B Defining Vectors

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I made an animation displaying what it looks like as the "orbits" increase:

243821
 
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Assuming your spiral is based on a model of a perfectly smooth sphere, you'd still need a reference to elevation.
Suppose instead of relying on a "reference to elevation" in the manner which I previously described (ie the center of the "sphere"), which consisted of a point that does not lie along the spiral, instead I rely only on points along the spiral itself to point to the vector. I make a straight line from point 0% to point 100%. Next I bisect line 0%-100% creating a point A. The vector is created first by a line from point A, through the referenced spiral point B to point C, a point in space. Points in space are referenced as Vector: 87.35775...% Distance: 686,739,974.97969...km Does this new convention "avoid reference to elevation?"
 
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I made a bit of goof-up previously so this post is to make a slight correction:

Suppose to create a spiral on the surface of a sphere I do the following:

Where sphere radius = 1, I place both points A and B at the same location on a rotating sphere surface. The sphere rotates at 2*pi radians per second.

Point A is fixed on the sphere at maximum distance from the rotation poles while point B is free to traverse the surface only with constant surface distance/time directly towards one of the rotation poles, but otherwise point B co-rotates with the same frequency 2*pi rad/sec frequency as point A from a non-rotating perspective.

The spiral would be drawn from the path through space taken by point B (from a non-rotating perspective) where path length AB including the sphere's 2*pi rad/sec rotation = 2 *(((((1+sqrt(5))/2)^2)/5)*6) = 2 * 3.141640... = 6.28328... which is slightly greater than 2*pi

If I look at the co-rotating distance along the surface after 1 second between points AB, this distance gives me a surface velocity for moving point B relative to point fixed A from the rotating perspective.
Assuming point B continues moving along the surface at a constant surface velocity away from point A, then point B forms the rest of the half-sphere spiral from the non-rotating perspective. Point C forms the other half-sphere spiral completing the whole sphere spiral by repeating the operation with the sphere rotating with the same angular frequency in the opposite direction, with point C having freedom of movement only towards the opposite pole as point B.

My questions are:

-would this be a valid method of describing the sphere-spiral?

-will the use of closer and closer approximations to pi which are greater than pi yield more and more accurate vector definitions when referencing points along the spiral in relation to the center of the spiral?
After looking at the spiral a bit more I began having nagging doubts that maybe I'd gotten it wrong... Perhaps after 1 rotation, point B actually takes a SHORTER path through space than point A and 2pi. After looking at the problem a bit more, I make 2 observations:

-Point B initially covers more distance per unit time than point A
-But after 1 complete sphere rotation, point A has actually covered more distance than point B, so technically parts of what I had written were wrong

-Both of the following plots show different portions of point B's path along the same curve.

243859


^This plot's half orbit arc length is slightly larger than pi, and represents point B initially traveling faster than point A.

243860


^This plot's half orbit arc length is slightly smaller than pi, and represents point B covering less distance than point A when measured after 1 sphere rotation
 
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514
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^This plot's half orbit arc length is slightly smaller than pi, and represents point B covering less distance than point A when measured after 1 sphere rotation
Well after even further investigation it turns out the situation is even more complicated... if point B's path length after one sphere rotation is close enough to 2pi but also greater than 2pi, then point B can travel a distance greater than 2pi and path length A after one complete sphere rotation...

2 *(((((1+sqrt(5))/2)^2)/5)*6)
^The above equation is not close enough to 2pi to achieve length B after one rotation >2pi

However, this arc length describing point B's arc length after 1 sphere rotation is greater than 2 pi...

ParametricPlot3D [ { [//math:cos(2*pi*500000000*t)*sin(pi*t)//] , [//math:sin(2*pi*500000000*t)*sin(pi*t)//] , [//math:cos(pi*t)//] } , { t , [//number:0.5//] , [//number:0.5000000020//] } ]

greater-than-2-pi.jpg


In other words, if points A and B are in a "race through space" to 2pi distance traveled starting from the equator of a rotating 1 radius sphere, & point A can't change its surface coordinates on the sphere & point B can only move at constant surface velocity along the co-rotating surface of the sphere directly towards one of the poles, then point B can "win a race" to 2pi distance traveled through space after one sphere rotation, but only if point B's constant surface velocity toward the pole is low enough...
 
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I want to calculate the limit, which is greater than 2pi, of how far through space point B can travel after exactly one sphere rotation, by only heading north from the equator with constant surface velocity? Is this limit certainly irrational? And if this sphere were approximately the size scale of the earth, what's the fastest surface velocity or surface angular speed in rad/sec I can roll "north" to "beat" someone who's just resting on the equator to 2pi arc length through space assuming 1 = approximated earth radius and the sphere rotates 2pi rad/sec?
 
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Does this new convention "avoid reference to elevation?"
Not sure if you still find this relevant, but yes, it would define elevation. It would complete reference to a 3rd dimension after (1st-D) in terms of no. of "orbits" and (2nd-D) in terms of % of distance from NP to SP since I'm assuming that we're still in a sphere of unitary size (r=1). In order to express 3D entirely within the realm of "orbits" it might be a good idea to express the 3rd-D reference in terms of total distance along the "orbits'" path from NP to SP since we would effectively be defining the radius of the sphere beyond r=1.
 

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