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The two towers of the Verrazano Bridge are not parallel: they are [???] to account for the curvature of the earth.

enter image description here

What is that word?

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    $\begingroup$ want to bet those towers sway more than 20mm? $\endgroup$ – agentp May 3 '17 at 2:27
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    $\begingroup$ @agentp - Personally, I'd bet the towers weren't built that accurately anyway. 20mm in 211m is 1 in 10000, or 0.3degrees. I don't have to hand any specs for construction tolerance for towers, but that sounds pretty tight to me. More meanderings from me on an answer to a duplicate question on ELU. $\endgroup$ – AndyT May 3 '17 at 11:39
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    $\begingroup$ ..."perpendicular to the geoid?" $\endgroup$ – Adam May 3 '17 at 22:51
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    $\begingroup$ Co-normal. Both are normal to the same surface, thus they are co-normal. $\endgroup$ – Hari Ganti Jan 2 '18 at 22:30
  • $\begingroup$ If the angle between the tower is not there by design to account for asymetric cable load on each side of each tower, but really just to account for the curvature of the earth, then in a physic sense general relativity tells us that the two towers are actually parallel, it is only the spacetime in which they reside which is warped by earth gravity. $\endgroup$ – Hoki Nov 15 '19 at 11:32
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angled, divergent, splayed... I'd probably go with angled depending how I was phrasing the description.

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    $\begingroup$ In general angled does not make sense in 3D. The term divergent might also be ambiguous because in one direction the lines converge and in the other, they diverge. Do you have any reference that is using these classifications? $\endgroup$ – MrYouMath May 3 '17 at 10:03
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In general, we can distinguish four cases.

  1. Case: parallel --> If you take the curvature into account, the pillars cannot be treated as parallel. While I would guess that the error is pretty small on this scale.

  2. Case: orthogonal --> Obviously not the case.

  3. Case: intersecting lines: If we assume that the axis of these pillars meets at the middle of the earth, we could call them intersecting lines. Note that for this case the lines should not be parallel and not be orthogonal.

  4. Case: skew lines: This case requires that the lines are not parallel, orthogonal and do not intersect. If you look at the axis of these pillars with high precision then it will be obvious the axis of these pillars do actually form skew lines.

Summary: Depending on your accuracy you would call the pillars by the following categories.

  • low accuracy: parallel
  • mid accuracy: intersecting lines or intersecting
  • very high accuracy: skew lines or skew
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  • $\begingroup$ How do you fit the words you suggested into the space the OP supplied? $\endgroup$ – Solar Mike May 3 '17 at 5:22
  • $\begingroup$ What do you mean by that? All these terms define a relative orientation of one line to the other. You just imagine a mid-axis for the pillars, the following steps in classification should be obvious. $\endgroup$ – MrYouMath May 3 '17 at 9:49
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    $\begingroup$ As to "what do i mean by that" - so the OP had this phrase : "they are [???] to account for the curvature of the earth" . Take your first word parallel - it fits, but is not correct... $\endgroup$ – Solar Mike May 3 '17 at 10:50
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If you say two lines (towers) are parallel, you state a relationship between, and only between, the two lines. The relationship you are looking for is not just between the two lines. Therefore, there is no word to describe that relationship.

  • You need to include, for instance, the curvature of the earth. You could state: the two towers are both perpendicular to the curvature of the earth.
  • Instead of curvature, how about center of gravity of the earth. The forces would like that. Then you could state: the two towers intersect each other at the center of gravity of the earth.

There is just no way to describe a three-way relationship with just two of the three parts.

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    $\begingroup$ As the Earth is not spherical, but an oblate spheroid, then the centre based on the curvature most likely won't coincide with the centre of gravity from all ponts. Also, use the OP's example sentence where the three items were included. $\endgroup$ – Solar Mike May 4 '17 at 13:39
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Orthogonal to imaginary surface of Earth.

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Radial from the center of the earth?

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Sorry I am only answering the title of your question, but this answer does not allow to go on and give you a recommendation for your second question (in the question body).

There is no "opposite" to parallel, in engineering, in physics, or even in mathematics or phylosophy.

Not what you wanted to hear but bear with me. The notion of opposite is only ever defined on a 1D domain. On a 1D domain (a line). If a word indicate a direction along this line, then a word can exist to describe the opposite direction.

Now take a 2D domain (let's start with a circle for now, but your non-parallel lines will fit there too). On the previous line, I only had 2 possible directions, each opposite to each other, where I could navigate. Now starting from the center, there is an infinity of possible direction I can take (I can divide them in 360 degrees, 6.28 radian or any other metrics but the important thing is in any case there are more than 2 directions).

"parallel" only indicate one posible orientation (and 2 directions along this line). There is no opposite to that orientation. Which one would it be, the orientation with an angle of 1°, an angle of 3°, an angle of 90°? None of the other orientation qualify for "opposite" (and none of them has an "opposite" either). There is no opposite to "parralel".

If the english language had defined a word exactly describing "anything NOT parallel", you'd have the strongest candidate for the opposite but even then it wouldn't exactly be true. One word would define a single point of the domain (of all the possible orientations) and the other would include the rest of the set (all the infinity of other possible orientations). I don't know if such an unbalance between 2 words would qualify them to be opposite. I am not aware of such a word anyway (bar "non-parallel" if you consider that a word) ...


That was the pedantic view. Now as an engineer, I sometimes use verbal shortcuts too, and in this context (as a verbal shortcut, knowingly not accurate but understandable enough for my interlocutors if provided in the right context ) I could see myself replying to "we need these things parallel for reason xx ..." with "No it's the opposite, we need them perpendicular/othogonal for reason zzz ...". It would obviously be a gross misuse of language, but with context it may be understood fine because parallel and orthogonal are special orientations among the domain, with special properties that other angled orientation do not have. (and some of these properties between parallel and orthogonal could be considered opposite, although even that would require a bit of handwaving)


And just for fun (or to confuse people): Say I have a trajectory, on a 2D domain, going along the positive side of the X axis (so parallel to X axis). What is the opposite of this object? Well for me if the orientation AND direction was defined, then the opposite object would be with the same orientation, but going in the opposite direction. So it would be a line going along the negative side of the X axis. Wait what ? ... It means that this object would also be parallel to X axis, and parallel to the initial object. It means, the opposite of a parralel trajectory is ... a parralel trajectory ... rhaaa, my brain is melting now ...


Last one: As said in comment, the premise of the question might even be unfounded. IF these 2 pillars are each exactly parallel to their local gravitation field, then they are parallel to each other in the spacetime. Only earth gravity is warping the spacetime in which they reside so they appear at an angle (yeah I thought "at an angle" is better than "non-parallel", still don't have a good word for you I'm affraid).

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Gravity is a force and a force has a direction. So the opposite to parallel for a gravitational body is gravitational direction or plumb.

From An Introduction to Fluid Mechanics By Chung Fang

Equation(3.1.3) indicates that a change in pressure can only take place in the direction parallel to that of the gravitational acceleration. An increase in pressure is obtained if the direction from one point to another is the same as that of g, and vice versa. If the direction is orthogonal to the direction of g, no pressure change takes place. Thus, the free surface of water on the earth’s surface is always perpendicular to the gravitational direction, for the pressure on the water free surface assumes a constant value, i.e., the pressure of the air above.

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