WO1992009967A1 - A system for imaging objects in alternative geometries - Google Patents
A system for imaging objects in alternative geometries Download PDFInfo
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- WO1992009967A1 WO1992009967A1 PCT/US1991/008919 US9108919W WO9209967A1 WO 1992009967 A1 WO1992009967 A1 WO 1992009967A1 US 9108919 W US9108919 W US 9108919W WO 9209967 A1 WO9209967 A1 WO 9209967A1
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Classifications
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
- G06T17/00—Three dimensional [3D] modelling, e.g. data description of 3D objects
Definitions
- the present invention relates to the field of imaging devices, for use in the educational,
- Lobachevskian geometry which is a hyperbolic geometry, one sees that it is not possible to change the size of a given figure without at the same time changing its shape which implies in effect the existence of an absolute standard of length.
- the "fourth" dimension is interpreted as time.
- Einstein's special theory of relativity has made it a commonplace assumption that the fourth dimension is time, while the curvature of physical space or the space-time continuum has become a common concept through popularization of
- geometries such as 4-D Euclidean
- 4-D Euclidean is often regarded as an important impediment, and many methods have been devised to supplant the missing direct intuition.
- Such methods include the display of objects in alternative geometries, such as 4-D Euclidean, through the use of projections, intersections or models in conventional space or 3-D Euclidean geometry in a great variety of forms.
- these take the form of some method of transformation which maps the figure which has been formally defined in an alternative geometry into a figure within 3-D Euclidean space, which we are then able to contemplate in the usual way.
- such a figure may consist of an intersection of the figure in the alternative geometry with Euclidean
- three-dimensional models which can in turn be rotated as solid figures in 3-D space and/or viewed in three-dimensional perspective.
- the form and complexity of the original figure in the alternative geometry may, by such methods, be revealed through first giving it rotation in its original space, changing the mode or center of projection, in that space and then observing the consequence as shown in the resulting object.
- 4-D object such as the Hypercube in 4-D space, and then computes and displays corresponding mappings in conventional space either as intersections, orthogonal projections, or stereographic
- Orthogonal projections are those in which the conventional coordinates are projected unaltered, while the fourth is ignored
- Stereographic projections are those in which the three-dimensional object is generated by rays or lines originating from a point at finite distance. Images so generated have been used to produce animated films of the 4-D cube or hypercube, and the 4-D sphere or hypersphere. In these films, as the objects are rotated, dramatic animated sequences are produced in full color and with computer-aided enhancements of light and shading. The resulting images aid greatly in approximating an intuitive sense of the four-dimensional objects themselves. However, these images do not reproduce the effect of light rays coming directly from the object in the alternative geometry, and hence, do not provide a view of the 4-D object itself, but show only the 3-D projections of the 4-D object.
- curvilinear geometry In order to visualize the curvature of a three-dimensional space properly, however, it is necessary to embed it in a linear space of four dimensions. With the availability of a wealth of computer technology for graphical design and imaging, and even for the experiencing of virtual reality in various modes, it seems clear that an instrument is needed to provide for greater visual intuition of objects in alternative geometries. Such an instrument would need to provide what is in effect a direct visualization of the objects themselves in the alternative geometry, rather than the projections, models and intersections of the prior art.
- geometries such as, but not limited to, four-dimensional Euclidean and three-dimensional
- intersections, or projections of the object in the alternative geometry are intersections, or projections of the object in the alternative geometry.
- invention to provide a system for the direct visualization of an object specifically in 4-D Euclidean geometry.
- invention to provide a system for the direct visualization of an object specifically in 3-D Lobachevskian geometry.
- visualization of an object which is also capable of interactive visualization and manipulation of visual images of the object in an alternative geometry.
- invention to provide a system which is capable of interactive visualization and manipulation of an object specifically in 4-D Euclidean geometry.
- invention to provide a system which is capable of interactive visualization and manipulation of an object specifically in 3-D Lobachevskian geometry.
- invention to provide a system capable of imputing planar rotation and translational motion to a user viewing an object in an alternative geometry.
- invention to provide a system for graphing an object specifically in 4-D Euclidean geometry.
- invention to provide a system for geometric construction of an object specifically in 3-D Lobachevskian geometry.
- inventions to include a system for imaging of objects in alternative gemetries as they appear directly in the space of the alternative geometry, including but not limited to 4-D Euclidean and 3-D Lobachevskian geometry.
- the present invention is a system and method of imaging on the human retina of a user an object in an alternative geometry.
- Input to the system are the characteristics of the alternative geometry and the object.
- the object points forming the object in the alternative geometry are transduced into corresponding 2-D image points in 3-D Euclidean space.
- a processing device of the system then assembles the transduced image points to form an image of the direct appearance of the object to a user's eye placed in the alternative geometry.
- the output of the processing device controls an output device for presenting the assembled image points forming the object to the user's eye.
- the imaging system and method displays the direct appearance of the object in the alternative geometries of 4-D
- the imaging system and method provides for interactive planar rotation and translation of the object and/or the viewing position of a user in the alternative geometries, for a fully interactive imaging system.
- the transducer operates by
- the transducer determines the corresponding angles in the 3-D space of the output device and the user's eye according to the geometrical relationships of 3-D Euclidean geometry from which the retinal coordinates of each image point are determined.
- a computer assembles the image points for delivery to a computer monitor screen for display of the object as it would directly appear in the
- Fig. 1 illustrates an image of the 4-D
- Fig. 2 illustrates the 3-D Euclidean space of the human eye.
- Fig. 3 illustrates the fact that a ray from a 4-D point lying outside a given 3-D space cannot intersect the 3-D space in more than one point.
- Fig. 4 illustrates a view in 3-D Euclidean space of the rotation of a shoe in 4-D space.
- Figs. 5(a) - 5(b) illustrate, respectively, the view to an un-aided 2-D eye of the letter "F” rotating in 3-D Euclidean space and the same view with the aid of the present invention.
- Fig. 6 illustrates the 3-D Euclidean or conventional space of the system eye of the transducer of the present invention and a point P in 4-D Euclidean space.
- Fig. 7 illustrates the un-aided human eye viewing a 3-D object from a particular position and orientation.
- Figs. 8(a) - 8(d) illustrate a 2-D eye viewing 2-D objects in 3-D Euclidean space with the use of the system of the present invention.
- Fig. 9 illustrates a flow diagram of the overall function of the system of the present invention.
- Fig. 10 illustrates in flow diagram form the general step-by-step functions of the system of the present invention.
- Fig. 10(a) illustrates a device in accordance with the present invention.
- Fig. 11 illustrates the geometrical
- Fig. 11(a) illustrates that the transducer of the present invention has the effect of projecting rays into the user's eye which form on the user's retina a direct image of the object which exists in the alternative space, and that the display screen for presenting the direct image to the human eye "cancels out.”
- Figs. 12 - 14 illustrate the geometrical relationships in 3-D space of a light ray forming an image point on the picture plane used by the transducer of the present invention.
- Fig. 15 illustrates the screen display and control panel for the GO/LOOK/TOUCH mode of the INTERACTIVE VISUALIZATION AND MANIPULATION mode of the present invention.
- Figs. 16(a) - 16(d) illustrate the functional modules of the control panel used in both the GO/LOOK/TOUCH and MOVE/TURN/TOUCH modes of the INTERACTIVE VISUALIZATION AND MANIPULATION mode of the present invention.
- Fig. 17 illustrates the flow diagram for the TOUCH test of the GO/LOOK/TOUCH mode of the present invention.
- Figs. 18(a) - 18(d) illustrate the screen displays and panel controls for an example of the GO/LOOK/TOUCH mode in the INTERACTIVE
- Figs. 19(a) - 19(c) illustrate the geometrical relationship of a light ray from an object point in 3-D Lobachevskian space which is transduced into 3-D space with the system of the present invention.
- Fig. 19(d) illustrates the geometrical
- FIGs. 20(a) - 20(h) illustrate the screen displays and panel controls in the
- GRAPHING/IMAGING mode of the present invention for graphing a point in 4-D Euclidean space.
- Figs. 22(a) - 22(d) illustrate the graphing of complex numbers in 4-D in the GRAPHING/IMAGING mode of the present invention.
- Figs. 23(a) - 23(d) illustrate the graphing of a hyperbolic cosine in 4-D in the GRAPHING/IMAGING mode of the present invention.
- Fig. 24(a) - 24(d) illustrate the rotation of the image in Fig. 22 in 4-D at angles of 60, 120, 150, and 180 degrees respectively.
- Figs. 25(a) - 25(e) illustrate the sectioning of the graph of Fig. 24 representing a data structure with a plane in any 4-D orientation.
- Fig. 26 illustrates the HYPERGRAPH device accepting laboratory input from two coupled resonant systems.
- Figs. 27(a) - 27(e) illustrate the virtual blackboard screen displays and panel controls of the BLACKBOARD mode of the present invention for geometric construction of objects in an
- Figs. 28(a) - 28(c) illustrate by screen displays the problem in constructing geometrical objects of an alternative geometry on a Euclidean blackboard.
- Figs. 29(a) - 29(b) illustrate the screen displays for construction of a Lambert
- Fig. 30 illustrates a Lambert quadrilateral in 3-D Euclidean space having sides x, y, z and m.
- Fig. 31(a) - 31(b) illustrates the drawing of a straight line though two points P and Q on a Lobachevskian blackboard in the BLACKBOARD mode.
- Fig. 32 illustrates in flow diagram form the steps of the system for the drawings of Fig. 31(a) and 31(b) in the BLACKBOARD mode of the present invention.
- Figs. 33(a) - 33(d) illustrate the drawings of Figs. 31(a) and 31(b) in the BLACKBOARD mode following the flow diagram of Fig. 32, and the corresponding results on the 3-D Euclidean
- a square is a representative figure in a hypothetical 2-D Euclidean space or world (i.e., a plane) and its counterpart, the cube, in the Euclidean space of three dimensions.
- a square is an impossible figure in the geometry of Lobachevski.
- a line moved a unit distance perpendicular to its length forms a square.
- a square moved a unit distance perpendicular to its surface forms a cube.
- a cube in 4-D space moved a unit distance perpendicular to its volume forms the 4-D object the Hypercube, as shown in Fig. 1, in which the image has been produced by means of the present invention.
- the geometrical progression can be recognized through the fact that the
- Hypercube of Fig. 1 has sixteen vertices, whereas the cube of 3-D space has eight vertices.
- space is not to be limited to its literal interpretation discussed herein but is to also be understood metaphorically or abstractly, as in discussion of "color space,” “tone space,” or the “configuration space” of a physical system.
- present invention is equally applicable by way of the visual images it produces, to such abstract, diagrammatic or metaphorical spaces as well.
- Fig. 2 represents the human eye as it is confined to 3-D Euclidean space.
- the eye as represented, is a system which is physically only 3-D in that, as shown in Fig. 2, it consists of a plane ⁇ (the retina) and a point outside that plane (the pupil, O).
- the eye is represented as the space S 0 .
- the space S 0 is shown as a box it in fact has no "edges," as shown in Fig. 3.
- the conventional space S 0 of the eye which is 3-D Euclidean space (i.e., conventional space). The problem exists in directly seeing an object that lies outside the 3-D Euclidean space of the eye.
- Fig. 2 shows the point P in the space of the alternative geometry outside the conventional space S 0 .
- Fig. 3 if a ray originates in a different space, corresponding to a different alternative geometry, and intersects the pupil it therefore cannot also intersect the retina and thus be visible.
- the present invention overcomes this problem with the use of a transducer, as described below.
- Fig. 4 shows a view in conventional space of the rotation of a shoe in 4-D space. As shown, the left shoe 1 does not seem to be
- Fig. 5(a) shows a hypothetical 2-D world, i.e., a plane.
- a person in such a 2-D space would be 2-D having a 2-D eye with a one-dimensional retina.
- the unaided 2-D eye as represented in Fig. 5(a), is looking at a levular (i.e., left- handed) letter "F" as it rotates.
- a levular (i.e., left- handed) letter "F" is looking at a levular (i.e., left- handed) letter "F" as it rotates.
- the left-handed "F” would appear only to be shrinking in size until it disappears and then reappearing as a dextral (i.e., right-handed) letter "F".
- dextral i.e., right-handed
- the present invention will also enable a person's 3-D eye to visualize directly a 4-D object in the 4-D space, as opposed to the 3-D projections, models, or intersections of the prior art.
- the present invention will also extend to any alternative geometry, such as 3-D
- the transducer used in the present invention replaces the pupil of the human eye with a "system eye" and functions as an interface between the conventional space S 0 of the eye, and the space S 1 of the alternative geometry in which the object is inserted.
- the transducer will allow the user's eye to in effect be inserted in any alternative geometry by becoming congruent with the space of the alternative geometry.
- the transducer has two functions. It first determines the angle at which any
- the conventional space of the system eye of the transducer is designated S 0 and the point P outside the space S 0 is in the space S 1 of the alternative geometry.
- the point O represents the pupil of the eye, the plane ⁇ the retina.
- the line OV represents the visual axis of the eye in the space S 1 looking through the
- the ray PO lies in the alternative space, and strikes the transducer or pupil of the eye at O.
- the transducer determines the angles ⁇ and ß which PO makes with the x and y axes of the eye's system, and generates a new ray OQ in the conventional space S 0 of the eye in such a way as to make the determined angles with the two axes.
- OQ belongs to the eye's space and strikes the retina to form a 2-D image point Q.
- the transducer is used to generate an image point in conventional space S 0 for any object point in the space S 1 of any alternative geometry. It is
- the rays PO and OQ are not the same lines, even though, in cases where the angles in both spaces are equal, it will appear that way. Also, when the angles are equal it is only because of the type of display employed and it is not necessary that they be equal. Only by using this optical transducer is it possible to obtain the image point Q on the retina. If the space S 1 is 4-D space then by transducing each object point P (x, y, z, v) of an object into an image point Q we form a retinal image or image on a picture plane of the object in 4-D Euclidean space. Each image point Q would have the
- Fig. 7 shows the unaided eye of a person in conventional space as the person
- 3-D object such as a cube from one angle and position.
- the person's retina only sees 2-D pieces of the 3-D object and intuitively, although subconsciously, determines where each of the pieces of the cube are and then there
- Fig. 8(a) shows a 2-D person looking through a transducer at the 3-D world.
- the 2-D person would note that the sense of distance looking at a 2-D object is preserved in 3-D space, in that an object will appear to reduce in size in proportion to its distance from the pupil as it moves down an axis perpendicular to 2-D space. More
- Fig. 8 (b) shows the 2-D eye having a one-dimensional retina looking at points P and Q, for simplicity, on 2-D objects (i.e., squares) in a 3-D space.
- Fig. 8(c) shows that rotation about the x-axis will not help to distinguish the y and z axes, but Fig. 8(d) shows that rotation about the y-axis will quickly separate one from the other and correspondingly the points p, q.
- the present invention utilizes a computer for the transducer.
- the computer acts as a versatile optical instrument for imaging objects in alternative geometries, whether visually or by other means of imaging.
- Fig. 9 depicts in diagrammatic form the overall function of the system of the present invention.
- the system includes an input system 10 for inputting the characteristics of a selected alternative geometry and an object in the space S 1 of the alternative geometry.
- Such input systems would include physical position and motion detectors, graphic devices or keyboards and voice instructions to the computer.
- the imaging system further includes a transducer 11 to place a user's eye virtually in the space S 1 of the alternative geometry and determine the directions of light rays which would reach the pupil of an eye so positioned. The transducer then would generate an altogether new set of rays in the conventional space S 0 (i.e., 3-D Euclidean space) of the eye, and their corresponding intersections with a retina or picture plane (i.e., image points).
- the system has a corresponding physical output system 12 in the conventional space S 0 for presenting the image formed in a selected form, including film, tape, computer screen display, or actuating any of the devices such as stereoscopic spectacles, stereophonic sound, manipulator handles, etc., designed to generate sensations of virtual
- the transducer is the most essential part of the new invention, and makes it possible for the display of direct physical images of virtual objects defined in spaces of alternative geometries.
- the functioning of this system will first be described in broad terms in Section II with respect to 4-D Euclidean geometry and more specifically in
- Section III related to the specific modes of operation of the system for both 4-D Euclidean geometry and 3-D Lobachevskian geometry. It will be understood that the present invention can be applied to any other alternative geometries, the 4-D Euclidean and 3-D Lobachevskian geometries being described herein as examples, without limiting the invention thereto.
- the mode of operation of the system stored in the mode configuration storage 14b is selected in Step 1 using the input device 14a of Fig. 10(a).
- Other modes of the system can include "GRAPHING/IMAGING” which permits drawing and viewing of graphs in a selected alternative geometry, and the "BLACKBOARD” mode which permits construction of geometric forms and provides appropriate computer-aided design (CAD) tools for that purpose.
- CAD computer-aided design
- Step 2 geometry and an object or objects in that geometry are next defined in Step 2 using the input device 14a or selected from the predefined alternative geometry storage 14c and the object storage 14d.
- Many characteristics of the system can be defined to, in effect, create, manage, and alter a world or environment in the space S 1 of the alternative geometry of any desired degree of complexity and detail.
- An essential characteristic is the
- representing inanimate or animate forms may be caused to move in any ways and with any designated velocities or interactions, either in response to operator instructions, or as governed by
- Step 4 additional translation and rotation of the object or objects may be made in Step 4, whether before displaying the objects, after display as part of a feedback, ergonomic or interactive input system, or for preprogrammed continuous display selected from the disposition storage 14e (i.e., location, rotation, etc.).
- More than one input device or station, utilized by more than one operator, may be attached to give instructions to a single system, by using the methods described here, and making provision for current priorities in system control. Additional characteristics might include, for example, color and animation. The selection of a number of objects along with color and texture of any detail will present a vivid sense of presence in an environment.
- the system eye of the transducer must be positioned in the 4-D space S 1 and also defined, as a 3-D
- Euclidean space S 0 i.e., conventional space
- the user's eye must be positioned in such a way as to locate the pupil and specify the orientation of the visual axis of the system eye within the original 4-D coordinate system.
- translation and rotation of the object and/or system eye in Step 4 may be changing in any manner as a function of the input device 14a or other input devices.
- the input device 14a or other input devices.
- Step 6 requires selection of the mode of output for the system and any scaling which must be figured into the
- Step 7 is the determination of the actual image points in space S 0 .
- the CPU 14f and an appropriate output device 14g and output storage 14h of the system carry out the generation of the signals necessary to generate any required screen display or other output.
- Step 8 provides controls through input device 14a, for example panel controls, for versatile redefinition by the user of the object and/or system eye positions (i.e., translation and/or rotation), as specified in Steps 2 and 3.
- Fig. 11 Here O represents the pupil of the eye, and P a point outside the eye's space, in this case assumed to be a point in a four-dimensional space S 1 .
- the present invention requires that the transducer determine the angle at which the ray PO meets an axis of the eye's system, for example, the x-axis, with which PO makes the angle ⁇ .
- the points O, the x coordinate and P serve to define a plane, which is not in general a plane in S 0 , yet serves to permit the measurement of the angle ⁇ , which is all that is required for the present technique.
- the transducer in accordance with the present invention will thus determine the angles a and ⁇ at which the ray OQ would meet the axes x and y in accord with the laws of the appropriate geometry, which in the case of this example are quite simple, but in other cases, such as 3-D
- Lobachevskian geometry may become complex.
- the transducer In passing a corresponding ray onto the Output System, the transducer in effect "changes hats," and determines the direction of the output ray under the laws of 3-D Euclidean (i.e.,
- the system eye is placed directly in the alternative space S 1 , and the line PO from object-point to pupil of the system eye is a straight line drawn in that space. Since S 1 the 4-D Euclidean case assumed for the present example has a Euclidean metric, the distance r from P to O is computed by use of the Pythagorean relation, with the fourth "v"
- the system eye though itself a three-dimensional system with conventional geometry, is assigned at its pupil a full set of four coordinates in S 1 , while its three axes may be aligned for convenience with three of the four orthogonal axes of S 1 . If the
- Steps 7 and 9 to use conventional algebra and trigonometry to determine the display to be presented by means of a given output device, for example, for presentation on a screen
- the system of the present invention may be used for VISUALIZATION AND MANIPULATION in either GO/LOOK/TOUCH or MOVE/TURN/TOUCH viewing modes.
- GO/LOOK/TOUCH mode motion is imputed to the user who views an object or objects in an alternative geometry, and the direction of view or visual axis is freely selected. Further, the user can translate and/or rotate so as to touch any point in the alternative geometry. The user has the experience of becoming capable of, for
- MOVE/TURN/TOUCH mode it is the object or objects chosen that are visualized and manipulated by moving and turning.
- control and viewing of the method and system are achieved using a computer and by the design of control panels and viewing screens on a conventional computer monitor, one example of which is shown in Fig. 15, for the GO/LOOK/TOUCH mode.
- a computer one example of which is shown in Fig. 15, for the GO/LOOK/TOUCH mode.
- what is here represented on the computer screen may take the form of ergonomic physical controls and readouts apart and/or remote from the computer.
- the present example of Fig. 15 can be implemented by HYPERSTACK cards used on the Macintosh computer using the MATHEMATICA program, and has been so implemented for convenience and flexibility for development purposes.
- HYPERCARD and MATHEMATICA or other systems generally available for use on personal computers may be of great value for certain applications, the present invention could also be implemented through the use of more advanced, rapidly
- Active “buttons” on the screen controlled by touching the screen as shown in Fig. 15, or use of a “mouse,” “trackball,” keypad or other device, accept the user's commands, and readouts on the screen keep track of current coordinates and orientations.
- the configuration of the panel can be changed so as to be appropriate for a given geometry or mode of use, such as the INTERACTIVE VISUALIZATION AND MANIPULATION mode, BLACKBOARD mode, and GRAPHING/IMAGING mode.
- Figs. 16(a) - (d) The modules of the representative control panel of Fig. 15 in the INTERACTIVE VISUALIZATION AND MANIPULATION mode are further identified in Figs. 16(a) - (d). These modules are described with reference to both the GO/LOOK/TOUCH mode and the alternative MOVE/TURN/TOUCH mode.
- the screen module 20, shown in Fig. 15, is a window within a computer monitor screen 15, which represents any screen or any form of display device, including a recording, printout, or the like.
- the positional readout module 25, shown in Figs. 15 and 16(a), includes indicators 26 which report current position and attitude of the system.
- indicators 26 which report current position and attitude of the system.
- MOVE/TURN/TOUCH mode they report the position and angular attitude of the object being manipulated. From left to right the indicators are of the x, y, z, and v coordinates of the center or reference point in the object; the two remaining indicators show values of u and w, angles of rotation about the yz and xv planes, respectively.
- the same indicators report the corresponding position and orientation of the user with respect to the coordinate system of the alternative geometry, as shown in Fig. 15.
- the translation module 30, shown in Figs. 15 and 16(b), includes any convenient device for managing the position of the object (in the
- buttons 31 or equivalently, a slider
- the operation controls consist of a mouse (or trackball), actuated buttons, or equivalently. touch-screen positions.
- buttons 32 serve to interpolate between button positions or (on a double-click of the mouse) to start an automatic continuous run to the corresponding end of the axis. Also in the present example, only one axis is under management at a given time; the current axis is indicated in the readout 33. The choice of axis is readily switched, through the
- the rotation module 35 shown in Figs. 15 and
- buttons 36 includes a radially arranged display which consists of active buttons 36.
- Each button takes the object (in the MOVE/TURN/TOUCH mode) or the user (in the GO/LOOK/TOUCH mode) by 30 degree steps, to the corresponding angular position in rotation about one coordinate plane.
- Buttons 38 permit continuous motion between the 30 degree steps, and when the mouse is double-clicked, set the object (or user) moving in a continuous rotation in the indicated direction.
- Buttons 39 permit choice of the plane about which rotation is occurring.
- Button 39a is the HOME button, used for returning the object or the user to the coordinate center or reference position in the alternative geometry.
- Button 36a is the TOUCH button; it serves to test whether an object visible in output is in fact within "reach," a certain established test
- the Control Module 40 shown in Figs. 15 and 16(d), includes buttons 41 which permit quick choice of an axis along which translation is to be controlled by the translation module 30.
- Area 42 is available for control buttons ("next screen",
- Step 1 GO/LOOK/TOUCH mode in 4-D Euclidean Geometry implemented by the steps of Fig. 10 and in the device illustrated in Fig. 10(a), and specifically implemented as shown in Fig. 15 on the Macintosh computer: Step 1
- the device mode selected is INTERACTIVE MANIPULATION AND
- VISUALIZATION and the specific mode selected within it is GO/LOOK/TOUCH.
- the combination enables a user to move around in space S 1 , of an alternative geometry.
- Every object point corresponds to a choice of four coordinates, forming a vector.
- the four coordinate axes are the unit vectors,
- any number of objects can be defined.
- the objects can be used as streets, buildings, gateways, or benchmarks of any sort. These objects can be viewed with respect to one another. For example, one object may be chosen for interactive manipulation, and the others used as background or benchmarks. Every object be will be interpreted, as here, as
- the objects are defined as a set of vectors in the form of: where the symbol represents the j-th point on the i-th object.
- an object i can consist of one or more points j.
- GO/LOOK/TOUCH requires that a coordinate system in space S 0 for the system eye of the transducer be defined.
- the system eye per se constitutes a three dimensional Euclidean space S 0 , but it is nonetheless located and oriented at all times in the coordinate system ⁇ of the 4-D space S 1 .
- the space S 0 of the system eye is located for convenience so that its origin (O, O, O) in conventional space, which is the pupil of the system eye, coincides with ⁇ , while its axes (x, y, z) coincide with the axes (x 5 , y 5 , z 5 ) of ⁇ .
- the pupil is the origin and the retina or picture plane is at a distance d 0 (i.e., focal length) from with coordinates ⁇ , ⁇ .
- d 0 i.e., focal length
- the transducer converts input information to outputs which ultimately lead to perceptions and responses on the part of the user.
- the monitor screen of the computer shown in Figs. 15 and 16 acts as a picture plane generating light rays which enter the eye of the user just as the original rays enter the system eye in ⁇ .
- the term d 0 i.e., focal length
- picture plane is used to mean a plane on which an image is formed in such a way as to project a similar image onto the retina of the user's eye; it has the ultiiate effect of projecting rays into the user's eye which form on the user's retina a direct image of the object which exists in the alternative space, as if the user's eye had been placed directly in that space.
- the ray PO from the external point P makes the angle ⁇ with the x-axis of the system eye of the transducer.
- the new ray OQ makes the same angle, and strikes the screen or picture plane at Q, which becomes, for example, a luminous point on a screen.
- a ray QS from point Q enters the user's eye at the pupil S, and passes at the same angle ⁇ to meet the retina of the user's eye at R. In this way, the screen has a role as intermediary, causing the ray QS to enter the user's eye at the same angle at which the screen received it.
- Ray QR is
- the present invention differs from other methods which picture four dimensional objects, in which the aim is to produce a picture or projection, such as a
- stereographic projection which is not, and does not purport to be, a view which the eye would receive if it were to look directly at the 4-D object.
- control panel is the central working input, where the purpose is to allow the user to move (virtually) about in the space ⁇ in a
- Step 3 Since the values entered in Step 3 will be parameters which govern the target location to which the user chooses to GO and LOOK in the space of the alternative geometry, the actual input will preferably be in physical form (joystick, gloves, helmet, etc.)
- the user In the GO/LOOK/TOUCH mode the user is able to choose the position of the system eye and the orientation of its visual axis. In effect, controlling the direction in which the user/viewer can LOOK, which is also the "forward" direction in which the user reaches in the TOUCH mode.
- the position of the system eye (or the object in the MOVE/TURN mode) can be manipulated interactively.
- the changing of the position of the system eye shall refer to the translation and rotation of the system eye and it visual axis. Translation shall refer to the producing of a change in location, and rotation as affecting the orientation of the visual axis. Defining the location of the system eye may be less crucial in other modes, where a default choice of eye location may often suffice.
- the system in full, interactive operation operates in a cycle, which begins with an input on the part of the user; the system then presents a
- the cycle is completed when the user interactively responds by calling for a new input appropriate to the user's own goals.
- the user may move from the original position specified in Step 3 to a designated system eye position by specifying the eye position in translation and rotation.
- the system eye can in effect GO any where in the 4-D space.
- Translation requires defining the new location of the origin for the pupil of system eye in the ⁇ coordinate system of space S 1 by the vector where:
- New determining points are determined for the objects in ⁇ of the 4-D space with respect to the new location in ⁇ of the system eye represented by Also, the origin ⁇ of ⁇ and the unit vectors are translated with respect to the translation of the position of the system eye.
- the new object vectors, unit vectors, and origin of ⁇ are
- the orientation of the visual axis of the system eye may also be changed by planar rotation.
- the visual axis of the system eye remains parallel to itself, and is in a sense looking at the same point at infinity, down the system eye's z axis.
- To re-orient the system eye requires rotation in the fourth
- a set of rotations used for the present example are designated as follows: through angle s about the yv plane, through angle t about the xy plane, through angle u about the yz plane, and through angle w about the xv plane.
- T s is the matrix representing rigid rotation about s, namely: cos s 0 0 sin s 0 0 0 1 0 0
- the resulting orientation of the system eye S 0 will need to be designated in terms of the angular positions of its axes, with respect to ⁇ .
- To determine the resulting orientation of S 0 requires translating back the unit vectors, which have already been rotated to their new positions, to the origin of S 0 . This is done by subtracting from their coordinates those of the origin of the coordinate system ⁇ as follows:
- the resulting coordinates of the unit vectors will be its projections on the respective axes of S 0 , and these will be the direction-cosines of the angles which determine the orientation of S 0 .
- the visual axis is the z-axis in the system eye's space S 0 .
- the resulting orientation of the visual axis of the system eye with respect to ⁇ is represented by the angles ⁇ , ⁇ , ⁇ , ⁇ and the position of S 0 in ⁇ by x 0 , y 0 , z 0 , v 0 .
- the system of the present invention enters the transducing function in Step 5.
- a light ray is imagined projected from to the pupil of the system eye.
- a plane is envisioned which contains that ray and the x-axis of the eye's system as shown in Figure 11; since does not in general lie in the eye's three-dimensional space S 0 , neither will this plane, but that does not matter with the use of the transducer.
- the angle that the light ray makes with the x and y axis must be computed. To do so, first the distance r ij from the to must be computed from
- angles ⁇ ij and ⁇ ij exist at the intersection of and the eye's x-axis and y-axis, respectively, for each object point.
- Step 5 has been completed when these two angles have been computed for all points .
- Step 8 The more the system is implemented with a high degree of computing speed, using the panel controls of the computer of Step 8 for continuous feedback to Step 4, the more rapidly the loop between Steps 8 and 4 can be closed. Thus, the operator will sense the reality of motion within four-dimensional space more immediately.
- Step 5 is completed for input to the transducer.
- the transducer is to generate image points Q ij having coordinates ⁇ , ⁇ on the retina or picture plane of the system eye for each object point .
- the transducer will use the incident angles ⁇ ij and ⁇ ij for each object point to determine a new light ray in the space S 0 of the system eye for visualization by a user.
- Step 6 requires selection of a mode of
- any mode by adjusting the perception of the system eye of space S 0 in any manner, either by altering the determination of the output ray to produce the effect of optical adjustments such as telescopic or wide-angled viewing, or by using an optical lens of any desired kind.
- the mode of utilization of the system output may have important consequences for actual computation in Step 7, it is assumed for the present example that the output will be used for ultimate presentation to some form of projector or screen, and that the choice in Step 6 will be one of scale. As such, the new rays are projected in the space S 0 of the system eye at the same angles.
- the output of the system will be to a human eye by way of the monitor screen of a computer, which as discussed earlier represents the retina of the system eye.
- the focal length d 0 of the system eye must be set equal to the reading distance at which the user will view the monitor which is normally approximately 20 inches in the user's world.
- Step 7 the light rays in space S 0 of the object points are determined to create corresponding image points Q ij on the retina or picture plane of the system eye.
- the angle ⁇ ' ij which each of the newly generated rays makes with the z-axis in S 0 is determined. It is
- angles ⁇ ' ij and ⁇ ' ij are taken as equal to ⁇ ij and ⁇ ij respectively, that is not the case with the angle ⁇ ' ij .
- the distance in S 0 for each light ray of an object point is determined.
- each image point represents a corresponding transduced object point and is represented by the coordinates ⁇ , ⁇ .
- This step sets the output controls for display and controls any further interactive manipulation of the object displayed.
- Any given device may have any number of alternative modes of assembling and utilizing the image points ( ⁇ ij, ⁇ ij ) .
- the output is to a monitor screen of a computer.
- assembled and transduced image points would pass to the output step.
- the user's impression in setting the controls of the panel of the computer described earlier in the GO/LOOK/TOUCH mode is one of adjusting a display ⁇ that is, looking at the object or objects in a certain readily adjustable way (GO and LOOK) .
- any change in the display in that mode is in fact sent back to Step 4, to generate an entirely new set of outputs. It is in this sense that the system is fully interactive.
- the TOUCH command (button 36a, Fig. 16(c)), controlled in Step 8, allows a user to virtually touch a point or object in an alternative
- the object can be touched only if it is in the same space S 0 so as the user and is within a predetermined reaching distance so that a user can step forward and touch the object.
- a touch test in the case of an object in 4-D space on the visual axis, two criteria are employed:
- the object may be touched using the TOUCH command.
- Fig. 17 A more detailed description of this test, as applied in the present example, is shown in the flow diagram of Fig. 17.
- the TOUCH test is selected.
- the object points to test are selected.
- the TOUCH criteria are
- the maximum test distance (d t ) for reaching to TOUCH the object is defined by d t ⁇ 2d 0 , and the point or points must be present in the space S 0 of the system eye.
- a processing device such as the computer discussed above, used in connection with the present invention in Step 8 will assemble the image points Q ij for output to an output in Step 9, which in this example is a computer monitor.
- the assembled image points will represent the object or objects as they appear in 4-D Euclidean space S 1 .
- the output can be configured for use with any display, whether it be video tape, film, or other types of conventional recording equipment, or an input to computerized drawing or CAD systems, any other form of further computer processing or utilization, including but not limited to film or video tape animation or editing equipment.
- the user is represented by the system eye of the transducer, which converts input information to output leading ultimately to perceptions and responses on the part of the user.
- the output could be printed graphs which are converted graphically to
- the computer monitor is the output device which acts as a picture plane receiving and displaying rays which enter the user's eyes just as the original rays enter the system eye in the alternative space S 1 .
- scales are under the user's control, but in normal mode the angles of entry in the system eye and the user's eye will be equal.
- the system ends up delivering to the operators eye, as a retinal image, exactly what the system eye itself sees.
- the overall system of the present invention functions as a computerized optical instrument for bringing the light rays of objects from the spaces of alternative geometries and presenting them
- Step 1 the device mode (Step 1) has been selected as GO/LOOK/TOUCH and the alternative geometry as 4-D Euclidean.
- a set of objects in space S 1 must be defined (Step 1)
- the objects can be 3-D but reside in the space of 4-D Euclidean geometry.
- the generation of the rays in 3-D Euclidean space discussed above in Steps 1 through 9 are used to determine what would appear on the retina of the system eye of the transducer.
- a gate-way of some sort is set up at the origin ⁇ of the coordinate system ⁇ of the four-dimensional space S 1 .
- the gateway is called in this example "The Arch at the Origin.”
- Two benchmarks are then set up to mark the course of two of the coordinate directions ⁇ one a marker at a remote distance along the z-axis (the visual axis), and the other at an equal distance along the v-axis extending into the fourth dimension.
- the two markers are called respectively the
- the distance d 0 from pupil to retina of the system eye is taken as a kind of natural unit. Initially, the two obelisks are at 1000 such units from the origin along their respective axes. If the retina of the system eye is to relay its image in turn to an observing human eye at equal angles and at normal viewing distance, then the natural unit for d 0 will be the viewing distance of about 20 inches, and the obelisks will be at about a third of a mile from the Arch. The bases of these three
- the Z-obelisk (0, 0, 1000, 0)
- the V-obelisk (0, 0, 0, 1000).
- the choice of where to view the objects is arbitrary. To gain a little perspective, the user is placed back a distance of 300 units on the z-axis, and at a height of 10 units. As such, the user's eye, represented by the system eye, will be located in ⁇ at coordinates (0, 10, -300, 0).
- Location of the objects in the fourth dimension is determined using trigonometry and the Pythagorean theorem, extended to include the fourth coordinate objects in the fourth dimension.
- the user can see the first image from the fourth dimension, which is the V-obelisk.
- the Z-obelisk lies behind it, and at a somewhat greater distance. The user cannot see the Z-obelisk because the images double up on the user's retina. Since the rays arriving from the V-obelisk have no x-component, the image points generated by the transducer to represent them will fall directly along the visual axis.
- the visual axis acts as both a (real) z-axis and as
- Fig. 18(b) shows the result of a four degree rotation, though not a "rotation" in the ordinary sense ⁇ not, that is, an axial rotation the user is accustomed to, but one about a plane.
- the rotation has been about the yv plane, meaning that points in that plane (i.e., lacking x or z coordinates) stand fast, just as points on the axis do in 3-D rotations, while the remaining points carry out the same rigid rotation they would have in three dimensions.
- the v-axis has manifested itself to the user.
- the user can make virtual motions in the fourth dimension.
- the user will move directly into the fourth dimension, down the v-axis, and actually touch the V-obelisk itself.
- the user will go to a point on the v-axis using the controls of Step 8.
- the system of the present invention will place the position of the system eye where the user chooses. For this example, a point is selected just far enough away from the obelisk to leave enough scope to survey the scene: a point at 900 units into the fourth dimension, just 100 units from the V-obelisk.
- the user next will set the x and z dimensions to zero, and leave just enough y value to place the system eye, in effect the user's eye, at a reasonable eye-level. This translation is shown in Fig.
- the obelisk stands before the user in Fig.
- the computed distance to the doorknob is 20.0432 units (more than 20, because the user's eye via the system eye is higher than the doorknob, but well within the test limit of 25 units.
- the first criterion is met.
- the system must still determine whether the doorknob is within the user's grasp. This is determined by generating a vector which measures the object's relation to the system eye's
- the difficulty is that the user is, in effect, still in a space orthogonal to the v-axis.
- the user is standing at a point on the fourth axis, but has not yet turned toward the object ⁇ "forward" for the user remains a direction at right-angles to the axis the user stands on. If the user attempts to step "forward" to touch the doorknob, the user will actually be walking at right-angles to it.
- the user must rotate about a plane which will serve to exchange the z-axis (the user's old forward direction) for the v-axis, to be the new "forward" direction toward the obelisk. Since the user has to exchange old z for new v, then x and y must stand fast: and the rotation must be about the xy plane. Again, this is generated using the techniques discussed above in Steps 1 - 9 of
- the vector for the doorknob is looked at. Where before the user found the vector (0.8231, -1.01, 0, 20), the user now finds the vector (0.8231, -1.01, 20, 0).
- the third and fourth components have indeed exchanged places. The interval of 20 units lying along the axis between the user and the doorway before were actually still in the fourth dimension, at right-angles to the user's visual axis and forward direction. Now with the planar rotation of the system eye, the object lies directly in front of the user.
- the doorknob is an object in the coordinate system ⁇ of the four-dimensional space S 1 . No place in all of S 1 is closed to the user. A single image on the retina is not the whole measure of the user's spatial perception. The user can use these a succession of these compacted 2-D images as collectively leading to valid perceptions which enable the user by orderly processes to GO where the user desires, LOOK in any direction and enter the space of any object. As the user becomes more experienced with the present invention, the user would learn to make the 90 degree planar turn at the outset, thereafter proceeding steadily and confidently into the fourth dimension.
- Step 5 the characteristics of the alternative geometry and the object or objects must be defined in a hyperbolic coordinate system A.
- the difference in the transducer is only in Step 5 which determines the angles of incidence of the light rays from the object; the lines and angles must now be determined within the alternative space S 1 of 3-D Lobachevskian geometry, and hence must operate by the rules of hyperbolic trigonometry (see for example, Wolfe, H. E., Introduction to Non-Euclidean Geometry (New York 1954), Chapter V).
- a decision must first be made concerning the coordinate system to be used for
- the transducer is essentially the same in the
- Step 2 in addition to the coordinate system ⁇ , unit vectors , , , and origin for the space S 1 , a 4-D Euclidean coordinate system R 1 must also be defined for the object, corresponding unit vectors , and origin of R 0 having coordinates (O, O, O, O).
- Step 2 the user would redefine for the object to be manipulated the components of the object in vector form, as vectors Other objects used as benchmarks or background will as before be defined by the vector
- Step 4 translation will now be with respect to the object and its coordinate system R 0 .
- Step 4 will now also include the locating of the origin Q in the coordinate system ⁇ , as .
- the object is translated in the coordinate system ⁇ , along with the unit vectors, by the relations:
- Step 3 the position of the system eye will have to be defined as shown above in the
- Step 4 rotation in Step 4 would also be different because now the rotation is to the object and not the system eye.
- the coordinate system R of the object will be
- planar rotations of s, t, u, w will be the same.
- one of the planar rotations must be selected for rotating the object For example, if the planar rotation is k degrees about s then the representative
- Steps 6 through 9 are carried out as in the GO/LOOK/TOUCH mode as described above in Section III.A.1 and known to one skilled in the art. 4. MOVE/TURN/TOUCH mode in
- Step 5 in generating the angles of incidence of the light rays from the object; the lines and angles must now be determined within the alternative space S 1 of 3-D Lobachevskian geometry, and hence must operate by the rules of hyperbolic trigonometry.
- These are well-known, however, and their inclusion is simply a matter of utilizing the alternative trigonometric relationships known to those skilled in the art, but are set forth in some detail in Section III.A.2 above for Lobachevskian 3-D in the GO/LOOK/TOUCH mode.
- the MOVE/TURN/TOUCH mode the differences from the GO/LOOK/TOUCH mode outlined in Section III.A.3 above for the 4-D MOVE/TURN/TOUCH mode, equally apply, and need not be further detailed.
- the "GRAPHING/IMAGING" mode allows a user to graph functions in alternative geometries, and thus to graph or visualize data input from measuring instruments, imaging devices, or from data sets of any sort in which it is useful to be able to study relationships among variables in an alternative geometry.
- the object is now a graph in the alternative geometry, which can be visualized and manipulated, as discussed above, using the transducer of the present invention in the
- GRAPHING/IMAGING mode makes this same advantage available in the case, for example, of sets of four related measurements, or of data from imaging devices looking at an object from more than one three-dimensional point of view.
- the present invention provides for graphing and imaging in four dimensions as a technique routinely available to engineers, scientists, the medical profession, artists, or any persons concerned with the structure of data sets or the visualization of spatial relations.
- the first problem is to explain the method for the graphing of a single point in four-dimensional space; in
- Figs. 19(a) and 19(b) show "HYPERGRAPH instructor", proposed as an example of the GRAPHING mode.
- the sequence of figures 19 (a) - 19 (h) were created as a HYPERCARD stack to be run on the Macintosh computer, though the same effects could be produced by other programming techniques, to be run on other computers, including in particular much faster interactive workstations.
- the diagrams produced, however, are computer generated, and arise only through the system of the transducer implemented in the present example on the Macintosh computer using the MATHMATICA program.
- Fig. 19 shows a point P plotted in 4-D Euclidean geometry with coordinates (2, 6, 4, 2) at a planar rotation of 45 degrees (i.e., "u") about the yz coordinate plane; the point (2,6) is first located in the x,y plane, and the point (4,2), in the z,v plane. The point itself is the intersection of planes run through these points parallel to the axial planes, as in the fig. 19(c). The transducer of the present invention serves to locate that intersection and to present it to the eye.
- the system of the present invention makes it possible to graph a number pair as a function of a number pair, i.e., to in effect map a plane into itself, as a surface in 4-D Euclidean space.
- the surface or graph can also be studied in the same way as any other 4-D objects by translation and planar rotation using the GO/LOOK/TOUCH and MOVE/TURN/TOUCH modes Of the INTERACTIVE VISUALIZATION AND
- Figs. 20(b) - 20(f) illustrate the graphing of the point P at (2, 6, 4, 2) shown in Fig. 20(a).
- a point roust be chosen by the user such as Q somewhere in the xy plane.
- the user projects the point Q back in to the space of the 4-D graph by drawing a plane though the point Q parallel to the zv plane, as shown in Fig. 20(c). All points on this plane share the xy coordinates 2, 6 through all values of z and v.
- the "HYPERGRAPH instructor utilizing the methods of the transducer previously discussed, makes this "drawing" possible. The result is shown in Fig. 20(d). Note in Fig.
- each of the points of the objects or graphs shown in 4-D space are transduced to corresponding image points in the space S 0 of the system eye in accordance with the flow diagram of Fig. 10 and the examples given for both modes in the selected geometry of 4-D Euclidean geometry.
- FIG. 10 of the device shown in Fig. 10(a) also makes it possible, for example, to visualize a Cartesian coordinate grid in 4-D space, and to plot in it figures representing functions involving four
- FIG. 21(a) a Cartesian coordinate plane in four dimensions is shown in Figure 21(a).
- the plane shown in Fig. 21(b) is a Cartesian coordinate system in 4-D space, made visible by the system and method of the present invention.
- the plane is shown together with conventional coordinate systems in
- Fig. 21(a) depicts 4-D coordinate axes for reference, whose center is in fact at the origin of Fig. 21(b). Projected onto the xy plane in Fig. 21(a), it gives the conventional grid in that plane, while projected onto the zv plane, both of whose axes are perpendicular to both x and y, it yields another Cartesian grid. Any point P located in the usual way with coordinates x,y in the first coordinate frame will project onto the 4-D system at a point S, and this in turn will project onto point R in the uv plane. In this way, all points of any chosen locus in the xy plane will map into a locus on the uv plane.
- the transducer makes it possible to graph the four-dimensional figure representing the function directly, and to present the loci in the z and w planes as projections from this one graph which presents to the system eye and the human eye the function as a whole.
- Figs. 22(a) - 22(d) show an example, produced uniquely by the system and method of the present invention in HYPERGRAPH mode, in which a function often studied in this context is graphed in four dimensions.
- Fig. 22(a) shown at the right is a set of lines in the (x,y) plane parallel to the x-axis, and hence representing successive fixed values of y.
- Fig. 22(c) the two projections of Figs. 22(a) and 22 (b) are combined to give a single visual impression of the 4-D spatial form which constitutes the 4-D graph of the complex hyperbolic cosine function.
- Fig. 22(d) the four coordinate axes are represented. The particular configuration which they take is the consequence of the point of view which has been chosen for these figures, in which the user's eye has been displaced from the origin and the visual axis tilted, so as to separate the axes visually and make the coordinate planes visible, as well as a rotation of 30 degrees about the (y,z) plane which has been given to the graph itself.
- Figs. 23(a) - 23(d) similarly show the graph of cosh (z) in four dimensions, and trace the projection by which one particular straight line in the z-plane maps into an ellipse in the plane of the dependent variable, by way of a space curve.
- Fig. 23(a) - 23(d) similarly show the graph of cosh (z) in four dimensions, and trace the projection by which one particular straight line in the z-plane maps into an ellipse in the plane of the dependent variable, by way of a space curve.
- Fig. 25(a) the new sectioning plane is represented, onto which the figure is to be projected.
- Figs. 25(b) and 25(c) are shown traced the two sets of curves in the (u,v) plane
- This example illustrates an important feature of this system, which permits sectioning the graph representing a data structure by means of a plane in any desired 4-D orientation, not necessarily in alignment with any of the original coordinate planes, and thereby to discover with the aid of visual perception unsuspected orders within the data.
- This is the visual counterpart of standard methods of coordinate rotations, as in "factor analysis" of data sets.
- Fig. 26 shows the HYPERGRAPH device accepting laboratory input data through inputs 81 from two coupled resonant systems 80 ⁇ here shown as electric circuits, but equally from any physical, chemical or biological system, industrial process, or medical imaging apparatus. It will be useful, in general, in the examination of data sets of any kind, however gathered. It will function as a module in standard laboratory instruments by which data are gathered and processed, such as the standard laboratory storage oscilloscope. Data examined in this way would be acquired, for example, through usual forms of analog/digital converter, stored in any convenient form of data-base storage 82, processed through any of the conventional technologies of filtering,
- module 83 Fourier transformation, etc. of module 83, and then output to output 84, a computer monitor, for study in any combination, rotated in four dimensions and viewed from a position chosen to best reveal
- Images scanned or otherwise made available in data form can be presented in this way, and presented by the system in an alternative geometry, and in particular, in a single image which includes a fourth dimension.
- the user could see aggregated on four axes two dimensional images captured by two cameras or other viewing devices, such as medical imaging devices, or study the relation of data on four channels of devices such as the EKG or EEG recorders. With practice, this permits the possibility of more complex viewing incorporating a new degree of
- the system will assign four coordinates to every vector, and even if the object drawn is 3-D, the system will place and draw it in 4-D space, remaining ready to respond to a further command which may call for a four-dimensional increment to the object.
- the BLACKBOARD acts, in effect, as a virtual blackboard for the construction and study of figures or objects in alternative geometries.
- One example of a control panel configured to implement this mode is shown in Figs. 27(a) through 27(e).
- the system shown is implemented using a HYPERCARD stack run on the Macintosh computer for the present example.
- the BLACKBOARD shown in Fig. 27(a) as a window 55 in a computer screen 50, may be any form of projection device or other medium for presenting visual images from electronic output. It may use commercially available technology to provide
- the xy analog readout module 60 shown in Figs. 27(a) and 27(b), reads x and y positions in rectangular coordinates on the screen 61, and angular position about the xy plane on the dial 62.
- the zv analog readout module 65 shown in Figs. 27(a) and 27(c), reads the zv coordinates on screen 66, and angular position about the zv plane on the dial 67.
- Quick shift from the xy analog readout module 60 and the zv analog readout module 65 and vice-versa is made with buttons 63 and buttons 68, respectively.
- the object handler module 70 shown in Figs.
- 27(a) and 27(d) contains a set of four buttons for left-right and up-down motions in the coordinates under current control. Either of the readout modules will be actively under the control of the handler at any given time; the active module is lighted.
- Buttons 71 and 72 provide continuous control in rotation, and when double-clicked, set the object under management into continuous rotational motion.
- Button 73 when activated brings onto a corner of the blackboard a numerical readout of the current position and orientation of any object under
- the chalk tray module 75 shown in Figs. 27(a) and 27(e), is a drawing program which contains the instruments such as a ruler and compass for drawing figures in the selected geometry using the transducer of the present invention ⁇ in this case, Euclidean four-dimensional space ⁇ including a tool box with objects as they appear in the alternative geometry. Each instrument is invoked by one of the buttons 76. Like the instruments of conventional
- a line on the board could thus lie along any direction in
- Button 77 on the tray invokes icons identifying the buttons on the tray, while button 78 in
- buttons 76 calls up a corresponding menu making the instrument's options accessible.
- the icons may be displayed immediately above the tray, while the menus, movable at the user's option, would normally appear to the side of the blackboard. Menus, icons, and the figures on the board may be arranged and managed in the manner of a desktop computer display.
- Normal file-handling routines would make it possible to input information to be displayed on the blackboard from any external source capable of providing an electronic output, and to output the figures produced on the blackboard to any system, storage divide or medium capable of accepting
- the user would first choose a line segment AB and erect perpendiculars at A and B having lengths equal to AB. However, when the user tries to construct the perpendicular C ⁇ it will miss the point D at which the user was aiming to complete the square. If the user next extends BD to meet C ⁇ at E, the Lambert Quadrilateral is created as shown in Fig. 28(b), and the angle CEB is less than a right angle required for a square.
- the present invention provides a transductive system eye to overcome the limitations of the human eye.
- the human eye is a Euclidean surface which is not able to visualize Lobachevskian objects in a consistent way, as shown in Figs. 28(a) and 28(b).
- the blackboard is the picture plane of the system eye of the transducer which intervenes between the two disparate geometries so that the blackboard, in effect, becomes a
- Fig. 29(a) shows the normal Euclidean
- the Blackboard is in the Euclidean mode, obeying the postulates of Euclidean geometry, with the result that the line JF is incapable of appearing straight to the unaided Euclidean eye.
- the user can change the mode of the blackboard to a Lobachevskian surface using the present invention. If the user does so, then the eye of the user is able through the system eye of the transducer to visualize the Lambert quadrilateral, as it would appear in the space of Lobachevskian geometry.
- the system eye sends light rays forming image points on a users retina
- the present invention transduces the object points of the Lambert quadrilateral in to corresponding image points in the space of the conventional eye.
- quadrilateral is an object which can be visualized and manipulated on the blackboard according to the flow of the GO/LOOK/TOUCH or MOVE/TURN/TOUCH modes of the VISUALIZATION AND MANIPULATION mode of the present invention, discussed above.
- Step 1 For example, if the BLACKBOARD and GO/LOOK/TOUCH modes are selected (Step 1) for a selected 3-D
- the user is unaware of processes occurring when the user selects an object of an alternative geometry from a chalk tray to be placed on the screen in an alternative geometry blackboard. It will be illustrative here to track through one particular example, that of drawing a straight line between two points directly designated in the Lobachevskian plane.
- the user may designate the two points, P and Q, through which a straight line is to be drawn as shown in Fig. 31(a).
- This selection may be made merely by pointing to a light-sensitive computer screen, or by selecting them by means of a mouse-click or on a digital drawing tablet, or by any other of a number of means.
- the transducer of the present invention operates to produce an image, which the user will immediately see, of the line passing through the selected points, as it would be seen by the system eye placed in 3-D Lobachevskian space. This result is shown in Fig. 31(b).
- the experience is that of having drawn a line in the 3-D Lobachevskian space, or on a blackboard substance which behaved in a Lobachevskian manner.
- Step 1 the system passes through the steps delineated in the flow diagram of Fig. 32. It first uses standard computer utilities to determine the screen coordinates of the selected points in Step 1 ⁇ these may be displayed by means of a "Chalk Tray" option as discussed above, and are shown here in Figs. 33(a) and 33(b). In Steps 2-3, the
- Step 4 makes an algebraic test to determine whether the two points are on the line which constitutes the "roof" of a Lambert
- Step 6(a) the system continues in Step 6(a) to use standard algebraic curve-fitting techniques to determine the equation in Lobachevskian coordinates of the roof of the appropriate Lambert quadrilateral. If the discriminant is negative, on the other hand, the required line will intersect the baseline, and the program accordingly follows Step 6(b), fitting the points to a side of a Lobachevskian right
- the Blackboard may be placed in "Euclidean" mode and this line plotted as it would appear in the Euclidean plane as shown in Fig. 33(d).
- this line must then be taken as an object in Lobachevskian space and translated by the transducer of the present invention, in the manner already described above in Section III.A.1 for the GO/LOOK/TOUCH mode, into the retinal coordinates to be plotted on the Lobachevskian Blackboard shown in Fig. 33(b), where they produce an image of the line through the given points, as shown in Fig. 33(b).
- the present invention provides a system for designing and imaging objects in alternative
- the present invention is not limited to 4-D Euclidean Geometry and 3-D Lobachevskian geometry.
- the system is equally applicable for Riemannian 3-D, curvilinear systems (differential geometry), higher-dimensionalities, etc.
- the system of the present invention can also be used in modes other than the modes described above, as long as the transducer of the present invention is implemented.
- the system can produce displays, videos, hard copy output, movies, stereoptic displays, stereographic displays, and stereophonic output, etc.
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- Physics & Mathematics (AREA)
- Engineering & Computer Science (AREA)
- Computer Graphics (AREA)
- Geometry (AREA)
- Software Systems (AREA)
- General Physics & Mathematics (AREA)
- Theoretical Computer Science (AREA)
- Processing Or Creating Images (AREA)
- Image Generation (AREA)
Abstract
Description
Claims
Priority Applications (1)
Application Number | Priority Date | Filing Date | Title |
---|---|---|---|
JP4502943A JPH06503194A (en) | 1990-11-27 | 1991-11-26 | Object image creation system in alternating geometry |
Applications Claiming Priority (2)
Application Number | Priority Date | Filing Date | Title |
---|---|---|---|
US61881390A | 1990-11-27 | 1990-11-27 | |
US618,813 | 1990-11-27 |
Publications (1)
Publication Number | Publication Date |
---|---|
WO1992009967A1 true WO1992009967A1 (en) | 1992-06-11 |
Family
ID=24479242
Family Applications (1)
Application Number | Title | Priority Date | Filing Date |
---|---|---|---|
PCT/US1991/008919 WO1992009967A1 (en) | 1990-11-27 | 1991-11-26 | A system for imaging objects in alternative geometries |
Country Status (5)
Country | Link |
---|---|
EP (1) | EP0559823A4 (en) |
JP (1) | JPH06503194A (en) |
AU (1) | AU9148691A (en) |
CA (1) | CA2097085A1 (en) |
WO (1) | WO1992009967A1 (en) |
Cited By (3)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
EP0702330A2 (en) * | 1994-09-14 | 1996-03-20 | Xerox Corporation | Layout of node-link structure in space with negative curvature |
US5585097A (en) * | 1992-03-24 | 1996-12-17 | British Technology Group Limited | Humanized anti-CD3 specific antibodies |
CN109191577A (en) * | 2018-10-22 | 2019-01-11 | 杭州睿兴栋宇建筑科技有限公司 | A kind of distribution BIM collaborative platform |
Citations (1)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US4807158A (en) * | 1986-09-30 | 1989-02-21 | Daleco/Ivex Partners, Ltd. | Method and apparatus for sampling images to simulate movement within a multidimensional space |
Family Cites Families (2)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US4944023A (en) * | 1987-05-19 | 1990-07-24 | Ricoh Company, Ltd. | Method of describing image information |
FR2638875B1 (en) * | 1988-11-04 | 1990-12-14 | Gen Electric Cgr | METHOD FOR SELECTING AN OBJECT FROM A N-DIMENSIONAL REFERENCE BASE AND VIEWING THE SELECTED OBJECT |
-
1991
- 1991-11-26 JP JP4502943A patent/JPH06503194A/en active Pending
- 1991-11-26 CA CA002097085A patent/CA2097085A1/en not_active Abandoned
- 1991-11-26 WO PCT/US1991/008919 patent/WO1992009967A1/en not_active Application Discontinuation
- 1991-11-26 AU AU91486/91A patent/AU9148691A/en not_active Abandoned
- 1991-11-26 EP EP19920902755 patent/EP0559823A4/en not_active Withdrawn
Patent Citations (1)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US4807158A (en) * | 1986-09-30 | 1989-02-21 | Daleco/Ivex Partners, Ltd. | Method and apparatus for sampling images to simulate movement within a multidimensional space |
Non-Patent Citations (2)
Title |
---|
FEINER et al., Visualizing n-Dimensional Virtual Worlds With n-Vision, Computer Graphics, 24(2) March 1990; p. 37-38. * |
See also references of EP0559823A4 * |
Cited By (7)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US5585097A (en) * | 1992-03-24 | 1996-12-17 | British Technology Group Limited | Humanized anti-CD3 specific antibodies |
US6706265B1 (en) | 1992-03-24 | 2004-03-16 | Btg International Limited | Humanized anti-CD3 specific antibodies |
EP0702330A2 (en) * | 1994-09-14 | 1996-03-20 | Xerox Corporation | Layout of node-link structure in space with negative curvature |
EP0702330A3 (en) * | 1994-09-14 | 1996-07-10 | Xerox Corp | Layout of node-link structure in space with negative curvature |
US5590250A (en) * | 1994-09-14 | 1996-12-31 | Xerox Corporation | Layout of node-link structures in space with negative curvature |
CN109191577A (en) * | 2018-10-22 | 2019-01-11 | 杭州睿兴栋宇建筑科技有限公司 | A kind of distribution BIM collaborative platform |
CN109191577B (en) * | 2018-10-22 | 2023-04-14 | 杭州睿兴栋宇建筑科技有限公司 | Distributed BIM cooperative platform |
Also Published As
Publication number | Publication date |
---|---|
CA2097085A1 (en) | 1992-05-28 |
AU9148691A (en) | 1992-06-25 |
JPH06503194A (en) | 1994-04-07 |
EP0559823A4 (en) | 1994-05-18 |
EP0559823A1 (en) | 1993-09-15 |
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