US3222984A

US3222984A - Coordinate transformation

Info

Publication number: US3222984A
Application number: US256176A
Authority: US
Inventors: Albert M Loshin
Original assignee: Individual
Current assignee: Individual
Priority date: 1957-10-02
Filing date: 1963-02-04
Publication date: 1965-12-14
Anticipated expiration: 1982-12-14

Description

Dec. 14, 1965 A. M. LOSHIN 3,222,984

COORDINATE TRANSFORMATION Original Filed Oct. 2, 1957 3 Sheets-Sheet 1 llllll 20 /6 as v\ a 6 ea 2w Kw INVENTOR. warez-m (a fl/rV BY Lu e, ,2

Dec. 14, 1965 A. y. QLOSHIIN 3,222,984

COORDINATE TRANSFORMATION Original Filed Oct. 2, 1957 3 Sheets-Sheet 2 J6 3 4a 7 J4 57 J! 37 INVENTOR.

Dec. 14, 1965 A. M. LOSHIN COORDINATE TRANSFORMATION Original Filed Oct. 2, 1957 3 Sheets-Sheet 5 INVENTOR. 448E 97 M lfif/l/IV BY M 4.

United States Patent O 3,222,984 COORDINATE TRANSFORMATION Albert M. Loshin, 2195 Central Road, Fort Lee, NJ. Original application Oct. 2, 1957, Ser. No. 687,844, now

Patent No. 3,094,783, dated June 25, 1963. Divided and this application Feb. 4, 1963, Ser. No. 256,176

11 Claims. (Cl. 8824) (Granted under Title 35, US. Code (1952), see. 266) The invention described herein may be manufactured and used by or for the United States Government for governmental purposes without payment to me of any royalty thereon.

This application is a division of my copending application Serial Number 687,844, filed October 2, 1957 now United States Letters Patent No. 3,094,783, dated September 25, 1963.

The invention relates generally to a method and means for reproducing figures such as plots of orthogonal graphs and at the same time varying the ratio of the horizontal and vertical dimensions. More particularly, the invention concerns a simple and direct technique whereby figures including plots of orthogonal graphs can be reproduced in such a manner that the relationship of horizontal to vertical dimensions can be changed to a new and desired ratio.

In the reproduction of graphs for analysis and comparison, a problem arises when the scales and dimensions vary on the graphs or figures to be compared. Also in the publication of prepared graphs a similar problem is encountered when it is desired to use a certain plot in a particular limited space of a definite size and shape in the printed report. The present invention overcomes these difficulties by presenting relatively low cost easyto-use methods for varying the relative dimensions of plots, graphs, drawings, or other figures by a process of reproduction thereby eliminating the necessity of tedious and time-consuming procedure of redrawing or replotting the figures in order to obtain the desired size.

Accordingly, it is a primary object of the invention to provide a method and means for reproducing graphs and drawings whereby either the vertical or horizontal dimension or both can be varied independently so that the resulting reproduction is of some desired shape and size or some new and different ratio of height to width.

Another object of the invention is to provide relatively simple and easy-to-use apparatus for accomplishing the desired ratio variation of the figure to be reproduced. The apparatus mechanically, photographically, or otherwise projects a given figure onto an angularly related plane thus effecting the desired change in dimensions proportional to the angular difference between the plane on which the original figure is mounted and the inclined copy board.

A particular advantage of this method of reproduction lies in the fact that by determining the angle between the planes, the new ratio can easily be computed because there exists the simple relationship of 1: cos 0, where is the angle between the planes in singly oblique projection. In double oblique projection where it is desired to vary both dimensions, the angle 0 is measured from a reference plane which lies perpendicular to the direction of projection.

These and other objects, features, and advantages will become more apparent after considering the following detailed description taken in connection with the annexed drawings and appended claims.

In the drawings:

FIGURE 1 illustrates a typical curve to be varied;

FIGURE 2 shows a view of the curve having the desired height to width ratio;

FIGURE 3 shows a schematic view of transformation;

FIGURE 4 shows schematically an example of doubly oblique projection;

3,222,984 Patented Dec. 14, 1965 FIGURES 5 and 6 depict a mechanical device for pro ducing the transformation; and

FIGURES 7 and 8 show photographic apparatus for accomplishing this result.

Referring now to FIGURE 1, there is shown a curve 12 plotted and drawn on graph paper. For the sake of simplicity assume that the figure is of essentially equal height and width, that is, the x and y axes are equal. Then, suppose it is desirable to transform this curve 12 to a curve 14 shown in FIGURE 2, where the y or horizontal dimension is 1.41 times longer than that of the x or vertical dimension.

To determine the angular inclination of copy plane 18 in FIGURE 3 with relation to the reference plane 16 in order to accomplish the desired ratio change, it is necessary to find the angle 0. Since the relationship of the two curves vary as the cosine of this angle then the required angle would be the one whose cosine was equal to 1/1.41 or approximately 45 Then, by inclining the

copy plane

18, 45 in relation to the reference plane 16 and projecting the curve 12 onto the copy plane, the resulting curve 14 will be increased in the horizontal or y direction by a factor of 1.41, while the vertical or x dimension remains the same.

It may be desirable in certain instances to simultaneously decrease one of the dimensions while increasing the other. In FIGURE 4 there is shown an example of this particular variation. Using the same curve 12 as the original, the copy plane 18 is inclined at 45 thus the horizontal dimension is increased the same amount as in FIGURE 3. However, since the original is not mounted on the reference plane 16 but, instead, is on a difierent plane 20 placed at an angle 0, 30 from the reference plane 16. Since the cosine of 30 is .866, then the vertical dimension of the curve 12 would be reduced by this factor resulting in a vertical dimension ratio of 1:.866 while the horizontal dimension ratio has increased to 111.41. This process may be called doubly oblique projection and results in a curve 22 which has changed from a horizontal to vertical ratio of 1:1 to a ratio of 1.41:.866.

This invention is particularly concerned with the method and means for accomplishing the projection of the original curve in such a manner as to result in the desired variation. A mechanical device for projecting a curve is shown in FIGURE 5 with an exploded detail of some of its parts in FIGURE 6.

The device is supported on a stable base 31 on which the frame and most of the supports are directly mounted. The principal frame member 32 serves as a support for the original-figure holder 33 which is located some distance above the base 31. A side frame 35 is employed to support the varied-figure holder 37 and since these two holders must be capable of being rotated or tilted with respect to one another, provision is made to make the varied-figure holder 37 singly oblique in respect to the original-figure holder 33. This is effected by mounting the varied-figure holder 37 on the side frame 35 with hinges 36.

Provision is made to lock said holder 37 in any desired position by a slotted and calibrated support member 39 and a position locking knob 40. Additional rigidity of the varied-figure holder 37 can be obtained, if necessary, by adjusting and locking one or more adjustable spot supports 41 which might be included for this purpose.

A flange 42 fixed to the base member 31 supports the principal supporting rod 43 which in turn supports the entire tracing apparatus. Two clamps 44 locked in place by suitable bolts, serve to mount the tracing apparatus on the principal supporting rod 43. These clamps 44 each hold a vertical cylindrical bushing 46 shown in detail in FIGURE 6. The swiveling rod 45 is mounted in these bushings and, as its name implies, turns freely around on its axis thus acting as a pivot for the remainder of the tracing apparatus. The pivot-guide rod connector 47, shown in detail in the FIGURE 6, connects the substantially horizontal guide rods 48 to the swiveling rod 45 and also restrains them from moving up and down when the clamps 44 are properly spaced. The guide rod connectors 47 should ride on the ends of the cylindrical bushings 46 with sliding contact, lubricated to reduce friction and wear.

The guide rods 48 rotate in one plane only and restrain the styliriders t and 54 to a single degree of freedomnamely along the axis of said guide rods 48. The substantially horizontal guide rods 48 are given rigidity by joining their ends with two substantially vertical connecting rods 49, thus forming a complete structural loop. The tracing stylus rider 50 is free to move in a plane to permit tracing a curve. The writing stylus rider 54 is additionally free to move vertically in response to movements of the stylus 50 on the plotted curve. This will permit the tracing of any point on the figure, and locates such point by what is tantamount to its polar coordinates with respect to the pivot.

The two

riders

50 and 54 are to have the same identical motion. In order for the writing stylus rider 54 to be laterally displaced along the guide rods as the tracing stylus rider 50 is moved, these two riders must be rigidly connected. This is accomplished by the use of control rods 51, control-follower plate 52, and the follower rods 53. The control-follower plate 52 also rides on the guide rods 48 to give the structure greater rigidity.

To effect the actual projection, vertical movement must be allowed so that the vertical writing stem 56 will contact, at all times, the oblique plane of the varied-figure holder 37. This is accomplished by letting the rider 54 also act as a bushing for the vertical writing stem 56. This stem 56, which is free to move up and down, is terminated with a writing stylus holder 57, whose writing point is kept against the varied figure by gravity. A weight 55 is added to the upper end of the writing stem 56 to insure that the writing stylus will follow the oblique converted-figure plane, and to supply sufficient writing pressure. A partially completed converted figure 38, and its associated original figure 34 are shown mounted on their

respective holders

37 and 33. In actual use, the stylus rider 50 would be moved manually keeping the tracing stylus along the original figure 34 and simultaneously the writing stylus 54 by means of its writing point will form the converted figure comparable to the original figure except that one of its orthogonal components is elongated.

The results as outlined above may also be obtained by the use of photographic means. FIGURE 7 depicts apparatus for accomplishing the projection Where the light source is immediate or close to the original figure.

Using the photographic method it is necessary to obtain a parallel beam of light. This may be accomplished as shown in FIGURE 7 by providing a point source of light 60 located at the focus of parabolic reflector 61. A nonrefiective shield 62 may be used to prevent a photographic hot-spot or high intensity spot of light from existing, and to eliminate direct radiation which would be radial rather than parallel. The parabolic reflector 61 has a flanged non-reflecting rim 63 which is padded with felt 64 or some other material so as not to harm the film 65. Ordinarily said film 65 would be a negative of the original figure to be transformed.

A flat transparent plate of glass 66 can be used to support the original film 65. Another fiat plate of glass (not shown) may be placed on top to protect and/ or flatten the film 65. The source of parallel light may be scanned or moved in the plane of the original film if it is not sufficiently large or uniform in intensity to cover the en tire film. The original figure on film 65 is projected on to an oblique plane 68 located on the other side of the original film, producing a latent image on the unexposed photosensitive paper or film 69 which lies on the oblique plane 68. A frame 70 with flat cover glass 67 might be used to advantage in framing the unexposed paper 69 so that unwanted distortion will be avoided. The light source 66 is kept lighted for a time adequate to permit proper exposure.

In FIGURE 8, which is an example of doubly oblique projection, a remote source of parallel light is specified. Since two transformations are possible in this case, the parallel source of light is not necessarily perpendicular to the plane on which the original film is held. It is, however, parallel with the plane of base and perpendicular to the common edge between 75 and plane 76, said planes being attached at their common edges by hinges 77. Since the parallel beam is quite remote only a portion of it is utilized, hence, would be less critical as to evenness of intensity.

The device as shown in FIGURE 8 consists of a base 75 onto which a front frame 76 is attached by means of hinges 77. An angular scale 78 is provided to hold said frame 76 at the desired angle and to indicate said angle in relation to the base 75 or calibrated directly in accordance with the corresponding transformation ratio. Clamps 79 are provided to hold the original film or plate 80 in its proper position. The rear frame 81 is attached by means of hinge 82 to upright member 83 which in turn is firmly attached to the base 75. The angular scale 84 is provided so that the angular displacement of the rear frame 81 can be determined and also to hold said frame at this desired angle. Clamps (not shown) are provided to hold the photosensitive paper or film 85 in place for exposure. This device can simultaneously reduce the axis perpendicular to the base and increase the axis parallel with the base. Such a double transformation is seldom warranted except for extending the possible range of ratio changes, particularly where the film grain would be the limiting factor.

Although the invention has been described with reference to particular embodiments, it will be understood to those skilled in the art that various changes and modifications can be made therein without departing from the invention.

What is claimed is:

1. A device for reproducing a figure represented by a semitransparency wherein a height to width ratio variation is obtained, said device comprising:

(a) a first surface upon which an original size figure is mounted,

(b) a tiltable surface mounted for angular movement about one axis only which axis is parallel to said first surface, and

(0) lens free photographic means for transferring said figure from said first surface to said tiltable surface including a parallel beam of light generated with a semi-shielded parabolic reflector and passing through the transparent sections of said original figure to selectively vary the intensity of different areas of the beam while the beam remains parallel, said intensityvaried parallel beam impinging on a light-sensitive coated material attached to said tiltable surface,

((1) thereby resulting in a figure of varied height to width ratio proportional to the difference in inclination of the two surfaces.

2. A device for reproducing a figure represented by a semitransparency wherein height to width ratio variations are obtained, said device comprising:

(a) a first tiltable surface upon which a semitransparent original size figure is mounted, said first surface being mounted for angular movement about one axis only,

(b) a second tiltable surface mounted for angular movement about a second axis only, said axes being orthogonal to one another, and

(c) lens free photographic means for transferring said original figure from said first surface to said second surface including a parallel beam of light passing through the transparent sections of said original figure to selectively vary the intensity of different areas of the beam while the beam remains parallel, said intensity-varied parallel beam impinging on a lightsensitive coated material attached to said second surface,

(d) thereby resulting in a figure of varied height to width ratio proportional to the difference in inclination of the two surfaces.

3. An optical transformation apparatus for reproducing by light projection plane figures including portions having image characteristics while at the same time varying the ratio of the orthogonal dimensions thereof, said apparatus comprising:

(a) a first light passing plane support for an original figure to be reproduced,

(b) a second plane support for a plane photosensitive element,

(0) means mounting said supports for selectively variable relative angular movement into positions in which the planes of said original figure and said photosensitive element are aparallel, and

(d) means to direct a collimated beam of light on the original figure on the first plane support with the beam oriented to project the light passed through and thereby variably attenuated by said original figure onto the aparallel plane photosensitive element on the second plane support without passing through a lens and while the variably attenuated light remains collimated,

(e) whereby the ratio of the orthogonal dimensions of the figure projected onto the photosensitive element can be selectively differed from the ratio of the orthogonal dimensions of the original figure by varying the angular relationship of the planes of the two supports.

4. An optical transformation apparatus for reproducing by light projection plane figures including portions having image characteristics While at the same time varying the ratio of the orthogonal dimensions thereof, said apparatus comprising:

(b) a second plane support for a plane photosensitive element,

(c) means mounting said supports for selectively variable relative angular movement into positions in which the planes of said original figure and said photosensitive element are aparallel, and

(d) means for providing a collimated beam of light,

(e) means to direct said collimated beam of light on the original figure on the first plane support with the beam oriented to project the light passed through and thereby variably attenuated by said original figure onto the parallel plane photosensitive element on the second plane support without passing through a lens and while the variably attenuated light remains collimated,

(f) whereby the ratio of the orthogonal dimensions of the figure projected onto the photosensitive element can be selectively differed from the ratio of the orthogonal dimensions of the original figure by varying the angular relationship of the planes of the two supports.

5. An optical transformation apparatus for reproducing by light projection plane figures including portions having image characteristics while at the same time varying the ratio of the orthogonal dimensions thereof, said apparatus comprising:

(a) a first light-passing plane support for an original figure to be reproduced,

(b) a second plane support for a plane photosensitive element,

(c) means mounting said first support for selectively adjustable angular movement about one axis only into positions in which the plane of said first support is aparallel to the plane of said second support, and

6. An optical transformation apparatus as set forth in claim 5 in which the axis about which the first support is angularly movable is perpendicular to the direction of the collimated light.

7. An optical transformation apparatus for reproducing by light projection plane figures including portions having image characteristics while at the same time varying the ratio of the orthogonal dimensions thereof, said apparatus comprising:

(b) a second plane support for a plane photosensitive element,

(c) means mounting said second support for selectively adjustable angular movement about one axis only into positions in which the plane of the second support is aparallel to the plane of the first support,

8. An optical transformation apparatus as set forth in claim 7 in which the axis about which the second support is angularly movable is perpendicular to the direction of the collimated light.

9. An optical transformation apparatus for reproducing by light projection plane figures including portions having image characteristics while at the same time varymg the ratio of the orthogonal dimensions thereof, said apparatus comprising:

(b) a second plane support for a plane photosensitive element,

(0) means mounting said first and second supports for individual selectively adjustable angular movements about separate single axes perpendicular to one another into positions in which the plane of the two supports are aparallel to one another, and

(e) whereby the ratio of the orthogonal dimensions of the figure projected onto the photosensitive ele- 7 ment can be selectively differed from the ratio of the orthogonal dimensions of the original figure by varying the angular relationship of the plane-s of the two supports.

10. An optical transformation apparatus as set forth in claim 9 in which the two perpendicular axes about which the two supports are angularly movable are perendicular to the direction of the collimated light.

11. An optical transformation apparatus for reproducing by light projection plane figures, including portions thereof having image characteristics, while at the same time varying the ratio of the orthogonal dimensions thereof, said apparatus comprising:

(a) a base;

(b) means for providing a parallel beam of light;

(c) a front frame including means to hold in planar condition a transparency forming an original figure to be reproduced;

(d) hinge means securing the front frame to the base for angular movement relative thereto about a first axis only relative to the base, said hinge means supporting the front frame and the transparency carried thereby in the path of the parallel beam of light so that said beam is variably attenuated by the transparency while maintaining its parallel condition;

(e) a rear frame including means to hold a plane photosensitive element in back of the front frame in a position such that the photosensitive element will be struck by the variably attenuated parallel beam (f) hinge means securing the rear frame to the base for angular movement relative thereto about a second axis only which second axis is orthogonal to the first axis; and (g) whereby the ratio of the orthogonal dimensions of the figure projected on to the photosensitive element can be selectively differed from the ratio of the orthogonal dimensions of the original figure by varying the angular relationship of the planes of the two frames.

References Cited by the Examiner UNITED STATES PATENTS 1,654,070 12/1927 Corlett et al. 8824 2,327,818 8/1943 Peterson 240-1.3 X 2,409,454 10/ 1946 Thomas 9573 X 2,409,603 10/1946 Wood 8824 2,540,417 2/1951 Bergmans et al. 8824 X 2,607,268 8/1952 B-artz 8824 2,643,576 6/1953 Merriam 8824 OTHER REFERENCES Society of Motion Picture Engineers, Vol. 47, No. 5, 1946, p. 388.

NORTON ANSHER, Primary Examiner.

Claims

1. A DEVICE FOR REPRODUCING A FIGURE REPRESENTED BY A SEMITRANSPARENCY WHEREIN A HEIGHT TO WIDTH RATIO VARIATION IS OBTAINED, SAID DEVICE COMPRISING: (A) A FIRST SURFACE UPON WHICH AN ORIGINAL SIZE FIGURE IS MOUNTED, (B) A TILTABLE SURFACE MOUNTED FOR ANGULAR MOVEMENT ABOUT ONE AXIS ONLY WHICH AXIS IS PARALLEL TO SAID FIRST SURFACE, AND (C) LENS FREE PHOTOGRAPHIC MEANS FOR TRANSFERRING SAID FIGURE FROM SAID FIRST SURFACE TO SAID TILTABLE SURFACE INCLUDING A PARALLEL BEAM OF LIGHT GENRATED WITH A SEMI-SHIELDED PARABOLIC REFLECTOR AND PASSING THROUGH THE TRAMSPARENT SECTIONS OF SAID ORIGINAL FIGURE TO SELECTIVELY VARY THE INTENSITY OF DIFFERENT AREAS OF THE BEAM WHILE THE BEAM REMAINS PARALLEL, SAID INTENSITYVARIED PARALLEL BEAM IMPINGING ON A LIGHT-SENSITIVE COATED MATERIAL ATTACHED TO SAID TILTABLE SURFACE, (D) THEREBY RESULTING IN A FIGURE OF VARIED HEIGHT TO WIDTH RATIO PROPORTIONAL TO THE DIFFERENCE IN INCLINATION OF THE TWO SURFACES.