Quaternions avoid much of the ugly trigonometry since quaternion rotation is along a great circle. They’re very useful for smooth rotations in computer graphics and many aerospace applications.
Ed _______________________ Ed Angel Founding Director, Art, Research, Technology and Science Laboratory (ARTS Lab) Professor Emeritus of Computer Science, University of New Mexico 1017 Sierra Pinon Santa Fe, NM 87501 505-984-0136 (home) an...@cs.unm.edu <mailto:an...@cs.unm.edu> 505-453-4944 (cell) http://www.cs.unm.edu/~angel <http://www.cs.unm.edu/~angel> > On Sep 20, 2021, at 9:19 AM, Steve Smith <sasm...@swcp.com> wrote: > > Answering my own earlier question (with props) it seems that a (smooth) globe > and piece of string would be good support for an intuitive approach. My > sister made a "hanging string lampshade" in 3rd grade by wrapping cotton > string soaked in glue around a large balloon and letting it dry. I > inherited it when she went into middle school and was "too cool" for it > anymore. I would lay in bed and stare at it, remembering the challenges (I > helped wrap the string) of getting the slick string to do *anything* but > following near-great-circles. The shadows it cast on the walls (and > furniture) might have been my first awareness of projective geometry... I > abandoned it when I went to college. > > I assume (REC?) that early age-of-exploration navigators were more > constrained by trade-winds and convenient resupply/port-of-call than > great-circles, though misunderstanding spherical geometry would be very > inconvenient. > > I also believe that Polynesian stick charts would have great-circleness (or > more-to-the-point shortest time paths) built into them by their nature, not > any arcane understanding of spherical (ellipsoidal) geometry. I'm sure they > have been studied quite thoroughly. > > > > On 9/19/21 3:36 PM, Stephen Guerin wrote: >> In the early 90s, my expat friends in Beijing would often debate which were >> the closer US Cities to Beijing by eyeballing the globe - wish we had this >> too for easy lookup back then. Of course, we could have used lat/longs and >> calculated haversine distances but we had pints to drink - and frankly it >> would have been difficult for me to find the algorithm in the nascent web >> back then ;-) >> >> Beijing (PEK) Great Circle Distances for 10 US cities >> <https://www.greatcirclemap.com/globe?routes=%20PEK-SEA%2C%20PEK-LAX%2C%20PEK-DET%2C%20PEK-SAN%2C%20PEK-MSP%2C%20PEK-YYZ%2C%20PEK-EWR%2C%20PEK-ORD%2C%20PEK-ANC%2C%20PEK-BGR%2C%20PEK-HNL> >> >> <image.png> >> >> >> _______________________________________________________________________ >> stephen.gue...@simtable.com <mailto:stephen.gue...@simtable.com> >> CEO, Simtable http://www.simtable.com <http://www.simtable.com/> >> 1600 Lena St #D1, Santa Fe, NM 87505 >> office: (505)995-0206 mobile: (505)577-5828 >> twitter: @simtable >> z <http://zoom.com/j/5055775828>oom.simtable.com <http://oom.simtable.com/> >> >> On Sun, Sep 19, 2021 at 3:22 PM Frank Wimberly <wimber...@gmail.com >> <mailto:wimber...@gmail.com>> wrote: >> That gives a good view, Stephen. Hawaii is farther south than I thought. >> Thanks. >> >> --- >> Frank C. Wimberly >> 140 Calle Ojo Feliz, >> Santa Fe, NM 87505 >> >> 505 670-9918 >> Santa Fe, NM >> >> On Sun, Sep 19, 2021, 3:19 PM Stephen Guerin <stephen.gue...@simtable.com >> <mailto:stephen.gue...@simtable.com>> wrote: >> I was going to post this similar site but Ed beat me with his :-) This one >> also has a globe view... >> >> https://www.greatcirclemap.com/globe?routes=CLT-HNL >> <https://www.greatcirclemap.com/globe?routes=CLT-HNL> >> _______________________________________________________________________ >> stephen.gue...@simtable.com <mailto:stephen.gue...@simtable.com> >> CEO, Simtable http://www.simtable.com <http://www.simtable.com/> >> 1600 Lena St #D1, Santa Fe, NM 87505 >> office: (505)995-0206 mobile: (505)577-5828 >> twitter: @simtable >> z <http://zoom.com/j/5055775828>oom.simtable.com <http://oom.simtable.com/> >> >> On Sun, Sep 19, 2021 at 2:03 PM Frank Wimberly <wimber...@gmail.com >> <mailto:wimber...@gmail.com>> wrote: >> Thanks, Ed. That's useful. >> >> --- >> Frank C. Wimberly >> 140 Calle Ojo Feliz, >> Santa Fe, NM 87505 >> >> 505 670-9918 >> Santa Fe, NM >> >> On Sun, Sep 19, 2021, 1:54 PM Edward Angel <an...@cs.unm.edu >> <mailto:an...@cs.unm.edu>> wrote: >> Close: https://www.airmilescalculator.com/distance/clt-to-hnl/ >> <https://www.airmilescalculator.com/distance/clt-to-hnl/> >> _______________________ >> >> Ed Angel >> >> Founding Director, Art, Research, Technology and Science Laboratory (ARTS >> Lab) >> Professor Emeritus of Computer Science, University of New Mexico >> >> 1017 Sierra Pinon >> Santa Fe, NM 87501 >> 505-984-0136 (home) an...@cs.unm.edu >> <mailto:an...@cs.unm.edu> >> 505-453-4944 (cell) http://www.cs.unm.edu/~angel >> <http://www.cs.unm.edu/~angel> >> >>> On Sep 19, 2021, at 1:48 PM, Frank Wimberly <wimber...@gmail.com >>> <mailto:wimber...@gmail.com>> wrote: >>> >>> Not to change the subject but... >>> >>> A large airliner recently, 15 minutes ago, flew over Santa Fe headed west. >>> My Flight Radar app tells me that it's a Boeing 777 going from Charlotte to >>> Honolulu at an altitude of 38000 feet. I wouldn't have thought that Santa >>> Fe was on a great Circle route between those two cities. But maybe it is. >>> >>> --- >>> Frank C. Wimberly >>> 140 Calle Ojo Feliz, >>> Santa Fe, NM 87505 >>> >>> 505 670-9918 >>> Santa Fe, NM >>> >>> .-- .- -. - / .- -.-. - .. --- -. ..--.. / -.-. --- -. .--- ..- --. .- - . >>> FRIAM Applied Complexity Group listserv >>> Zoom Fridays 9:30a-12p Mtn UTC-6 bit.ly/virtualfriam >>> <http://bit.ly/virtualfriam> >>> un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com >>> <http://redfish.com/mailman/listinfo/friam_redfish.com> >>> FRIAM-COMIC http://friam-comic.blogspot.com/ >>> <http://friam-comic.blogspot.com/> >>> archives: >>> 5/2017 thru present https://redfish.com/pipermail/friam_redfish.com/ >>> <https://redfish.com/pipermail/friam_redfish.com/> >>> 1/2003 thru 6/2021 http://friam.383.s1.nabble.com/ >>> <http://friam.383.s1.nabble.com/> >> >> >> .-- .- -. - / .- -.-. - .. --- -. ..--.. / -.-. --- -. .--- ..- --. .- - . >> FRIAM Applied Complexity Group listserv >> Zoom Fridays 9:30a-12p Mtn UTC-6 bit.ly/virtualfriam >> <http://bit.ly/virtualfriam> >> un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com >> <http://redfish.com/mailman/listinfo/friam_redfish.com> >> FRIAM-COMIC http://friam-comic.blogspot.com/ >> <http://friam-comic.blogspot.com/> >> archives: >> 5/2017 thru present https://redfish.com/pipermail/friam_redfish.com/ >> <https://redfish.com/pipermail/friam_redfish.com/> >> 1/2003 thru 6/2021 http://friam.383.s1.nabble.com/ >> <http://friam.383.s1.nabble.com/> >> >> .-- .- -. - / .- -.-. - .. --- -. ..--.. / -.-. --- -. .--- ..- --. .- - . >> FRIAM Applied Complexity Group listserv >> Zoom Fridays 9:30a-12p Mtn UTC-6 bit.ly/virtualfriam >> <http://bit.ly/virtualfriam> >> un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com >> <http://redfish.com/mailman/listinfo/friam_redfish.com> >> FRIAM-COMIC http://friam-comic.blogspot.com/ >> <http://friam-comic.blogspot.com/> >> archives: >> 5/2017 thru present https://redfish.com/pipermail/friam_redfish.com/ >> <https://redfish.com/pipermail/friam_redfish.com/> >> 1/2003 thru 6/2021 http://friam.383.s1.nabble.com/ >> <http://friam.383.s1.nabble.com/> >> >> .-- .- -. - / .- -.-. - .. --- -. ..--.. / -.-. --- -. .--- ..- --. .- - . >> FRIAM Applied Complexity Group listserv >> Zoom Fridays 9:30a-12p Mtn UTC-6 bit.ly/virtualfriam >> <http://bit.ly/virtualfriam> >> un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com >> <http://redfish.com/mailman/listinfo/friam_redfish.com> >> FRIAM-COMIC http://friam-comic.blogspot.com/ >> <http://friam-comic.blogspot.com/> >> archives: >> 5/2017 thru present https://redfish.com/pipermail/friam_redfish.com/ >> <https://redfish.com/pipermail/friam_redfish.com/> >> 1/2003 thru 6/2021 http://friam.383.s1.nabble.com/ >> <http://friam.383.s1.nabble.com/> >> >> .-- .- -. - / .- -.-. - .. --- -. ..--.. / -.-. --- -. .--- ..- --. .- - . >> FRIAM Applied Complexity Group listserv >> Zoom Fridays 9:30a-12p Mtn UTC-6 bit.ly/virtualfriam >> <http://bit.ly/virtualfriam> >> un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com >> <http://redfish.com/mailman/listinfo/friam_redfish.com> >> FRIAM-COMIC http://friam-comic.blogspot.com/ >> <http://friam-comic.blogspot.com/> >> archives: >> 5/2017 thru present https://redfish.com/pipermail/friam_redfish.com/ >> <https://redfish.com/pipermail/friam_redfish.com/> >> 1/2003 thru 6/2021 http://friam.383.s1.nabble.com/ >> <http://friam.383.s1.nabble.com/> >> >> >> .-- .- -. - / .- -.-. - .. --- -. ..--.. / -.-. --- -. .--- ..- --. .- - . >> FRIAM Applied Complexity Group listserv >> Zoom Fridays 9:30a-12p Mtn UTC-6 bit.ly/virtualfriam >> un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com >> <http://redfish.com/mailman/listinfo/friam_redfish.com> >> FRIAM-COMIC http://friam-comic.blogspot.com/ >> <http://friam-comic.blogspot.com/> >> archives: >> 5/2017 thru present https://redfish.com/pipermail/friam_redfish.com/ >> <https://redfish.com/pipermail/friam_redfish.com/> >> 1/2003 thru 6/2021 http://friam.383.s1.nabble.com/ >> <http://friam.383.s1.nabble.com/> > > .-- .- -. - / .- -.-. - .. --- -. ..--.. / -.-. --- -. .--- ..- --. .- - . > FRIAM Applied Complexity Group listserv > Zoom Fridays 9:30a-12p Mtn UTC-6 bit.ly/virtualfriam > un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com > FRIAM-COMIC http://friam-comic.blogspot.com/ > archives: > 5/2017 thru present https://redfish.com/pipermail/friam_redfish.com/ > 1/2003 thru 6/2021 http://friam.383.s1.nabble.com/
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