Tag: Geodesic

Davening Direction from Alaska

As mentioned briefly in the “Why Some Zmanim Never Occur” post, the Lubavitch Jewish Center of Anchorage, Alaska has an interesting davening (prayer) direction question. As explained in the “Bearing to Yerushalayim and Zmanim Map” (additional technical information is available in the “Technical Information about the Bearing to Yerushalayim Map” post), there are two opinions about the proper direction to face during Tfilah (prayers). The Levush and others are of the opinion that one should face the rhumb line direction to Yerushalayim, while the Emunas Chachamim and others are of the opinion that one should face the initial bearing of the great circle route. Both of these opinions would clearly use the shortest rhumb or great circle lines to Yerushalayim. As an example, a person standing on Har Hazaisim (Mount of Olives) a mile east of Har Habayis would not daven east using the logic that continuing east for approximately 21,175 miles[1] would reach Yerushalayim via global circumnavigation, when Har Habayis is just one mile to the west. The shul in Anchorage is located at a longitude of -149.861°. The antipode of Har Habayis is on the -144.764851 longitude line – 180° of longitude from Har Habayis. People on different sides of this line (this line is also the halachic dateline according to Rav Yechiel Michel Tucazinsky and others) will daven in opposite directions. The Anchorage shul with a longitude of -149.861° is 169.69 miles (273.1 km) west of the -144.764851° longitude line[2]. For this reason, the shul should daven not east, but west (255.78° from the north, or slightly south-west) if using the common rhumb line calculation, or north (355.67° from the north, or slightly west of north) if using the great circle calculation. Alaska is the only location in the world that has the -144.764851° longitude line cut across dry land, and is therefore the only non-ocean location to face this issue.

Notes:
1. Based on a rhumb line along the 31° 50′ circle of latitude as opposed to 24,901 miles at the equator) using a WGS84 Reference ellipsoid.
2. Rhumb line calculation as calculated by Movable Type using the same latitude for both points. The ellipsoidal shape of the earth is factored into this calculation. The short distance means that the 169.66 mi (272.98 km) great circle distance is very close to the rhumb line distance.

Technical Information about the Bearing to Yerushalayim Map

Vintage Map with CompassI have been asked a number of questions recently about the directional accuracy of the Bearing to Yerushalayim and Zmanim Map calculations. The rhumb line calculations are very straight forward, and I will not spend much time on it in this article. As a side note, for those interested in the subject, I would like to mention that Rabbi Gavriel Goetz recently published a very comprehensive pamphlet Gevuras Moishe on the subject and it is well worth reading.
My original calculations and implementation of the great circle route (geodesic line) used simple spherical trigonometry to calculate the initial bearing (and distance) based on a sphere using an authalic mean radius of 6371 km. These calculations were very similar to the method used by Rabbi Yehuda Herskowitz’s article in Yeshurun volume III page 586. This model of the earth is more than accurate enough for these calculations (you would need a very accurate “nose alignment” and the total lack of shuckling to even get within a degree of being correct). Google Maps API, Yahoo Maps and Microsoft’s Live Maps for simplicity sake, all use very similar calculations as mentioned in the cfis blog quoting Morten Nielsen

Google Maps / Virtual Earth / Yahoo Maps(?) all use a spherical datum based on WGS84. That is, it has the same center, orientation and scale as WGS84, but has no flattening. The radius of the sphere is the same as the semi-major axis of WGS84 (6378137 meters).

Using a sphere, all the above-mentioned mapping APIs use spherical law of cosines formula for the calculations, that yield an identical result to the more complex Haversine formula.

Curious how accurate these calculations really are, being based on a perfect sphere and not the oblate ellipsoid/spheroid that the earth actually is, I started looking into the subject a little more. The most accurate current geoid model of the earth is the WGS84 model. The geodesist/mathematician Thaddeus Vincenty published the Vincenty formulae for this type of calculation that is said to be accurate to about one-half millimeter, more than adequate for our calculation. Chris Veness implemented this formula in JavaScript and released it under the LGPL, making it very simple to use. I slightly modified his work to allow easier interaction with the Google Maps API. The implementation that I used when I finally published the map on December 30, 2007, uses the Vincenty formula with the WGS84 geoid model (the Vincenty formula can easily be used with different goeiod models). The table below shows a comparison of the results for the initial bearing and distances of the different calculations based an a Lakewood, NJ (latitude: 40.095965°, longitude: -74.22213°) to Har Habayis (latitude: 31.77805°, longitude: 35.235149°) example.

Calculation Method Initial Bearing Distance
6,371 km (authalic mean radius) sphere using spherical trigonometry 53.86555° 9224.67442 km
Google Maps API 6,378.137km (WGS84 equatorial mean radius) sphere using spherical trigonometry N/A 9235.00819 km
Vincenty formula using a WGS84 geoid 53.81786° 9244.61686 km
rhumb line 95.37152° 9891.16074 km

Bearing to Yerushalayim and Zmanim Map



Did you ever find yourself unsure of the correct bearing (direction) to Yerushalayim (Jerusalem)? The new Zmanim/Bearing to Yerushalayim map will help you find it very easily. The map draws two lines from your location to Yerushalayim. The blue line uses the commonly used (לבוש Levush and others) rhumb line (a straight line on a Mercator projection map), while the green line shows the great circle route (Rabbi Aviad Sar Shalom Basilea הרב אביעד שר שלום באזילה in his אמונת חכמים Emunas Chachamim and others) on a map. While it appears that the circle line is not a straight line, on a globe, the shortest (and straightest) line between two points follows the great circle route. Future posts will detail the technical aspects of the work involved. For additional information of the correct bearing for tefila, see Rabbi Yehuda Herskowitz’s article in Yeshurun volume III page 586 and Rabbi Gavriel Goetz comprehensive pamphlet Gevuras Moishe on the subject.

How to Use

Find your exact location on the map by dragging the marker to your exact location or by clicking anywhere on the map. You can also click on the search button and enter your location (such as “Lakewood, NJ” (the default), “Vilnius, Lithuania”) or just enter a zip code (Canadian or British postal codes work as well) to zoom in on your location. Changing to satellite view and zooming in farther will make the job easier (assuming that you can identify your roof). Once the red marker is centered on your location, click the red marker to show your exact latitude, longitude and bearing to Yerushalayim in an info window. There are two additional tabs that show today’s zmanim. You can click on the “Link” button to refresh the page with the URL that contains coordinates to the currently displayed map with its zoom level so that it can be bookmarked or emailed.
I would like to thank Rabbi Yehuda Herskowitz for his excellent article and additional information that he provided.

Known Issues

Google Maps API Bugs (All Fixed by Google)

  1. Zmanim data sometimes fails to show in Firefox. This seems to be a bug in the Google Maps API when dealing with synchronous AJAX calls.
  2. Locations close to 180° from Yerushalayim (Alaska is a good test case) sometimes shows an incorrect rhumb line bearing (though the circle route shows correctly). The information in the info widow is correct. This is a known issue with the Google Maps API.
  3. Related to the above issue, sometimes the lines are drawn a “world away” and horizontal scrolling of the map is required to see the lines. Again this is a known Google Maps issue.

To Do (related to Zmanim)

  1. Support Daylight savings time (supported by the timezone webservice used, but not yet implemented in the map).
  2. Link to a yearly calendar.
  3. Elevation lookup (already available in the yearly calendar).
  4. Allow changing of the date.

Note that it is not clear that the Levush was of the opinion that the rhumb line should be used.