Abstract:
Sundials and solar compasses including are disclosed. Some embodiments include an omni-directional lens, which can focus a sun beam into a sharp spot with a long depth of field. By projecting the spot onto a cylindrical panel, both the day of the year and the time of the day can be read off simultaneously with very high accuracy. Because of the simultaneous displaying of time and date, no equation-of-time correction is required. If the time is known, the true north can be determined with high accuracy, and the device becomes a reliable and easy-to-use solar compass.

Description:
BACKGROUND 
   The sundial is one of the earliest scientific instruments created by mankind. Thousands of years ago, almost every culture of the world independently discovered the principle of determine the day in a year and the time in a day by observing the position of the sun, and invented some type of the sundial. Although the mechanical clock was invented in the 16th century, until late 19 th  century, the sundial was still used as a reliable time piece in Europe. Even more recently, novel types of sundials are being invented. Most known sundials use a gnomon, which is an opaque piece of solid material that can project a shadow on a panel. The gnomon can be a long and thin opaque stylet or an opaque sphere. Because the sun has a finite radius, the shadow is typically fussy. Some sundials use a hole and or equivalently, a mirror to project a bright spot on the panel instead of a dark shadow. Because the angle of rotation of the sun often exceeds 180 degree, the hole can only be effective for a small range of angles, thus its usefulness is limited. To improve the sharpness of the image, some sundials use a concave mirror or a cylindrical mirror to focus the sun beam. However, the focus surface is a special curved surface in the three-dimensional space, which must be strictly arranged and followed. And, similar to the case of curved mirrors, if the sun beam is seriously off the axis, the sharpness of the image is low. 
   Another problem with the traditional sundials is that the angular position of the sun depends on the day of the year. The difference of the solar time and the average time is represented by the well-known equation of time. The error could be a large fraction of an hour. Therefore, the accuracy of the sundial is limited, especially the stylet and cylindrical-mirror type. Usually, a conversion table or conversion chart is attached to a sundial for the equation-of-time correction. 
   The magnetic compass is widely used for determining directions. However, the position of the magnetic North Pole is off about 10° from the true North Pole, and the magnetic South Pole is off about 25° from the true South Pole. In the United States, the error (magnetic inclination) could be as large as 20°. The magnetic inclination also varies year by year. Furthermore, the magnetic compass is greatly affected by the ferromagnetic materials in the neighborhood of a compass, e.g., iron ore in the ground or any steel or iron pieces. 
   In 1834, W. A. Burt invented the solar compass which uses the position of the sun to determine the true north. Because of its reliability and accuracy, since the middle of the 19 th  century, the US government defined the solar compass as the standard for land surveying. The solar compass is also used in the military for reliably determining the directions in the battle field. However, the operation of known solar compasses is complicated and requires the calculation of the local solar time versus the local standard time at the time of measurement, and requires elaborate manual adjustments. When a gnomon is used, the same inaccuracy problem with the sundials, the fussiness of the image and the equation of time, is present. 
   It is well known that a convex lens can focus sunlight into a sharp spot. However, it works only when the position of the sun is aligned with the axis of the lens. When the sun is slightly off the axis, the image is distorted. If the sun is seriously off the axis, the image is grossly distorted and eventually disappears. Furthermore, the depth of field is usually quite shallow. The use of convex lens in solar compass requires manual adjustment to align the axis with the sun. 
   SUMMARY 
   As illustrated in  FIG. 1 , aspects of the disclosed subject matter include an optical device for projecting the center of the sun from any direction to form a sharp spot of light onto a panel. It comprises two concentric spheres. The outer sphere with radius R 1  is made of a transparent material with index of refraction n 1 , and the inner sphere with radius R 2  is made of another transparent material of index of refraction n 2 . Under the condition n 1 &gt;n 2 , and R 1 &gt;R 2 , the parallel light comes from any direction will be focused on a spot at a distance f at the opposite side of the sphere. Such a lens is called omni-directional. The omni-directional lens does not generate an image of the sun in the strict sense. Instead, it generates a distribution of light intensity with a sharp center spot which can be easily identified by naked eye, or by a light sensor. The position of the center of the sun can be determined much finer than the apparent diameter of the sun, which is about 0.5 degree (32′). The focal length is not a sharp, fixed number. Instead, it has a range. In this sense, such a lens has a large depth of field. 
   The center of the light spot has a much higher intensity than the direct sunlight. It makes the center spot very easy to be identified. To avoid burning the panel, at least one of the spheres is made of a heat absorbing material, for example, doped with copper sulfate. With copper sulfate, only the blue light can go through the lens. Furthermore, by using a panel with dark blue background, the brightness of the area exposed to direct sunlight is substantially reduced, and the bright blue spot projected through the omni-directional lens becomes even more eye-catching. 
   Because of the large depth of field, the center of the sun can be projected on a cylindrical panel without loosing its sharpness over the entire area. A precise printout of the path of the sun can be easily made, which can provide a highly accurate reading. Because both the day of the year and the time of the day can be identified, the correction due to the equation of time is done automatically and accurately. By using two panels per year (from one solstice to another solstice), the daylight saving time can be marked directly. 
   If the time and the date are known, the instrument can be configured as a compass. The principle of the solar compass is not new. However, to use existing solar compasses, the instantaneous position of the sun must be calculated from astronomical data one by one, and the operator must wait the predetermined time to come. This consumes a lot of time and requires a profound knowledge on astronomy. For the solar compass based on the omni-directional lens, the astronomical information is explicitly marked on the panel. Therefore, it operation is independent of time, very intuitive, and easy to use. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a diagram showing how an omni-directional lens according to some embodiments of the disclosed subject matter focus sunlight beams. 
       FIG. 2  is a side section view of an omni-directional lens according to some embodiments of the disclosed subject matter. 
       FIG. 3  is a front isometric view of a stationary sundial using an omni-directional lens according to some embodiments of the disclosed subject matter. 
       FIG. 4  is a front isometric view of a portable sundial using an omni-directional lens according to some embodiments of the disclosed subject matter. 
       FIG. 5  is a diagram of a panel according to some embodiments of the disclosed subject matter, which includes markings for the first half of the year. 
       FIG. 6  is a diagram of a panel according to some embodiments of the disclosed subject matter, which includes markings for the second half of the year. 
       FIG. 7  is a front isometric view of a solar compass using an omni-directional lens according to some embodiments of the disclosed subject matter. 
   

   DETAILED DESCRIPTION 
   The principle of the omni-directional lens is shown in  FIG. 1 . It is a sphere comprises two concentric components. The outer sphere  101  with radius R 1  is made of a transparent material with index of refraction n 1 . The inner sphere  102  with radius R 2  is made of another transparent material of index of refraction n 2 . When a light beam  103  impinges on the lens with an offset h from the axis, owing to the difference in the index of refraction, the beam is refracted four times on the four surfaces,  105 ,  106 ,  107 , and  108 . Detailed mathematical analysis shows that under the conditions n 1 &gt;n 2  and 
                 R   2       R   1       &gt;         n   1     -     n   2           (       n   1     -   1     )     ⁢     n   2           ,         
all rays with the same h will converge at a point Q ( 109 ) on the axis with a finite distance f ( 110 ) from the center of the sphere P. In general, for rays with different h, the focal length f is different. Since the rays with the same offset h converge at the same spot at the axis, it creates a light spot with very high intensity. However, all the rays with different h will be divergent and have much lower intensity comparing with that of the central spot. Because the lens is spherically symmetric, parallel light rays coming from any direction will be focused the same way. Therefore, the effect of light focusing is omni-directional.
 
   The theoretically predicted focusing effect is experimentally verified. For example, for n 1 =1.50 (Lucite) and n 2 =1.33 (water), a lens with R 1 =25 mm and R 2 =9.3 11 mm generates a focal distance from 60 mm to 75 mm. In other words, the average focal length if 67.5 mm and the depth of field is 15 mm. 
   For applications in sundial and solar compass, it is not necessary to have the omni-directional focusing effect over the entire sphere. For the longitude, 360 degree is required. However, since the tropic circle is 23.5 degrees from the equator, a 60 degree latitude range is sufficient. 
   As shown in  FIG. 2 , the raw material for the lens is a solid Lucite sphere,  201 . The first step is to drill a cylindrical hole  202 . The second step is to cut the interior sphere  203  with a special cutting and a special grinding tool. Then, a Lucite plug  204  is placed and sealed with silicone gel. The pug  204  has a hole  205  which is filled with silicone gel. The cavity  203  is than filled with aqueous solution of copper sulfate using a syringe through the silicone gel  205 . The remaining air is letting out by another syringe through the same piece of silicone gel. After filling, the silicone gel will provide a good seal for the liquid. Finally, the lens is mounted on a metal handle  206  through the thread  207 . 
   An example of a stationary sundial using a spherical omni-directional lens is shown in  FIG. 3 . The cylindrical shell  301  with a replaceable panel  302  is supported by the foot  303 , which is adjusted to the latitude of the location. The upper surface ABCD is leveled. On the base plate  304 , erects a vertical post  305 . The omni-directional lens with handle  306 , is fixed to the post by a screw  307 . The center of the lens,  308 , is located at the center of the ABCD plane. 
   The panel  302 , preferably having a blue background and dark-blue of black markings, is designed according to the local longitude to correct for the difference between the local solar time and the local standard time. To ensure accurate readings, it is preferable to have a Spring panel (from the Winter solstice of the last year to the summer solstice of the current year), and a Fall panel (from the Summer solstice to the Winter solstice). If the size of the panel is a relatively large, the accuracy of the sundial could easily reach a single day except in the neighborhood of the solstices. Therefore, the daylight saving time can be marked on the panel. 
   The above sundial can be used only at a specific location. A sundial can be used for any location, a portable sundial, is shown in  FIG. 4 , with adjustments for both latitude and longitude. The semi-cylindrical penal holder  401  holds the replaceable panel  402 . The semi-cylindrical panel holder is mounted on a semicircular base plate  403 . Plate  403  can be rotated around the axis  404  of the handle of the omni-directional lens, to adjust for the difference of local solar time and local standard time. The angle of adjustment can be read off from the slot  405  against the rectangular piece  406 . The rectangular piece  406  in turn can be rotated around the axis  407  to make adjustment for the latitude. The height of the center of the lens  408  is aligned with the middle of the panel  402 . 
   The design of the panels is shown in  FIG. 5  and  FIG. 6 . The equation-of-time correction is fully implemented. If the panel is large, every day of the year, including the weekdays, can be displayed. Therefore, the starting date and the ending date of daylight saving time can be marked. 
   By letting the device to rotate horizontally, the sundial with an omni-directional lens can be configured as an accurate and easy-to-use solar compass. An example of the design of a solar compass is shown in  FIG. 7 . The base plate  701  is sitting on three feet  702 , two of the three are screws to adjust the plate with the help of the level  703 . Screw  704  is used to adjust the latitude arc  705 , which sets the inclination of the rectangular bar  706 . The omni-directional lens  707  is supported on the rectangular bar  706 , and the handle of the lens is acting as the axis of rotation of the panel holder  708 . The longitude arc  709  is adjusted by the screw  710 . The tip  711  is the south pointer; and the tip  712  is the north pointer. It is worth noting that for a solar compass, it is not necessary to display the 12 hours. For the normal working days, 10 hours (for example, 8 am to 6 pm) is sufficient. The panel could span over 150 degrees rather than 180 degrees. 
   To use it, first adjust the base plate using screws  720  with the help of level  703 . Then, adjust the latitude and longitude to match the location of measurement using screws  704  and  710 . To find the true north, just rotate the base place such that the sun beam is focused on the current local time and the current date. The tips should point to the true North and true South. It is worth noting that when rotating the base place, both the date reading and the time reading would change. This will provide a consistency check. 
   Although the disclosed subject matter has been described and illustrated with respect to embodiments thereof, it should be understood by those skilled in the art that features of the disclosed embodiments can be combined, rearranged, etc., to produce additional embodiments within the scope of the invention, and that various other changes, omissions, and additions may be made therein and thereto, without parting from the spirit and scope of the present invention.