DE102012109105A1 - Method for determining planar projection of property and data projected on virtual spherical surface, involves providing value distribution on spherical surface or hemisphere surface, where value distribution is relevant to property - Google Patents

Abstract

The method involves providing a value distribution on a spherical surface or a hemisphere surface, where the value distribution is relevant to the property. A virtual camera is provided with an aperture stop. The camera is aligned, so that a straight line runs perpendicular to the image plane of the camera. The straight line runs through the center point of the aperture stop. A desired two-dimensional-planar projection is selected from the group such as gnomonic projection, stereographic projection, equidistant projection, equal-area projection, parallel projection and central projection.

Description

The invention is in the field of simplified calculation and thus accelerated visualization of radially distributed data, values, properties, measurements or features by means of a projection. In particular, the invention is in the field of accelerated graphical representation of projected property distributions, or projections using a standard computer.

In this context, "radially" distributed data or properties are to be understood as meaning any spatial distributions of data or properties which, starting from a point in the directions considered, are each represented by a single property value. Whether this value sits on a spherical surface, e.g. the earth's surface, or whether it sits on a cube surface, or for example corresponds to the star distribution in the universe seen from the earth, is insignificant thereby. It is also irrelevant whether the sphere surface belongs to a real sphere or a virtual projection sphere. For an illustrative illustration of the "only one individual property value" using the example of the mentioned star distribution, it is emphasized that the projection in question typically does not identify radially consecutive stars, or the radially consecutive values (stars) only as one point, i. represented as a value. Alternatively, radially consecutive data can of course also be added up. In other words, once only the single visible part can be drawn (for example, a star) and, alternatively, all stars in that direction can also be represented as star density along that direction. In the application of electron microscopy with detection of the backscattered electrons from the sample (EBSD) one does exactly the latter, by representing the number of points as intensity (mathematical "folding"). In other words, each point is e.g. described by a Gaussian distribution and all Gaussian distributions are added up. Here and in the following, the terms "data", "values", "properties", "measured variables", "features" and "parameters" are understood as synonyms for designating a state assigned or assignable to the respective point on the projection surface.

Spatially distributed data, measured values, parameters, as for example for the description of the orientation of crystal surfaces in crystallography, for the description of the intensity distribution of X-ray reflections in the X-ray structure analysis, for the description of the distribution of rainfall on the earth, the star distribution from one point, etc. are typically projected onto a spherical surface as a radial distribution if they are not already distributed as data on a sphere. This phenomenon is a classic problem of cartography. A projection of this spherical surface into the plane is called stereographic projection or even angle-accurate azimuthal projection. The mathematical derivation of a stereographic projection has been known for 2000 years (150 BC). In that case, a property distribution projected on the spherical surface (spherical projection) is projected onto the equatorial plane - or alternatively and completely equivalent to the azimuthal plane. The resulting projection is called angled, because the angle at which any plane passing through the center of the sphere intersects is preserved even in the stereographic projection. Numerous approaches are known for transforming a property distribution - projected onto a spherical surface - into a similar (or relative) plane surface with as little distortion as possible. In an azimuthal projection, the property parameters assigned to a real or a virtual spherical surface are transferred to corresponding surface segments of a plane either true to the real area ratios, true to the real radial distance ratios or conformal to the projection plane, i. projected. Unless otherwise stated, the term "projection" will always be used to refer to "azimuthal projection".

For the area-equivalent transformation (Engl. - equal area projection) into a level different representation possibilities are used. For example, in the azimuthal Lambert projection - as an alternative name for the equal area projection - each point of the spherical projection is converted into a projection point in the azimuthal plane. Thereby a radial property, e.g. the direction of vectors, starting from the surface of a sphere whose center coincides with the projection origin, imaged surface-to-surface in a flat surface. In the construction, the secant lengths are transmitted from the "north pole" or from a first vertex to the azimuthal plane present at this first vertex of the projection sphere.

For better understanding or interpretation, these data must often be rotated in the room. The rotation of the data requires a respective recalculation of all the projection data, which in turn can be very time consuming.

The calculation of the projection data requires a proportionally increasing expenditure of time as the size of the underlying measurement data increases. For example, for crystal orientation measurements by means of EBSD (Engl. - Electron backscatter diffraction) almost always several million Recalculate projection points. The amount of time to be estimated using a standard computer (standard computer system) is in seconds. The delay resulting from the calculation period prevents continuous presentation of the projection data during spatial rotation using conventional computing techniques.

• a) Die vermessene Eigenschaft – als 3D-Daten oder auf einen beliebigen Körper projiziert – rotiert um eine feststehende spiegelnde Kugeloberfläche, so dass sich die gedrehte Eigenschaft auf der feststehenden Kugel bzw. Halbkugel darstellt. Dabei wird die gedrehte Eigenschaft selbst gar nicht visualisiert, sondern nur deren Reflektion auf der Kugeloberfläche.
• b) Die vermessene Eigenschaft wird wirklich auf einer Kugeloberfläche projiziert und diese Projektion dann bei feststehendem Projektionspunkt (Kameraposition) gedreht.
• c) Die radiale Eigenschaftsverteilung wird aus geometrischen Objekten, z.B. Zylindern, Kreisen usw., selbst erzeugt (zusammengestellt). Durch einen Trick wird alles das ausgeblendet, was nicht dargestellt werden soll, d.h. man betrachtet die erzeugte Anordnung geometrischer Objekte z.B. durch eine Aperturblende, die den nicht erforderlichen Teil, der aus Sicht des Nordpols über die Äquatorebene hinaus geht, ausblendet (ausschließliche Darstellung der oberen Halbkugel). Alle Daten (Objekte) sind damit nach wie vor vorhanden und müssen nicht aufwendig weggeschnitten werden.
• d) Die Eigenschaft wird – wie in a–c) – erzeugt und dann wird nicht die sphärische Projektion gedreht, sondern der Projektionspunkt (Kamera mit vorgeschalteter Blende) um den Mittelpunkt der Projektionskugel bewegt. Damit bewegt man anstelle von sehr vielen Objekten – z.B. für c) können das einige hundert oder tausend Objekte sein – nur die abbildende Kamera.

In the laboratory measuring technique or in the practical use of such projection techniques for analytical purposes, the following main steps are taken:

• a) The measured property - projected as 3D data or onto any body - rotates around a stationary specular spherical surface so that the rotated property is on the fixed sphere or hemisphere. The rotated property itself is not visualized at all, but only its reflection on the sphere surface.
• b) The measured property is actually projected on a spherical surface and this projection is then rotated at a fixed projection point (camera position).
• c) The radial distribution of properties is generated (composed) from geometric objects, eg cylinders, circles, etc. By means of a trick, all that is hidden, which is not to be represented, ie one looks at the generated arrangement of geometric objects, for example by an aperture diaphragm, which fades out the unnecessary part, which goes from the viewpoint of the North Pole beyond the equatorial plane (exclusive representation of the upper hemisphere). All data (objects) are thus still available and need not be cut away consuming.
• d) The property is - as in a-c) - generated and then does not rotate the spherical projection, but the projection point (camera with front panel) moves around the center of the projection sphere. So you move instead of very many objects - for example, for c) can be a few hundred or a thousand objects - only the imaging camera.

According to the described approaches, all the elements of the spherical surface can be completely detected by allowing the area curved in space to be viewed from all sides and from all distances. This is made possible via an appropriate application specific interface (API) by selecting a specific distance and direction from the center of the projection sphere. Examples of such APIs are Glide (3dfx), OpenGL (Khronos Group) or Direct X (Microsoft). For the stereographic projection of the property distribution, the projection point, sometimes referred to as an eye point in historical treatises - in modern representations but often as a south pole - always lies on the sphere surface. Actually, his position is irrelevant as long as he is just an element of the sphere surface. The stereographic projection always takes place on the plane which is perpendicular (orthogonal) to the line formed by the projection point (position of the camera) and the center of the projection sphere and itself passes through the center of the sphere. It represents the equatorial plane as the projection plane, if it also passes through the center of the projection sphere and the position of the camera is always referred to as the south pole.

• (1) Gemäß einer ersten Ausführungsform wird ein Verfahren zum Ermitteln einer Planarprojektion von auf einer Kugeloberfläche verteilt vorliegenden Eigenschaftswerten vorgeschlagen, das die folgenden Schritte umfasst: – Bereitstellen einer Eigenschaftsverteilung auf der Oberfläche einer (Projektions-)Kugel oder auf der gekrümmten Oberfläche einer (Projektions-)Halbkugel, d. h. einer Kugelhalbschale; – Bereitstellen einer virtuellen Kamera; – Ausrichten der Kamera so, dass eine senkrecht zur Bildebene der Kamera verlaufende Gerade, die durch den Mittelpunkt der kreisförmigen Blende (Aperturblende) verläuft, auf den Mittelpunkt der Kugel trifft; – Auswählen einer gewünschten 2-D-Planarprojektion aus der Gruppe: gnomonische Projektion, stereografische Projektion (konforme azimutale Projektion, bzw. Engl. – conformal projection), abstandstreue, d.h. radial abstandstreue Projektion (Engl. – equidistant projection), flächentreue Projektion (Engl. – equal area projection), Parallelprojektion (Engl. – orthographic projection) und Zentralprojektion (Engl. – perspective projection); – Auswählen einer als Funktion der gewünschten Projektion abhängigen Kameraposition als Abstand der Kamera zum Projektionskugelmittelpunkt, wobei ein bezogen auf die Kamera hinter dem Kugelmittelpunkt liegender Durchtrittspunkt der Geraden durch die Kugeloberfläche einen ersten Scheitelpunkt der Kugeloberfläche definiert; – Erfassen eines Bildes mit der Kamera von zumindest einem Ausschnitt einer halben Kugeloberfläche, die den ersten Scheitelpunkt umfasst; – Ausgeben eines (2-dimensionalen) Kamerabildes als Abbildung der (gekrümmten) Kugeloberfläche, das der gewünschten 2D-Planarprojektion entspricht.

Against this background, a projection method according to claim 1, a graphic representation according to claim 14 and a visualization of measurement, analysis and / or simulation data according to claim 15 is proposed, which is independent of the data volume and that with the aid of conventional computer technology, in particular with the aid of a graphics processor (Graphics Processing Unit (GPU)) can be accelerated.

• (1) According to a first embodiment, a method for determining a planar projection of property values distributed on a spherical surface is proposed, which comprises the following steps: providing a property distribution on the surface of a (projection) sphere or on the curved surface of a (projection -) hemisphere, ie a hemisphere of a sphere; - Providing a virtual camera; - Aligning the camera so that a plane perpendicular to the image plane of the camera straight line passing through the center of the circular aperture (aperture stop), meets the center of the ball; - Selecting a desired 2-D planar projection from the group: gnomonic projection, stereographic projection (conformal azimuthal projection, or English conformal projection), distance-independent, ie, radial distance projection (English - equidistant projection), area-true projection (Engl - equal area projection), parallel projection (English - orthographic projection) and central projection (English - perspective projection); Selecting a camera position which is dependent on the desired projection as the distance of the camera to the projection sphere center, wherein a passing point of the straight line through the spherical surface, defined relative to the camera, defines a first vertex of the spherical surface; Capturing an image with the camera of at least a portion of a half sphere surface comprising the first vertex; Outputting a (2-dimensional) camera image as an image of the (curved) spherical surface corresponding to the desired 2D planar projection.

Advantages of this embodiment result from the accelerated representation of a surface-like projection by minimizing the amount of computation involved, since the actual property distribution is calculated only once, mapped onto the spherical surface, and then only has to be imaged by means of a camera. In this way, property distributions can be rotated virtually more quickly and quasi in real time and displayed at different viewing directions and angles of a spherical surface used as a projection surface on an electronic display, for example a display, a screen or with the aid of a projector, or printed out with the aid of a printer.

• (2) Gemäß einer zweiten Ausführungsform wird ein Verfahren vorgeschlagen, weiterhin umfassend: – Bewegen der virtuellen Kamera und/oder der Kugel relativ zueinander, wobei der gewählte erste Abstand beibehalten wird, und – Wiederholen des Erfassens eines Bildes.

Further advantages are the practically sufficient precision of the achieved surface fidelity or similarity, at the same time enormously accelerated, because simplified implementation. The actual and one-off calculation of the property distribution is practically as computationally intensive as before, but this one-time calculation is used to represent different viewing directions and distances - ie for different types of projection - so does not have to be constantly new, or is the corresponding software (eg OpenGL) implemented and the GPU hardware side executed. This allows a considerable reduction of the computational effort and a resulting time savings or virtually delay-free representation of different projections.

• (2) According to a second embodiment, a method is proposed, further comprising: moving the virtual camera and / or the sphere relative to each other while maintaining the selected first distance, and repeating the capture of an image.

• (3) Gemäß einer dritten Ausführungsform wird ein Verfahren vorgeschlagen, wobei das Erfassen ein zumindest abschnittsweises Aufnehmen derjenigen halben Kugeloberfläche umfasst, die den ersten Scheitelpunkt einschließt. Mit anderen Worten wird dabei ein vollständiges oder ein teilweises Bild (Teil-Bild) derjenigen halben Kugeloberfläche erfasst bzw. aufgenommen, die den ersten Scheitelpunkt einschließt. Die sich daraus ergebenden Vorteile betreffen eine weitere Beschleunigung der Darstellung besonders interessierender auf der Kugeloberfläche verteilt vorliegender Eigenschaften oder Parametersätze.
• (4) Gemäß einer vierten Ausführungsform umfasst das vorgeschlagene Verfahren weiterhin das Aufnehmen eines vollständigen Bildes jener halben Kugeloberfläche, die den ersten Scheitelpunkt nicht einschließt, das heißt jener, die noch nicht dargestellt wurde. Die sich daraus ergebenden Vorteile betreffen eine zusätzliche Minimierung des Rechenaufwandes und die zusätzliche Beschleunigung der Darstellung aller auf einer Kugeloberfläche verteilt vorliegenden Eigenschaften oder Parametersätze. Beispielsweise ergeben sich besondere Vorteile für die Vereinfachung relevanter Mess- oder Simulationsergebnisse auf dem Gebiet der Meteorologie und der Klimaforschung, ebenso aber auch der Kristallografie und Materialforschung. In dem Fall werden zwei sich diametral gegenüberliegende Kameras zur Abbildung genutzt, oder die Verteilung als zweite Projektionskugel mit entsprechend ausgerichteter Eigenschaftsverteilung und Kamera eingesetzt. Es wird ein Verfahren zur beschleunigten digitalen Darstellung von auf einer Kugeloberfläche verteilten Daten vorgeschlagen, wobei die Daten ausgewählt sind unter einer anisotropen Eigenschaft (Elastizitätstensor, Intensitätsverteilung in einem Beugungsbild usw.) oder einer anisotrop verteilten Eigenschaft (Intensität einer ausgewählten Kristallrichtung als Verteilung im Raum, d.h. wie bei der kristallographischen Texturanalyse).

Advantages of this embodiment result for the complete detection of the distributed present properties (values) or parameter sets, since a representation captures at most the property distribution of a half-space - as projected content of a hemisphere. Numerous property distributions are centrosymmetric, so that lower and upper hemispheres share the same content. In this case, the representation of a hemisphere to describe the property is sufficient. For non centrosymmetric distributions, eg distribution of the continental surfaces on the earth, or the stars in the universe, two hemispheres have to be shown for a complete representation. A substantial advantage results for the frequently existing requirement of a descriptive visual representation of complex connections. For crystallography, there are particular advantages in the determination of preferred orientations or the subsequent alignment of a microscopic structure, eg taking into account a loading direction imposed on the structure. In this way, an actually oblique cut or bevel of a material sample (real cutting plane) can be rotated so that the representation is orthogonal to a surface normal of the structure, ie, as if the bevel again made well-aligned and measured.

• (3) According to a third embodiment, a method is proposed, wherein the capturing comprises an at least partial recording of that half spherical surface which encloses the first vertex. In other words, a complete or partial image (partial image) of the half spherical surface that includes the first vertex is acquired. The resulting advantages relate to a further acceleration of the representation of particularly interesting properties or parameter sets distributed on the spherical surface.
• (4) According to a fourth embodiment, the proposed method further comprises taking a complete image of that half spherical surface which does not include the first vertex, that is, that which has not yet been illustrated. The resulting advantages relate to an additional minimization of the computational effort and the additional acceleration of the representation of all properties or parameter sets distributed on a spherical surface. For example, there are particular advantages for simplifying relevant measurement or simulation results in the field of meteorology and climate research, as well as crystallography and materials research. In this case, two diametrically opposed cameras are used for imaging, or the distribution is used as a second projection sphere with a correspondingly aligned property distribution and camera. A method for accelerated digital display of data distributed on a spherical surface is proposed, the data being selected from an anisotropic property (elasticity tensor, intensity distribution in a diffraction image, etc.) or anisotropically distributed property (intensity of a selected crystal direction as distribution in space, ie as in the crystallographic texture analysis).

• (5) Gemäß einer fünften Ausführungsform erfolgt das oben erwähnte Bewegen von Kugeloberfläche und Kamera relativ zueinander und das Wiederholen dieser Schritte so lange, bis allen auf der Kugeloberfläche verteilten Eigenschaften bestimmte Zielkoordinaten, d. h. Flächenkoordinaten zugeordnet sind. Die erzielten Vorteile sind die bereits vorstehend benannte Beschleunigung durch eine Vereinfachung notwendiger Rechenschritte. Dabei können die Zielkoordinaten in Abhängigkeit von der jeweils aktuellen Datenverteilung gewählt werden. Ebenso kann die Zuordnung von Flächenkoordinaten auch nur für einen interessierenden Teil der insgesamt verteilt vorliegenden Daten vorgenommen werden.
• (6) Gemäß einer weiteren Ausführungsform ergibt das vorgeschlagene Verfahren zum Ermitteln einer Planarprojektion einer auf einer Halbkugeloberfläche (Kugelhalbschale) vorliegenden Eigenschaftsverteilung eine flächenähnliche Projektion der Kugeloberfläche, wenn der erste Abstand der 2,15-fache bis 2,3-fache Wert des Radius der Kugel, insbesondere der 2,268-fache Wert des Radius der Kugel ist.
• (7) Gemäß einer weiteren Ausführungsform ergibt das vorgeschlagene Verfahren zum Ermitteln einer Planarprojektion einer auf einer Halbkugeloberfläche (Kugelhalbschale) verteilt vorliegenden Eigenschaft eine abstandsähnliche Projektion der Kugeloberfläche, wenn der erste Abstand ein Wert im Bereich vom 1,9-fachen bis zum 2,1-fachen des Radius ist, insbesondere der 2-fache Radius ist bzw. der Durchmesser der Kugel ist.
• (8) Gemäß einer weiteren Ausführungsform ergibt das vorgeschlagene Verfahren zum Ermitteln einer Planarprojektion einer auf einer Kugeloberfläche verteilt vorliegenden Eigenschaft eine stereografische Projektion der Kugeloberfläche, wenn der erste Abstand ein Wert im Bereich vom 0,85-fachen bis zum 1,15-fachen des Radius der Kugel ist, insbesondere der Radius der Kugel ist.
• (9) Gemäß einer weiteren Ausführungsform ergibt das vorgeschlagene Verfahren zum Ermitteln einer Planarprojektion einer auf einer Kugeloberfläche verteilt vorliegenden Eigenschaften eine gnomonische Projektion der Kugeloberfläche, wenn der erste Abstand einen Wert im Bereich des – 0,1-fachen bis zum 0,1-fachen des Radius der Kugel ist, insbesondere wenn der erste Abstand den Wert 0 annimmt.
• (10) Gemäß einer weiteren Ausführungsform wird ein Verfahren zum Ermitteln einer Planarprojektion einer auf einer Kugeloberfläche verteilt vorliegenden Eigenschaft vorgeschlagen, wobei die Eigenschaft eine Größe ist, ausgewählt unter einer zur zumindest ausschnittsweisen Charakterisierung der Erde oder eines anderen Himmelskörpers dienenden Größe, die gemessen oder errechnet (simuliert oder modelliert) sein kann, umfassend eine astronomische, meteorologische, geologische, glaziologische, hydrologische, geografische, ozeanografische, klimatologische, biologische, ökologische, agronomische, soziologische, demografische, politische, militärische, kulturelle, ökonomische, telemetrische, funktechnische, avionische, Satelliten oder andere Raumflugkörper, oder Funknetze, insbesondere Mobilfunknetze betreffenden Eigenschaft.

Advantages of this embodiment are the quick and simplified presentation Complex measurement situations from application areas of optical imaging with the help of marker substances.

• (5) According to a fifth embodiment, the above-mentioned moving of the spherical surface and the camera relative to each other and the repetition of these steps takes place until all the properties distributed on the spherical surface are assigned specific target coordinates, ie surface coordinates. The advantages achieved are the above-mentioned acceleration by simplifying necessary calculation steps. The target coordinates can be selected depending on the current data distribution. Likewise, the assignment of area coordinates can also be made only for a part of interest of the total distributed data available.
• (6) According to a further embodiment, the proposed method for determining a planar projection of a property distribution present on a hemisphere surface (spherical half-shell) yields a surface-like projection of the spherical surface when the first distance is 2.15 times to 2.3 times the value of the radius Sphere, in particular, is 2.268 times the radius of the sphere.
• (7) According to a further embodiment, the proposed method for determining a planar projection of a property distributed on a hemisphere surface (spherical half-shell) yields a distance-like projection of the spherical surface when the first distance has a value in the range from 1.9 times to 2.1 times the radius is, in particular, the 2-fold radius is or the diameter of the ball.
• (8) According to another embodiment, the proposed method for determining a planar projection of a property distributed on a spherical surface yields a stereographic projection of the spherical surface when the first distance is in the range of 0.85 to 1.15 times the Radius of the ball is, in particular the radius of the ball is.
• (9) According to a further embodiment, the proposed method for determining a planar projection of a distributed properties on a spherical surface gives a spherical projection of the spherical surface when the first distance has a value in the range of -0.1 times to 0.1 times is the radius of the sphere, in particular if the first distance assumes the value 0.
• (10) According to a further embodiment, a method for determining a planar projection of a property distributed on a spherical surface is proposed, wherein the property is a variable selected from a variable serving for at least partially characterizing the earth or another celestial body, which measures or calculates (simulated or modeled), including astronomical, meteorological, geological, glaciological, hydrological, geographic, oceanographic, climatological, biological, environmental, agronomic, sociological, demographic, political, military, cultural, economic, telemetric, radio, avionics, Satellites or other spacecraft, or radio networks, in particular mobile telephony network property.

• (11) Gemäß einer weiteren Ausführungsform wird ein Verfahren zum Ermitteln einer Planarprojektion einer auf einer Kugeloberfläche verteilt vorliegenden Eigenschaft vorgeschlagen, wobei die Eigenschaft ausgewählt ist unter einer physikalisch oder chemisch anisotropen Eigenschaft oder einer anisotrop verteilten physikalischen oder chemischen Eigenschaft., die zu Zwecken der beschleunigten Verarbeitung und/oder der Visualisierung auf eine virtuelle Kugeloberfläche projiziert wird.

Advantages arise, for example, for the improved understanding of complex relationships, the expedient design of radio networks, the easier solution of logistics issues or the optimized decision-making in other spheres of human activity.

• (11) According to a further embodiment, a method for determining a planar projection of a property distributed on a spherical surface is proposed, wherein the property is selected from a physically or chemically anisotropic property or an anisotropically distributed physical or chemical property, for accelerated Processing and / or the visualization is projected onto a virtual sphere surface.

• (12) Gemäß einer weiteren Ausführungsform wird ein Verfahren zum Ermitteln einer Planarprojektion einer auf einer Kugeloberfläche verteilt vorliegenden Eigenschaft vorgeschlagen, wobei die Eigenschaft eine einer Achse eines Kristalls zuordenbare physikalische oder chemische Größe, eine Elektronendichte oder eine Aufenthaltswahrscheinlichkeit eines Elektrons oder eines anderen Teilchens, z.B. eines Positrons oder eines Kerns, ist. Mit anderen Worten kann die Eigenschaft eine anisotrope physikalische oder chemische Größe (z.B. die Richtungsabhängigkeit der Stärke einer chemischen Bindung) sein. So kann sie beispielsweise ausgewählt sein unter: einer Materialeigenschaft, umfassend Härte, E-Modul, Bruchschlagzähigkeit, elektrische Leitfähigkeit, Wärmeleitfähigkeit, Konzentration, chemischer Aktivität, Elektronendichte, Aufenthaltswahrscheinlichkeit eines Elementarteilchens und anderen.

Advantages of this embodiment relate to the accelerated graphical representation and / or visualization on a flat surface of any feature distributed on a curved surface, in particular the rapid display of data distributed on a projection spherical surface, if for better understanding of existing measurement situations between individual views or clippings the projection surface is changed back and forth.

• (12) According to a further embodiment, a method for determining a planar projection of a property distributed on a spherical surface is proposed, wherein the property is a physical or chemical size, an electron density or a residence probability of an electron or another particle, eg a positron or a nucleus, is. In other words, the property may be an anisotropic physical or chemical quantity (eg, the directional dependence of the strength of a chemical bond). For example, it may be selected from among a material property including hardness, Young's modulus, fracture toughness, electrical conductivity, thermal conductivity, concentration, chemical activity, electron density, residence probability of an elementary particle, and others.

• (13) Gemäß einer weiteren Ausführungsform umfasst das vorgeschlagene Verfahren weiterhin ein Darstellen von ermittelten Flächenkoordinaten als Digitalbild.

Advantages arise, for example, for the accelerated representation of measurement signals of a detector for an electron microscope, in particular a detector for the detection of backscattered or emitted under the influence of high-energy radiation particles, and the entire field of crystallography and their application, for example in materials research and materials testing.

• (13) According to a further embodiment, the proposed method further comprises displaying determined surface coordinates as a digital image.

• (14) Gemäß einer weiteren Ausführungsform wird eine grafische Darstellung vorgeschlagen, die eine Planarprojektion umfasst, die gemäß einem der vorstehend genannten Verfahren gewonnen, oder von einem solchen hergeleitet wurde.

Advantages of this embodiment result in easier visualization of existing complex relationships and the possibility of directly incorporating obtained results into models, application-specific user interfaces or software modules for operating and controlling measurement and analysis technology, in particular electron microscopes, scanning electron microscopes, tomographs and X-ray apparatuses.

• (14) According to another embodiment, a graphical representation is proposed which comprises a planar projection obtained according to one of the aforementioned methods or derived from such.

In particular, the graphical representation can take the form of a screen image or a video sequence, in the form of a computer printout on paper or on film or else with light, for example with the aid of at least one laser beam guided over a display surface.

• (15) Gemäß einer weiteren Ausführungsform wird die Visualisierung von Mess-, Analyse-, oder Simulationsdaten gemäß einem der benannten Verfahren vorgeschlagen. Dabei können zusätzliche Farb- oder Musterkodierungen für einzelne Projektionsarten oder für Zahlenbereiche der dargestellten Werte vorgenommen werden. In ihrer Gesamtheit verbessern die vorgeschlagenen Ausführungsformen beispielsweise die Nutzerfreundlichkeit von Anwendersoftware zur Gerätesteuerung, die Anschaulichkeit komplexer Messsituationen oder erleichtern das Verständnis komplexer Zusammenhänge.
• (16) Gemäß einer weiteren Ausführungsform wird eine Anordnung mit mindestens einem Chip und/oder Prozessor vorgeschlagen, wobei die Anordnung derart eingerichtet ist, dass ein Verfahren zum Ermitteln einer Planarprojektion gemäß einer der vorstehend umrissenen und/oder nachfolgend detaillierter beschriebenen Ausführungsformen ausführbar ist.

Advantages exist, for example, in an increased user-friendliness of software for controlling X-ray and / or electron microscopes, in particular in the possibility of improved human-machine communication in the control of such devices.

• (15) According to another embodiment, the visualization of measurement, analysis, or simulation data according to one of the named methods is proposed. Additional color or pattern encodings can be made for individual projection types or for numerical ranges of the displayed values. In their entirety, the proposed embodiments improve, for example, the user-friendliness of application software for device control, the clarity of complex measurement situations or facilitate the understanding of complex relationships.
• (16) According to a further embodiment, an arrangement with at least one chip and / or processor is proposed, wherein the arrangement is set up such that a method for determining a planar projection according to one of the embodiments outlined above and / or subsequently described in more detail is feasible.

• (17) Gemäß einer weiteren Ausführungsform umfasst die vorgeschlagene Anordnung weiterhin eine Grafikkarte.

By way of example, the chip and / or processor can comprise an ASIC adapted for the respective field of application. This results in advantages for the rapid representation of each selected projection, in particular the rapid output of the camera image of the spherical surface, which corresponds to the desired planar projection.

• (17) According to another embodiment, the proposed arrangement further comprises a graphics card.

• (18) Gemäß einer weiteren Ausführungsform wird ein Computerprogramm vorgeschlagen, das es einer Datenverarbeitungseinrichtung ermöglicht, nachdem es in Speichermittel der Datenverarbeitungseinrichtung geladen worden ist, ein Verfahren zum Ermitteln einer Planarprojektion gemäß einem der Ansprüche 1 bis 13 durchzuführen.

Advantages result from the fact that commonly used computers and computer systems are typically already equipped with a graphics card that includes a graphics processor (GPU). These components are adapted for fast data processing, especially with the aid of graphics programs. For this reason, the execution of the proposed method in the interplay of graphics card, graphics processor, computer and computer screen allows a particularly rapid representation of 2-dimensional images according to the method made projections and allows for example when embedded in the appropriate control software, the effective device control of particle beam-based Analysis systems, such as scanning electron microscopes or X-ray microscopes.

• (18) According to another embodiment, a computer program is proposed, which allows a data processing device, after being loaded into storage means of the data processing device, to perform a method for determining a planar projection according to one of claims 1 to 13.

• – Bereitstellen einer Werteverteilung auf einer Kugeloberfläche;
• – Bereitstellen einer virtuellen Kamera, umfassend eine Aperturblende;
• – Ausrichten der Kamera auf den Mittelpunkt der Kugel;
• – Auswählen einer gewünschten 2D-Planarprojektion;
• – Auswählen einer als Funktion von der gewünschten Projektion abhängigen Kameraposition in einem definierten ersten Abstand der Kamera zum Mittelpunkt der Kugel;
• – Erfassen eines Bildes mit der Kamera von zumindest einem Ausschnitt einer halben Kugeloberfläche;
• – Ausgeben eines Kamerabildes der halben Kugeloberfläche oder zumindest des einen Ausschnitts der halben Kugeloberfläche, entsprechend der gewünschten 2D-Planarprojektion.

• (19) Gemäß einer weiteren Ausführungsform wird ein computerlesbares Speichermedium vorgeschlagen, auf dem ein Programm gespeichert ist, das es einer Datenverarbeitungseinrichtung ermöglicht, nachdem es in Speichermittel der Datenverarbeitungseinrichtung geladen worden ist, ein Verfahren zum Ermitteln einer Planarprojektion gemäß einem der Ansprüche 1 bis 13 durchzuführen.

Advantages result from the fact that the method steps or method sequences specified above or subsequently executed further can be carried out. Such computer programs can be made available for download (for a fee or free of charge, freely accessible or password-protected) in a data or communication network, for example. A computer program or a combination of modular programs provided in this way can then be utilized by a method comprising the following steps:

• Providing a value distribution on a spherical surface;
• Providing a virtual camera comprising an aperture stop;
• - Align the camera to the center of the sphere;
• - selecting a desired 2D planar projection;
• Selecting a camera position dependent on the desired projection at a defined first distance of the camera to the center of the sphere;
• Capturing an image with the camera of at least a section of half a spherical surface;
• Outputting a camera image of half the spherical surface or at least of a section of the half spherical surface, according to the desired 2D planar projection.

• (19) According to another embodiment, there is provided a computer-readable storage medium having stored thereon a program that allows a data processing device, after being loaded into storage means of the data processing device, to perform a planar projection determination method according to any one of claims 1 to 13 ,

• – Bereitstellen einer Werteverteilung auf einer Kugel- bzw. Halbkugeloberfläche;
• – Bereitstellen einer virtuellen Kamera, umfassend eine Aperturblende;
• – Ausrichten der Kamera auf den Mittelpunkt der Kugel;
• – Auswählen einer gewünschten 2D-Planarprojektion;
• – Auswählen einer als Funktion von der gewünschten Projektion abhängigen Kameraposition in einem definierten ersten Abstand der Kamera zum Mittelpunkt der Kugel;
• – Erfassen eines Bildes mit der Kamera von zumindest einem Ausschnitt einer halben Kugeloberfläche;
• – Ausgeben eines Kamerabildes der halben Kugeloberfläche oder zumindest des einen Ausschnitts der halben Kugeloberfläche, entsprechend der gewünschten 2D-Planarprojektion.

• (20) Gemäß einer weiteren Ausführungsform wird ein Verfahren vorgeschlagen, bei dem ein Computerprogramm aus einem elektronischen Datennetz, wie beispielsweise aus dem Internet, auf eine an das Datennetz angeschlossene Datenverarbeitungseinrichtung heruntergeladen wird, wobei das Computerprogramm, nachdem es in ein Speichermittel der Datenverarbeitungseinrichtung geladen worden ist, das Durchführen eines Verfahrens zum Ermitteln einer Planarprojektion gemäß den vorstehend beschriebenen Ausführungsformen gestattet.

Advantageously, the proposed storage medium can be used to carry out the method according to the invention for determining a planar projection, the method comprising the steps outlined above in more detail:

• - Providing a value distribution on a spherical or hemisphere surface;
• Providing a virtual camera comprising an aperture stop;
• - Align the camera to the center of the sphere;
• - selecting a desired 2D planar projection;
• Selecting a camera position dependent on the desired projection at a defined first distance of the camera to the center of the sphere;
• Capturing an image with the camera of at least a section of half a spherical surface;
• Outputting a camera image of half the spherical surface or at least of a section of the half spherical surface, according to the desired 2D planar projection.

• (20) According to another embodiment, a method is proposed in which a computer program is downloaded from an electronic data network, such as from the Internet, to a data processing device connected to the data network, the computer program having been loaded into a storage means of the data processing device is, performing a method for determining a planar projection according to the embodiments described above allowed.

The above-described embodiments may be arbitrarily combined with each other. Other embodiments, modifications and improvements will become apparent from the following description and the appended claims.

According to typical embodiments, the projection of a radial property distribution (eg, the distribution of an intensity) or the distribution of individual points according to the proposed method is not as usual on the calculation of the individual points, but by converting the data into an image (graph), which by suitable image immediately appears in the desired projection.

Since the property distribution does not change by rotation, its spherical projection or the conversion described above in [0007] must be generated or calculated only once. This results in the respective desired projection directly from the first calculation or representation of the spherical projection. This has the advantage that no recalculation of the projection data into the plane is required, but only the image is rotated on the display medium used (e.g., screen, display, projector surface, printout). The required computational effort is negligible and not perceptible by the user.

Typically, the proposed method uses to represent (i.e., generate the desired plane projection by virtual camera imaging) of a platform and programming language independent graphics interface. For example, the platform- and programming language-independent graphics interface OpenGL (Open Graphics Library) can be used. The radial property distribution is calculated only once and converted into one or more bitmaps that describe the overall space and are used for display as well as for projection using the selected graphics interface.

The accompanying figures illustrate exemplary embodiments and, together with the description, serve to explain the principles of the proposed method.

1 shows the camera positions for different projections of a hemisphere surface.

2 shows in the form of an alternative representation of the resulting at different camera positions projections.

3 shows one and the same property distribution (intensity distribution of a reflex, radial energy radiation of the sun (sun spots), distribution of meteorite impacts on the moon etc.), shown as a central perspective projection (A) and as a stereographic projection (B).

4 shows the schematic representation of the construction of the projection plane 3 according to the conventional Lambert projection and the virtual projection plane used by simple mapping 4 according to the proposed method.

5 shows the camera positions for the gnomonic (left), stereographic (center), and area-true (right) projection generated by simple mapping for a property distribution projected onto a hemisphere surface.

According to the proposed method, an azimuthal Lambert projection of the property distribution can be achieved by simply changing the position of the OpenGL camera.

For this purpose, according to a typical embodiment, the position of the OpenGL camera is shifted out of the center of the projection sphere by twice to 2.5 times the value of the radius of the projection sphere. Typically, the position of the OpenGL camera is shifted from the center of the projection sphere by 2.1 times to 2.3 times the radius of the sphere, in particular by the value of 2.268 × R, where R is the radius of the sphere (projection sphere ).

Typically, the geometric projection takes place in the azimuthal plane at the vertex, with the point on the spherical surface is transmitted with its secant distance in the projection surface. This corresponds approximately to one of the camera position P _e from a central perspective image of the spherical surface on a parallel to the azimuthal plane surface. In this case, the vertex S or Augpunkt that lying on the spherical surface point which is the camera diametrically opposite or furthest away from the camera position P _e .

The construction of the projection plane 4 is in 4 shown schematically. Here it can be seen that the designated vertex N corresponds to that point on the half of the projection sphere facing away from P _e , which results from the impingement on the spherical surface of the extension of a line running from the camera position P _e through the center of the sphere.

The image resulting from viewing the inner surface of the opposite hemisphere corresponds to the azimuthal Lambert projection for the designated camera position P _e with slight deviations. The occurring local deviations of the pixels, generated by a central projection by means of camera, of which the exact azimuthal lambert projection result from the set of rays and amount to a maximum of 0.6% for the preferred camera position P _e .

Since both the sphere and the camera can be freely rotated with respect to the center of the sphere, this results in a delay-free equalization of the property distribution shown on the sphere surface with a largely exact projection.

As can easily be understood, spherical projections can be used to represent undistorted the radial distribution of a property on a spherical surface. Likewise, spherical projections can also be used by not looking at the convexly curved surface (from the outside) but at the concavely curved surface (from the inside) as usual. 1 shows the projection types resulting from a view into a hemisphere for different viewpoints (camera positions). If one positions a camera at the "south pole" of the sphere shown, the distorted image, which now becomes visible on the inner sphere surface, corresponds to the stereographic projection of the pattern imaged on the sphere surface (property distribution).

This relationship is again in 2 illustrated. In particular shows 2 alternative to 1 a comparative representation of resulting projections, which can be generated by positioning an OpenGL virtual camera with a view to a spherical projection (half projection sphere) at the respective positions. The projections from positive direction (+) look at the outer (convexly curved) surface of the sphere, while those from the negative direction (-) look at the inner (concavely curved) surface of the half sphere. Except for the orthographic parallel projection, all other images represent central projections.

By moving the (virtual) camera to different positions, the representation of the gnomonic, the radially distance-like and - as shown below - even a surface-like projection is possible: This subsequently surprisingly simple context of different types of projection results solely from the position of the viewer, or of a selected camera position in relation to the ball center point: When the camera is positioned at point G, the result is gnomonic, when positioned at point S (South Pole) the stereographic (conformal), for the position of the camera at the point ED results in the (radial) distance-like and surprising for the position in the point EA the proposed here area-like projection.

The images resulting for different projections with the same property distribution are in 3 using the example of the comparison of the stereographic projection to the central projection (spherical surface from the outside). In particular, two equal property distributions are shown, once shown as a central perspective projection (A) and shown on the other hand as a stereographic projection (B).

Surprisingly, it has been found that an image of a central projection, which is very similar to the classical Lambert projection, can be obtained in a much simpler manner than heretofore if the projection point P _{e is brought} to a certain distance from the center of the sphere.

This shows 4 more accurate. It is shown how, in accordance with conventional embodiments, to obtain a central projection according to Lambert, a radial property, for example the direction of vectors, is plotted from a point (Origin). This point is designated 1 here. The projection plane is the surface of the sphere whose center coincides with the projection origin (spherical projection). In order to map the information from the spherical surface to a flat surface, the azimuthal Lambert or even area area or area-consistent projection is usually used. The secant lengths are transferred from the "North Pole" to the flat surface. The north pole is considered to be the point at which the projection plane (tangential plane) "touches" the sphere surface - the corresponding points (and their associated data) thus coincide. The resulting outwardly inevitably increasing radial distortion of the Lambert projection can be seen on the basis of the size differences of the squares, which result from the difference between the points distributed regularly (equidistantly) at a 15-degree distance from one another on the projection spherical surface.

Is in 5 shown again for three types of projection. In particular, illustrated 5 using the example of thirteen (15) points offset along the equator of a projection sphere (properties) the positions of the projection point P _e , ie the position of a camera for generating planar images of the gnomonic projection (left), the stereographic projection (center) and the coextensive projection (right). At the same time, the simplified azimuthal Lambert projection proposed here is also shown on the right 4 outlined.

As explained, according to the proposed method, the virtual camera used to produce the planar image of the projection is placed at a distance of 2.228 times the radius of the projection sphere to the center of the projection sphere. In this case, as specified above, this distance may also be slightly smaller or larger than the 2.268 times the radius in order to obtain the character of the area-true projection. With the 2.268-fold radius, only the distance is determined under which the radial deviation to the exact construction is the lowest. The computation of the camera image (i.e., the projection) is done using conventional computer programs or calculation routines known to those skilled in the art, which are executed by the graphics processor (GPU). It requires computationally much less effort than the classic Lambert projection. This results in implementation of the corresponding calculation routines with the help of the coordinated interaction of a graphics processor (GPU) and a graphics card or a corresponding computer hardware significant advantages for an accelerated result representation.

• a) die nun erhaltene Abbildung ist gemäß dem Strahlensatz eine geometrisch ähnliche Abbildung, denn in Höhe der in 4 mit 4 bezeichneten Ebene wären beide Projektionen praktisch fast identisch, und
• b) die Projektionen sind eben nur fast identisch, denn es treten Abweichungen auf, die im Falle eines Abstandes des Projektionszentrums P_e vom Mittelpunkt der Kugel, bzw. der Halbkugel des 2,268-fachen des Radius der Kugel bzw. Halbkugel allerdings maximal nur 0.6% betragen. Für andere -Abstände sind die Abweichungen entsprechend größer.

The resulting similarity of the proposed simplified projection to the classical Lambert projection arises for two reasons:

• a) the image obtained is a geometrically similar image according to the set of rays, because at the level of 4 With 4 level, both projections would be almost identical, and
• b) the projections are just almost identical, because deviations occur, which in the case of a distance of the projection center P _e from the center of the sphere, or the hemisphere of 2.268 times the radius of the sphere or hemisphere, however, only a maximum of 0.6% be. For other distances the deviations are correspondingly larger.

This means that with a virtual camera positioned at the point P _e , the view in the direction of the projection plane 3 and 4 located (upper) hemisphere 2 corresponds to the azimuthal Lambert projection. A rotation of the pattern (property distribution) on the sphere or a rotation of the camera around the origin point 1 always results in a Lambert projection without requiring an additional, explicit recalculation, whereby the computational conversion is performed by the graphics processor.

The advantages of the proposed method thus exist on the one hand in the possibility of a delay-free rotation by means of different projection of radial Property distributions, as well as on the other hand in the intuitive operation of a standard example in the texture research used projection of data at extremely low computational effort.

As a result, the data interpretation is significantly simplified, which causes a significant reduction in the workload.

A projection of about 800,000 data values requires about 15 seconds of computing time (dual-core Intel i5-460M). Every rotation of the data would require the same amount of time over again with the classic Lambert projection. In contrast, with the method proposed here, after a first calculation, there is no longer a delay, since no further calculation of the projection data has to be made, but rather the property distribution described for the first time as a spherical projection by means of the corresponding software, e.g. OpenGL, only rotated and - depending on the desired projection - is imaged by a camera shifted to a certain point.

The method proposed here enables the accelerated digital representation or mapping of physical measured values, parameters, and / or properties in the form of a digital image with the aid of conventional presentation tools or computer programs. In this context, a digital image is understood to be an image constructed from discrete pixels, it being possible to associate surface coordinates with the pixels in each case. In this case, individual pixels or all individual pixels continue to be assigned different information, for example in the form of different bit values (image depth, bit depth). The digitally generated or stored image may be a black and white image, a grayscale image, a bi-color, tri-color, or other color image or a color separation image or graphic, such as an RGB bitmap. The digital format of the electronically storable or stored present image data (graphic format) is accordingly freely selectable and can be selected for example under TIFF, GIF, BMP, JPG or any other digital formats.

Typically, digital data are determined with the aid of the described method, which are to be used for the visual representation of complex relationships of practically significant measurement situations, for example from physical measurement technology. Exemplary fields of application include materials research; the crystallography; microstructure research, in particular X-ray structure analysis, for example X-ray diffraction analysis; geology, especially geodesy, cartography; transport, in particular aerospace and flight control in the broadest sense; the meteorology; photometry or surveying.

The mentioned presentation tools or computer programs are typically object-oriented software programs and aids, such as those provided by the program module OpenGL.

Particular applications of the described projection method are in the field of materials science and materials science, for example in materials science using EBSD, in X-ray diffraction analysis by XRD and texture research, generally in the field of quality control, mining, mineralogy, geology and chemistry Cartography, and in particular for geography and meteorology.

Particular advantages of the described embodiments relate to applications in the field of materials science, in particular the reliability study, material testing and defectoscopy.

Further advantages result from a rapid representation of the properties distributed on a spherical surface, or from data projected onto a virtual spherical surface, even if the observation position or changed value set is changed. Overall, the advantages are thus in the simple computational implementation using known and hardware-side, for example, at processor level already implemented algorithms and the resulting extremely short computation times.

While specific embodiments have been illustrated and described herein, it is within the scope of the present invention to properly modify the illustrated embodiments without departing from the scope of the present invention. The following claims are a first, non-binding attempt to broadly define the invention.

LIST OF REFERENCE NUMBERS

N

North Pole, vertex;

EA, P _e

Camera position for (area-consistent) Lambert projection;

G

Camera position for gnomonic projection;

S

Camera position for stereographic projection (conformal);

ED

Camera position for distance-stable projection;

1

Projection sphere center or origin position of a virtual camera in the center of the spherical surface with gnomonic projection;

2

Upper hemisphere, or half of the projection ball facing away from the camera;

3

azimuthal projection plane of the (area-equivalent) Lambert projection;

4

virtual projection plane according to the proposed method;

Claims (20)

A method for determining a planar projection of a property distributed on a spherical surface, comprising: Providing a value distribution concerning the property on a spherical surface or a hemisphere surface; Providing a virtual camera comprising an aperture stop; - Aligning the camera, so that a perpendicular to the image plane of the straight line passing through the center of an aperture diaphragm used to hide meets the center of the ball; - Select a desired 2D planar projection from the group: gnomonic projection, stereographic projection, distance fidelity, i. Radial distance projection, areal projection, parallel projection and central projection; Selecting a camera position depending on the desired projection at a first distance of the camera from the center of the sphere, a passage point of the line lying behind the center of the sphere defining a first vertex of the spherical surface relative to the camera; Capturing an image with the camera of at least a portion of a half sphere surface comprising the first vertex; Outputting a camera image as an image of the spherical surface corresponding to the desired 2D planar projection. The method of claim 1, further comprising Moving the virtual camera and / or the sphere relative to one another while maintaining the selected first distance, and - repeating the capture of an image. The method of claim 1, wherein detecting comprises at least partially receiving the half sphere surface including the first vertex. The method of claim 3, further comprising: capturing a complete image of that half spherical surface that does not include the first vertex, that is, that which has not yet been detected. The method of claim 2, wherein the moving and repeating takes place until all distributed on the spherical surface properties are assigned surface coordinates of a projection plane. Method according to one of the preceding claims, wherein the planar projection results in a surface-like projection of half the spherical surface, if the first distance of 2.15 times to 2.3 times the value of the radius of the ball, in particular 2.268 times the value of the radius of the ball is. Method according to one of the preceding claims, wherein the planar projection results in a distance-like projection of half the spherical surface, if the first distance is a value in the range of 1.9 times to 2.1 times the radius, in particular 2 times the radius or the diameter of the ball is. A method according to any one of the preceding claims, wherein the planar projection gives a stereographic projection of the spherical surface if the first distance is a value in the range of 0.85 to 1.15 times the radius of the sphere, in particular if the first distance is Radius of the ball is. Method according to one of the preceding claims, wherein the planar projection results in a gnomonische projection of the spherical surface, when the first distance is a value in the range of - 0.1 times to 0.1 times the radius of the ball, in particular if the first distance assumes a value of 0. Method according to one of the preceding claims, wherein the property is a size selected from a simulated or modeled variable measured to at least partially characterize the earth or another celestial body, comprising an astronomical, meteorological, geological, glaciological, hydrological, geographic, oceanographic, climatological, biological, ecological, agronomic, sociological, demographic, political, military, cultural, economic, telemetric, radio, avionics, satellites or other spacecraft, radio or mobile telephony networks. A method according to any one of the preceding claims, wherein the property is selected from a physically or chemically anisotropic Property or anisotropically distributed physical or chemical property. The method of any one of claims 1 to 7, wherein the property is a physical or chemical size, an electron density or a probability of residence of an electron or other particle attributable to an axis of a crystal. A method according to any one of the preceding claims, further comprising: - Representing the determined surface coordinates as a digital image. A graphical representation comprising a planar projection obtained according to one of the methods of claims 1 to 13 or derived therefrom. Visualization of measurement, analysis, or simulation data according to one of claims 1 to 13. Arrangement with at least one chip and / or processor, wherein the arrangement is set up such that a method for determining a planar projection according to one of claims 1 to 13 can be executed. Arrangement according to claim 18, characterized in that the arrangement further comprises a graphics card. Computer program that allows a data processing device, after it has been loaded into storage means of the data processing device to perform a method for determining a planar projection according to one of claims 1 to 13. A computer-readable storage medium having stored thereon a program that allows a data processing device, after being loaded into storage means of the data processing device, to perform a planar projection determination method according to any one of claims 1 to 13. Method in which a computer program according to claim 18 is downloaded from an electronic data network, such as for example from the Internet, to a data processing device connected to the data network.

Images