The Azimuth Project
The Double-Cylindrical PointFocus (Rev #12)

The idea that nobody has wanted - so far

The parabola has the well-known property of reflecting axis-parallel rays to a point, as can be seen in this video clip. If we rotate the parabola around its axis, we create a parabolic disc. It has the well-known property of reflecting parallel rays (= planar wave fronts) to a point - if they are incident along the axis direction of the disc - which can be seen in this video clip.

Now, it is a beautiful (and little known) geometrical fact of great practical consequence that we can avoid the astronomical costs associated with creating a large parabolic disc, and harness the work-power of the sun in a more efficient way by bending two flat mirror sheets into the shape of two parabolic cylinders. If these parabolic cylinders are properly configured, the incoming parallel light rays (= planar wave fronts) will create an exact point focus after two successive reflections. This is due to

The Double Cylindrical Point Focus Principle:

If the focal line of the first cylinder is identical to the generating line of the parabola that is the intersection of the second cylinder with a plane perpendicular to its axis, then the incoming rays will be reflected to a perfect point.


The basic ray trace of the Double-Cylindrical Point Focus


The DCPF principle was discovered on 16 November 1976 by Ambjörn Naeve while working with Lloyd Cross, a well-known physicist and holographer who co-founded the School of Holography and the Multiplex Company in San Francisco, California. The original insight was recorded on this piece of paper, and a geometric proof of the DCPF principle can be found here.

Between 9-12 December 1976, Lloyd Cross and Ambjörn Naeve built the first point-focusing mirrors based on the DCPF principle. A short description of this intense event, including some pictures, can be found here.


Lloyd and Ambjörn with the first DCPF mirror


A sketched ray-trace of the first DCPF mirror


A computer-based animation of the double-cylindrical point-focusing mirrors in action, created by Ambjörn Naeve using the Graphing Calculator, can be seen in this video clip, and a live “burning demo” with a DCPF mirror-pair, built by Tomas Elofsson in 1989, can be seen in this video clip.

On 1 July 1989, in Gusum, Sweden, the DCPF mirror-pair shown in the latter video clip (1.5 m^2 primary mirror) was used in an experiment by Ambjörn Naeve and Tomas Elofsson that succeeded in melting limestone (2560 ˚C) in free air. In the pictures below, Tomas Elofsson is shown tuning the mirrors and melting the limestone sample:


Tomas Elofsson tuning the primary mirror


Tomas Elofsson tuning the secondary mirror


Tomas Elofsson melting the limestone sample in a open carbon crucible


A collection of video clips on the double-cylindrical point focus

The DC pointfocus videos by MathRehab on YouTube


Four major advantages of the double-cylindrical point focus

in comparison with the classical parabolic disc point focus

1) It is easier to build in large sizes, since cylindrical surfaces are curved in only one direction and therefore can be BENT into shape from a flat sheet, hence avoiding the “astronomical costs” that occur when a doubly-curved mirror has to be CAST into a fixed shape.

2) The focal point can be positioned outside of the solar influx area between the mirrors, where it is freely available to perform work, as in the DCPF wheelbarrel design.

3) The focal point can be moved by rotating the second cylinder around the focal line of the first cylinder. This fact can be used to create various forms of solid-state heat engines, for example by successively heating and cooling a series of bi-metal coils.

4) The mirror configuration can be approximated with planar strips, the number of which grows LINEARLY with the overall size of the construction – as opposed to the parabolic disc, where the number of planar approximators grows QUADRATICALLY with construction size.


Possible applications of the double-cylindrical point focus

As has been demonstrated above, the DCPF provides a cheap and efficient way to create “a huge burning glass”, which can achieve very high temperatures. Therefore, it enables local (= rural) development of a multitude of technologies that presently require advanced and expensive high-tech labs, such as:

Solar-powered steel plants that recycle scrap metal into a valuable resource for the local community. A discussion and some designs of what a solar steel plant could look like can be found here.

• Cheap and efficient super-heated external combustion engines (steam, sterling, …), whose work-power (like that of all heat engines) is directly proportional to the difference (in ˚K) between input and output temperatures.

• Highly efficient solar electricity generators, which concentrate the energy on high-performance photo-voltaic cells that must be water-cooled to avoid melting.


The idea that nobody wanted (1976 - 2001)

The story (some parts of which are in Swedish) of 25 years of trying to get people interested in the double-cylindrical point focus - from its discovery in 1976 and up until 2001 - can be found here. One of the chapters in this story took place in July 1995 in Gunnarskog in Värmland/Sweden, where Ambjörn Naeve and Tomas Elofsson recorded a video with their daughters acting as solar energy consultants, helping a farmer to melt horse shoes and load batteries by concentrating light on a solar panel.


P.O. Åhlander driving the DCPF solar horse cart


Ylva Naeve and Amanda Elofsson as solar energy consultants


These activities created a big stir in the community, with the local newspaper Värmlandsbygden publishing a cover story on the DCPF with a frontpage headline stating: “They have the solar power that melts mountains” (“De har solkraften som smälter berg”). Some local politicians got involved, interested in creating jobs for the local community, and it looked like things would start moving when an application for funding was sent to the Swedish Innovation and Industry Development Agency (NUTEK), which at that time was responsible for strengthening regional growth by furthering industrial applications of innovations. However, the application was dismissed on grounds that included the following reasons (translated from Swedish):

“A problem with the suggested design is that the temperature becomes very high. This means that the heat losses are increased, which makes the [DCPF] technique less interesting in energy applications. Moreover, concentration of light by mirrors is part of the research that is already carried out within the solar energy domain - although with different configurations than the DCPF, which lead to lower temperature levels and lower heat losses.”

By that time it had become painfully obvious - from this response as well as from other responses similar to it - that we were facing a serious pedagogical problem, and that there was a lot of learning of basic physics that needed to happen in several places in society for things to move forward.


Towards a global learning environment for solar energy

What problems or issues does the DCPF address?

The DCPF can contribute to the design of a more decentralized and sustainable energy technology that will be necessary as our technology must evolve beyond its present oil-burning stage.

The DCPF enables a more simple and less expensive approach to high-temperature solar energy applications that makes it possible to apply and maintain them in “low-tech” rural areas that are facing increasing difficulties of raising capital for development. For example, using the DCPF, low cost, medium temperature “scrap-mirror-strips” (that are “left-overs” from ordinary mirror production) can be used to create effective solar cookers, water-purifiers, and sea-water-desalination systems.

Who would benefit the most from the DCPF and how?

The DCPF is not a traditional “product” but a piece of globally important information for the sustainable survival of humanity in the rapidly emerging “post-oil-burning” age. If a big corporation wants to market solar energy concentrators, it can afford the large investments (“astronomical costs”) involved in creating a big parabolic disc reflector. However, this does not apply to smaller operators in poorer communities (such as rural areas in developing countries), who cannot afford these kinds of investments. About half of the people on this planet live under such conditions, and they are the principal beneficiaries of the DCPF. But we all benefit from a global increase of energy-independence, which can enable the Big Switch that humanity is facing for sustainable survival: Decreasing our dependence on globalized production processes, supported by localized information (secrets, patents, …) and increasing our dependence on more localized production processes supported by globalized (= globally relevant) information.

What are the initial steps required to get the DCPF off the ground?

1) Create an online Interactive Learning Environment around the DCPF principle, where “solar engineer wannabes” from across the globe can learn how to apply this technique to solve local production problems. The beginnings of such an online ILE can be found here.

2) Start an “ethical branding” movement among technical universities, that can work in partnerships with rural communities and developing countries in order to field-test various materials and designs. Such testing will be crucially important in order to optimize the DCPF constructions for a variety of different local conditions.

3) Create a pool of globally relevant and open ideas that can be contributed to under clear licensing conditions. Such licensing can build on Creative Commons. The default license should be: “Free to use, with proper accreditation to the source.“

Optical properties of Concis and Quadrics

Caustics involving conic reflectors
Point-focusing properties of Conics
Point-focusing properties of Quadrics
Practical burning demos of the DCPF mirrors


Tracking the sun - as it is seen from the surface of the earth


The Technology Enhanced Mathematics Rehabilitation Clinic

What is Mathematics?

Mathematics – in the sense of the Greeks, began with Thales around 600 BC – as part of an attempt to try to understand the world in a sensible (= non-magical) way. In fact, it was a part of a “scientific awakening” of the human mind against the tyranny of the priests. People started asking questions such as “what is the basic stuff that the universe is made of” – and asking such questions, not to the gods and religious authorities, but to nature itself. This was the birth of the scientific process.

/////// a few words about the difference between mathematics and science.

The word ‘mathematics’ is said to go back to Pythagoras (around 500 BC), who called his most advanced disciples ‘mathematikoi’. In the present context, this word will be interpreted in the following way:

Mathematics is the study of the totality of structures that the human mind is able to perceive.

Within the language of itself, mathematics can be described as the study of homomorphisms (= structure-preserving transformations) into the mind from its environment - including the mind itself.


Mathematics = Hom(Universe, Mind)


Life is structure, and since mathematics is the language of structure, it is the language of life, the ultimate ruler of the subspace within which life can be talked and reasoned about.

Hence, mathematics is concerned with the effective and efficient representation of structure. The stronger (= more effective and efficient) the representation, the shorter the computation that has to be performed in order to achieve a certain result. This shortness is referred to by mathematicians as “mathematical elegance.”


Euclid alone has looked on Beauty bare
Let all who prate of Beauty hold their peace,
And lay them prone upon the earth and cease
To ponder on themselves, the while they stare
At nothing, intricately drawn nowhere
In shapes of shifting lineage; let geese
Gabble and hiss, but heroes seek release
From dusty bondage into luminous air.

O blinding hour - - O holy terrible day,
When first the shaft into his vision shone
Of light anatomized! Euclid alone
Has looked on Beauty bare. Fortunate they
Who, though once only and then but far away,
Have heard her massive sandal set on stone.

[Edna St. Vincent Millay, 1920]




Inom alla andra verksamheter erfar människan en känsla av att krypa längs med marken;
hon förnimmer plågsamt sin otillräcklighet, sina delvisa kunskaper och sin brist på överblick.

Endast inom matematiken erfar människan en känska av att sväva i rymden;
hennes ängslan gäller att falla ned.

Den matematiska kalkylen spinner en underbar spindelväv,
högt över död och avgrunder,
vars trådar löper ut och löper samman i en rymd utan gräns.
Detta är matematikens höga rena omänsklighet.

På denna spindelvävsstege klättrar människan mot himmelen.
Månen och stjärnorna fångar hon i sin fjärilshåv.
Rund och glödande lyser månen i håvens botten,
stjärnornas silverfiskar kastar sin mjölk.

Rakt upp i det svarta djupet klättrar klättraren.
Under honom glimmar en mareld av stjärnljus;
stegen sviktar i en kosmisk vind.
O ingenting finnes som är likt denna svindel!

Ju högre han klättrar dess mindre blir han.
En dag när luften är fylld av solrök och vallflickors rop,
är han alldeles försvunnen.

[Willy Kyrklund]



CyberMath: A 3D interactive environment for mathematics