<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class="">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.<div class=""><br class=""></div><div class="">Ed<br class=""><div class="">
<span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; "><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; "><div class="">_______________________</div><div class=""><br class="Apple-interchange-newline">Ed Angel<br class=""><br class=""></div><div class="">Founding Director, Art, Research, Technology and Science Laboratory (ARTS Lab)<br class="">Professor Emeritus of Computer Science, University of New Mexico<br class=""><br class="">1017 Sierra Pinon</div><div class="">Santa Fe, NM 87501<br class="">505-984-0136 (home)<span class="Apple-tab-span" style="white-space: pre; "> </span> <span class="Apple-tab-span" style="white-space: pre; "> </span><a href="mailto:angel@cs.unm.edu" class="">angel@cs.unm.edu</a></div><div class="">505-453-4944 (cell) <span class="Apple-tab-span" style="white-space: pre; "> </span><span class="Apple-tab-span" style="white-space: pre; "> </span><a href="http://www.cs.unm.edu/~angel" class="">http://www.cs.unm.edu/~angel</a><br class=""></div></span></span>
</div>
<div><br class=""><blockquote type="cite" class=""><div class="">On Sep 20, 2021, at 9:19 AM, Steve Smith <<a href="mailto:sasmyth@swcp.com" class="">sasmyth@swcp.com</a>> wrote:</div><br class="Apple-interchange-newline"><div class="">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8" class="">
<div class=""><p class="">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.<br class="">
</p><p class="">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. <br class="">
</p><p class="">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.<br class="">
</p><p class=""><br class="">
</p>
<div class="moz-cite-prefix">On 9/19/21 3:36 PM, Stephen Guerin
wrote:<br class="">
</div>
<blockquote type="cite" cite="mid:CAOmOqnL+fH7MN=mkN=+vpqJ4NjQFQFEOyFkQrBvRKk4JyCvV0g@mail.gmail.com" class="">
<meta http-equiv="content-type" content="text/html; charset=UTF-8" class="">
<div dir="ltr" class="">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 ;-)<br class="">
<br class="">
<a href="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" target="_blank" moz-do-not-send="true" class="">Beijing (PEK) Great
Circle Distances for 10 US cities</a><br class="">
<br class="">
<span id="cid:part2.6AA9EAAC.1016AC79@swcp.com"><image.png></span><br class="">
<br class="">
<br class="">
<div class="">
<div dir="ltr" data-smartmail="gmail_signature" class="">
<div dir="ltr" class="">
<div class="">
<div dir="ltr" class="">
<div class="">
<div dir="ltr" class="">_______________________________________________________________________<br class="">
<a href="mailto:stephen.guerin@simtable.com" target="_blank" moz-do-not-send="true" class="">Stephen.Guerin@Simtable.com</a>
<div class="">CEO, Simtable <a href="http://www.simtable.com/" target="_blank" moz-do-not-send="true" class="">http://www.simtable.com</a><br class="">
<div class="">1600 Lena St #D1, Santa Fe, NM 87505
<div class="">
<div class="">office: (505)995-0206 <span style="font-size:12.8px" class="">mobile:
(505)577-5828</span></div>
<div class=""><span style="font-size:12.8px" class="">twitter:
@simtable</span></div>
<div class=""><span style="font-size:12.8px" class=""><a href="http://zoom.com/j/5055775828" target="_blank" moz-do-not-send="true" class="">z</a><a href="http://oom.simtable.com/" target="_blank" moz-do-not-send="true" class="">oom.simtable.com</a></span></div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<br class="">
</div>
<br class="">
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">On Sun, Sep 19, 2021 at 3:22
PM Frank Wimberly <<a href="mailto:wimberly3@gmail.com" target="_blank" moz-do-not-send="true" class="">wimberly3@gmail.com</a>>
wrote:<br class="">
</div>
<blockquote class="gmail_quote" style="margin:0px 0px 0px
0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
<div dir="auto" class="">That gives a good view, Stephen. Hawaii is
farther south than I thought. Thanks.<br class="">
<br class="">
<div class="">---<br class="">
Frank C. Wimberly<br class="">
140 Calle Ojo Feliz, <br class="">
Santa Fe, NM 87505<br class="">
<br class="">
505 670-9918<br class="">
Santa Fe, NM</div>
</div>
<br class="">
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">On Sun, Sep 19, 2021, 3:19
PM Stephen Guerin <<a href="mailto:stephen.guerin@simtable.com" target="_blank" moz-do-not-send="true" class="">stephen.guerin@simtable.com</a>>
wrote:<br class="">
</div>
<blockquote class="gmail_quote" style="margin:0px 0px 0px
0.8ex;border-left:1px solid
rgb(204,204,204);padding-left:1ex">
<div dir="ltr" class="">I was going to post this similar site but
Ed beat me with his :-) This one also has a globe
view...<br class="">
<br class="">
<a href="https://www.greatcirclemap.com/globe?routes=CLT-HNL" rel="noreferrer" target="_blank" moz-do-not-send="true" class="">https://www.greatcirclemap.com/globe?routes=CLT-HNL</a><br class="">
<div class="">
<div dir="ltr" class="">
<div dir="ltr" class="">
<div class="">
<div dir="ltr" class="">
<div class="">
<div dir="ltr" class="">_______________________________________________________________________<br class="">
<a href="mailto:stephen.guerin@simtable.com" rel="noreferrer" target="_blank" moz-do-not-send="true" class="">Stephen.Guerin@Simtable.com</a>
<div class="">CEO, Simtable <a href="http://www.simtable.com/" rel="noreferrer" target="_blank" moz-do-not-send="true" class="">http://www.simtable.com</a><br class="">
<div class="">1600 Lena St #D1, Santa Fe, NM
87505
<div class="">
<div class="">office: (505)995-0206 <span style="font-size:12.8px" class="">mobile:
(505)577-5828</span></div>
<div class=""><span style="font-size:12.8px" class="">twitter:
@simtable</span></div>
<div class=""><span style="font-size:12.8px" class=""><a href="http://zoom.com/j/5055775828" rel="noreferrer" target="_blank" moz-do-not-send="true" class="">z</a><a href="http://oom.simtable.com/" rel="noreferrer" target="_blank" moz-do-not-send="true" class="">oom.simtable.com</a></span></div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<br class="">
</div>
<br class="">
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">On Sun, Sep 19, 2021
at 2:03 PM Frank Wimberly <<a href="mailto:wimberly3@gmail.com" rel="noreferrer" target="_blank" moz-do-not-send="true" class="">wimberly3@gmail.com</a>>
wrote:<br class="">
</div>
<blockquote class="gmail_quote" style="margin:0px 0px
0px 0.8ex;border-left:1px solid
rgb(204,204,204);padding-left:1ex">
<div dir="auto" class="">Thanks, Ed. That's useful.<br class="">
<br class="">
<div class="">---<br class="">
Frank C. Wimberly<br class="">
140 Calle Ojo Feliz, <br class="">
Santa Fe, NM 87505<br class="">
<br class="">
505 670-9918<br class="">
Santa Fe, NM</div>
</div>
<br class="">
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">On Sun, Sep 19,
2021, 1:54 PM Edward Angel <<a href="mailto:angel@cs.unm.edu" rel="noreferrer" target="_blank" moz-do-not-send="true" class="">angel@cs.unm.edu</a>>
wrote:<br class="">
</div>
<blockquote class="gmail_quote" style="margin:0px
0px 0px 0.8ex;border-left:1px solid
rgb(204,204,204);padding-left:1ex">
<div class="">Close: <a href="https://www.airmilescalculator.com/distance/clt-to-hnl/" rel="noreferrer noreferrer" target="_blank" moz-do-not-send="true" class="">https://www.airmilescalculator.com/distance/clt-to-hnl/</a><br class="">
<div class="">
<span style="border-collapse: separate; font-family: Helvetica; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-variant-east-asian: normal; font-variant-position: normal; font-weight: normal; letter-spacing: normal; line-height: normal; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;" class=""><span style="border-collapse: separate; font-family: Helvetica; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-variant-east-asian: normal; font-variant-position: normal; font-weight: normal; letter-spacing: normal; line-height: normal; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;" class="">
<div class="">_______________________</div>
<div class=""><br class="">
Ed Angel<br class="">
<br class="">
</div>
<div class="">Founding Director, Art, Research,
Technology and Science Laboratory
(ARTS Lab)<br class="">
Professor Emeritus of Computer Science,
University of New Mexico<br class="">
<br class="">
1017 Sierra Pinon</div>
<div class="">Santa Fe, NM 87501<br class="">
505-984-0136 (home)<span style="white-space:pre-wrap" class=""> </span> <span style="white-space:pre-wrap" class=""> </span><a href="mailto:angel@cs.unm.edu" rel="noreferrer noreferrer" target="_blank" moz-do-not-send="true" class="">angel@cs.unm.edu</a></div>
<div class="">505-453-4944 (cell) <span style="white-space:pre-wrap" class=""> </span><span style="white-space:pre-wrap" class=""> </span><a href="http://www.cs.unm.edu/~angel" rel="noreferrer noreferrer" target="_blank" moz-do-not-send="true" class="">http://www.cs.unm.edu/~angel</a><br class="">
</div>
</span></span>
</div>
<div class=""><br class="">
<blockquote type="cite" class="">
<div class="">On Sep 19, 2021, at 1:48 PM, Frank
Wimberly <<a href="mailto:wimberly3@gmail.com" rel="noreferrer noreferrer" target="_blank" moz-do-not-send="true" class="">wimberly3@gmail.com</a>>
wrote:</div>
<br class="">
<div class="">
<div dir="auto" class="">Not to change the subject
but...
<div dir="auto" class=""><br class="">
</div>
<div dir="auto" class="">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.<br class="">
<br class="">
<div dir="auto" class="">---<br class="">
Frank C. Wimberly<br class="">
140 Calle Ojo Feliz, <br class="">
Santa Fe, NM 87505<br class="">
<br class="">
505 670-9918<br class="">
Santa Fe, NM</div>
</div>
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