of our
method is the sunlighting volume that we denote ∏(P, T). We describe this notion and we
show how to implement it in an architectural and urban design process.
RÉSUMÉ
Nous proposons un formalisme géométrique qui permet d'exprimer et de résoudre tous les
problèmes d'ensoleillement : les problèmes directs pour évaluer l'ensoleillement d'une scène,
et les problèmes inverses pour satisfaire des contraintes d'ensoleillement en situation de
conception. Le cœur de notre formalisme est la pyramide d'ensoleillement. Nous décrivons
cette notion et nous précisons sa mise en œuvre pour la conception architecturale ou urbaine.
INTRODUCTION
We distinguish the direct sunlighting problems from the inverse ones. The former consist in
determining the sunlighting states of a given scene using both shadows and shadings ; the
latter consist in resolving the geometrical conditions that enable a scene to achieve a given
sunlighting constraint in a design context (sunlighting as formgiver [1]).
In this paper, we propose a geometrical framework based on integral geometry [2] for
managing all sunlighting problems. This framework offers three main benefits : it gives
homogeneous expressions and solutions of the direct sunlighting problems of shadings and
shadows ; it enables the generalisation of these problems either in space or time ; it offers
solutions to the inverse problems of sunlighting.
The heart of our framework is the sunlighting volume that we denote ∏(P, T). We define this
notion in the first section, within a set of geometrical notions that make up our method. In the
second and third sections, we give solutions to the direct and inverse problems. Finally, we
show how these methods can be implemented in an existing CAD system, and why they offer
a powerful framework to cope with sunlighting problems in architectural and urban design.
1. THE INTEGRAL GEOMETRY OF SUNLIGHTING
Let ∑ be a geometrical scene located at the latitude l. We denote p any point of ∑ and P any
continuous set of points in ∑. We consider Dl the set of all the apparent solar directions at the
latitude l, and we denote t any direction included into Dl (i.e. any instant of the solar year).
We define a time period T as any continuous set of instants t, that is consequently a
composition of intervals of instants among days and months. T can likewise be seen as a
geometrical patch of Dl. Finally, we define the sunbeam R(p, t) as the half line starting from p
in the direction of t. We define the inverse sunbeam R(p,- t) starting from p in the opposite
direction of t (which is the perceived natural sunbeam).
1.1 Sunlighting volumes
Let us consider now sunlighting volumes settled as continuous set of sunbeams. First, we
define the sunlighting pyramid #(p, T), that is the set of sunbeams starting at the point p for
all the instants t of the time period T : #(p, T) = {R(p,t), t Œ T}. In the same way, we define
the sunlighting prism v(P, t), that is the set of sunbeams defined for all the points p of P and
for the single instant t : v(P, t) = {R(p,t), p Œ P}. Finally, we define the complex sunlighting
volume ∏(P, T) as the set of sunbeams starting from all the points p of the set P for all the
directions t of the time period T, that is : ∏(P, T) = {R(p,t), p Œ P, t Œ T}. Notice that ∏(P, T)
can be equally considered as the union of simple sunlighting pyramids #(p, T) when p draws
P, or as the union of prisms v(P, t) when t draws T, that is (fig. 1) :
∏(P, T) = 
  pŒP
U #(p, T) = 
  t ŒT
U v(P, t) (1)
T
p P
t
P
∏(P,T)"(  ,T)p v (P,  )t
Figure 1: The three kinds of sunlighting volumes
1.2 Cores of sunlighting volumes (solar envelopes)
We define ∂∏(P, T) the core of ∏(P, T) as the intersection of the sunlighting prisms v(P, t)
when t draws T, that is :
∂∏(P, T)  = 
  t ŒT
I v(P, t) (2)
The core ∂∏ embodies the set of all the points of ∑ that are affected by all the sunbeams
defined by all points of P during all instants of T. In other terms, any point of ∑ within ∂∏
will have its shadow within P whatever is the instant of T. This means that the shadows of any
volume built in ∂∏ will never spill over the edge of P during T (fig. 2). This is the definition
of Ralph Knowles' solar envelopes, as a container to regulate development within limits
derived from the sun's relative motion [3]. We offer here a geometrical framework for
defining and using this notion usually handled with empirical geometrical methods.
P
∂∏(P,T)
M
M
P
Figure 2: M defined in the core of a sunlighting volume (left) and its shadows during T (right)
2. DIRECT SUNLIGHTING PROBLEMS : EXPRESSION AND SOLUTIONS
Direct problems of shadows and shadings can be easily expressed, generalised either in time
and space, and geometrically solved using integral geometry.
2.1 Generalised shadows
Let Ω(P, t) be the shadow of P in the scene ∑ for the instant t. By definition, Ω(P, t) is the
intersection between the sunlighting prism v(P,-t) and ∑, that is : Ω(P, t) = v(P,-t) « ∑. If we
consider the time period T instead of the instant t, we can express the generalised shadow of P
during T, that is the union of all the shadows of P for all the instants t of T :
Ω(P, T) = 
  t ŒT
U Ω(P, t) = 
  t ŒT
U (v(P,-t) « ∑) = ∏(P,-T) « ∑ (3)
Therefore, the generalised shadow of P during T is given by the intersection between the
sunlighting volume ∏(P,-T) and the objects of ∑. This is the exact set of points of ∑ that are
exposed to the shadow of P during at least one instant of T.
In the same way, we prove that the exact set of points of ∑ that are exposed to the shadow of
P during all the instant of T are those that are in the core of the sunlighting volume ∂∏(P,-T).
Then we denote the core of shadow : ∂Ω(P, T) = ∂∏(P,-T) « ∑ (fig. 3 left).
Obviously, if P is an opening instead of a shading, then the generalised shadow is the union of
all sunspots generated by P during T (generalised sunspot), and the core of shadow is the set
of points that are always sunlit through P during T (core of sunspot, see fig. 5).
2.2 Generalised shading
Let M(p, T) be the shadings of the scene ∑ on the point p for the period T. M(p, T) is the
intersection between the simple sunlighting pyramid #(p, T) centred in p and the volumes of
the scene : M(p, T) = #(p, T) « ∑. If we want to determine the shadings of the scene ∑ on the
set of points P for the time period T, that is all the points of ∑ that may shade at least one
point of P during T, then we obtain :
M(P, T) = 
  pŒP
U M(p, T) = 
  pŒP
U (#(p, T) « ∑) = ∏(P, T) « ∑ (4)
The generalised shadings on P during T are given by the intersection between the sunlighting
volume ∏(P, T) and the objects of the scene (fig. 3 right). We also prove that any point within
the core of ∏(P, T) shade at least one point of P for all the instants of T. In other words, any
object within ∂∏ is a permanent shading on P during T.
∏(P,-T) ∂∏
P
T
∏(P',T)
∂∏
-T
P'
Figure 3: Generalised shadows (left) and generalised  shadings (right)
2.3 Discussion
Geometrically speaking, sunlighting problems have to be considered in a 4D space (3D plus
time). However, graphical simulation means have led to solve these problems by reducing
them to 3D volumes, using a discretization of space or of time, and then, by projecting this 3D
volumes onto 2D charts (axonometric projections for shadows at a discrete instant, and
perspective projections for shadings on a discrete point). When computer is used, the same
methods are usually transposed in a digital manner.
We have shown in this section that both problems of shadows and shadings i) can easily be
generalised either in time and space, and ii) can be expressed and solved in an homogeneous
way, as an intersection between the scene and a given volume ∏(P, T). Whatever are P and T
(single point, single instant or continuous sets of points and instants), we define the sets of
points of ∑ that involve simultaneously the points of P and the instants of T. Only the sign of
T changes (the direction of the sunbeams).
3. INVERSE PROBLEMS OF SUNLIGHTING
Direct simulation tools developed for integrating solar constraints within architectural and
urban design appraise the solar properties of geometrical forms that are fully ascertained.
Supposing the forms do not achieve previous fixed solar goals, the process of simulation must
be fully repeated until a good solution is found. One should consider two major drawbacks of
such an analytic approach : it imposes a trial and error process inappropriate for design
purpose ; it can neither provide the best solution that may achieve the goals, neither indicate
the whole set of solutions if any, neither specify if there are no solution.
The paradigm of inverse simulation has been proposed in order to overtake these drawbacks
[4]. The inverse problems of sunlighting can be expressed in a general way : given a set of
points P and a time period T, these consist in achieving the following constraint : « P must
check on the sunlighting state S during T ». In a more formal way, let us consider a
sunlighting constraint as a logical proposal (P, S, T). Solving the inverse problem consists in
making the sunlighting proposal true. Therefore, the solutions of a proposal (P, S, T) are the
transformations of the scene ∑ that make P be S during T. These are openings and closings of
the scene, that is geometrical transformations handled by architects during the design process.
Let’s study the solutions for the elementary proposals (P, sunlit, T) and (P, sunless, T).
3.1 Elementary proposals (P, sunlit, T)
The solutions for these proposals are the openings of the scene ∑ such as all the points of P
become sunlit for all the instants of T. Therefore, the set of solutions of the proposal
(P, sunlit, T) is the set of all the openings of the scene that contain the volume ∏(P, T). It
exists a single minimal solution O = ∏(P, T) « ∑, that is the opening by the exact intersection
between the scene and the sunlighting volume. Obviously, if this intersection is already
empty, the proposal is already true and the opening has no effect.
3.2 Elementary proposals (P, sunless, T)
The solutions for these proposals are the closings of the scene ∑ such as all the points of P
become sunless for all the instants of T. There exists an infinite set of solutions that are all the
objects that intersect all the sunbeams of ∏(P, T) (fig. 4). Therefore, as well as a generator of
sunlighting volumes, an implementation of the method within a 3D CAD engine should
consider a method for exploring this set of solutions. In [5], we propose such a method by
handling a plane that interactively intersects the volume ∏(P, T) and displays the shadings.
A point should be discussed here. If P is exactly the same for all the closings M defined with
∏(P, T), we notice that each generalised shadow Ω( M, T) is distinct. Obviously, the shadings
M close to the plane of P create generalised shadows close to P, while the shadings far from
the plane of P create very large generalised shadows. That means that each solution M of the
proposal (P, sunless, T) can be distinguished by its own generalised shadow (intersection
between ∏( M,-T) and the scene ∑). This gives a useful means for checking up the effects of
each closing achieving a (P, sunless, T) proposal while exploring the set of its solutions.
P P
M
P
∏(P,T)
Ω(M,T)
Figure 4: M is a section of ∏(P,T) (left and middle). The intersection of the shadows of M for
all instants of T is equal to P (right).
4. IMPLEMENTATION
4.1 Computing 3D boundaries of sunlighting volumes
This geometrical problem is not self-evident. In [5] we propose a solution for computing the
boundary of ∏(P, T) when P is a convex plane polygon. In that case, we show that any section
between ∏(P, T) and a plane Q parallel to the plane of P can be figured as the Minkowski
sum between P and the projection of T onto Q (that is, a computable geometrical operation).
Moreover, the intersection between ∏(P, T) and Q owns a geometrical structure that enables
to determine all the external beams of ∏. Using this approach, we have developed a generator
of 3D sunlighting volumes usable in a CAD system [5].
4.2 Using sunlighting volumes within a CAD system
A suitable CAD system for helping designers to cope with sunlighting problems only needs to
compute the intersections between some defined sunlighting volumes and the objects of the
scene. No more charts or geometrical constructions are needed. Only intuitive orders or
questions are used such as : « Is this face sunlit during the afternoon in winter ? », or « What
is the shadings of these buildings in the morning in summer ? », and « I want this room to be
sunless during T ». Such a system can make a diagnosis and it can provide an intelligent way
for managing sunlighting constraints throughout the evolution of the design. The constraints
are visualised as any other 3D volumes and their solutions can be explored interactively.
For instance, figure 5 shows how to design a window by controlling is total sunspot within a
room for a given time period. Moreover, we can simultaneously adjust the whole limit of the
spot given by the window on T (generalised sunspot) and the limit of the set of points that are
always sunlit by the window during T (core of sunspot).
∂∏(P,T)
P
T
Figure 5: Interactive design of a window considering the generalised sunspot and its core
In addition, the framework enables to solve the composition of sunlighting proposals in a
geometrical way. The point is to consider that the volumes ∏(Pi, Tk) associated to the
proposals (Pi, Sj, Tk) embody a geometrical representation of the corresponding constraints.
The set of solutions of two constraints is found at the intersection or in the geometrical
difference between the associated volumes. In figure 6 (left), we show an opening O defined
such as P is sunlit for all instants of T, and a shading M such as the same P is sunless during
T' (disconnected from T). We verify the achievement of these constraints for all instants of T
and T' with direct simulation (figure 6, middle and right).
P
O
M
           
Figure 6: Solutions of two sunlighting proposals (left) and verifications (middle and right)
4.3 Rendering and meshing
Other possibilities of the framework should be mentioned. For instance, it can help the pre-
processing for rendering very large scenes, like urban scenes. Supposing that T is the entire
solar year, and P any face of the scene, then only the faces that intersects ∏(P, T) have to be
taken into account for computing shadows on P (whatever is the considered instant). In the
same way, the sunlighting volumes may be helpful for computing the discontinuity meshing
of a scene. Indeed, the volume ∏(P, T) gives the limit of the points that are always exposed to
P during T (the core) and the limit of the points that are never exposed to P during T. For
solar assessment, the only required meshing is between these two limits.
REFERENCES
1. Lam, M.C. : Sunlighting as Formgiver for Architecture. Van Nostrand, New York, 1986
2. Serra, J. : Image analysis and mathematical morphology. Academic Press, London, 1982
3. Knowles, R., Villecco M. : Solar Access and Urban Form. AIA Journal, Feb 1980, 42-49
4. Schoeneman, C., Dorsey, J., Smits, B., Arvo, J., Greenberg, D. : Painting with Light. Proc.
of Siggraph'93, 1993, 143-146
5. Siret, D.: Propositions pour une approche déclarative des ambiances dans le projet
architectural. Application à l'ensoleillement. Ph D. Dissertation, Nantes, 1997


Inverse simulation of sunlighting: a geometrical framework for architectural and urban design

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