Geometrically, translating a figure means moving every point on that figure exactly the same distance and in exactly the same direction. You show the rotated points moving in different directions. Also, there seems to be an arithmetic error in the linear distance that point 322 moved.
My first thought when I read the problem was to translate the 322-321 line so that 321 coincided with 14, then rotate the 321-322 line so that the lines were collinear. That would not minimize the error, but it would put all of the remaining error in one point. But it would not meet the conditions that a solution was supposed to meet.
As you pointed out, the lines have different directions, so I
Geometrically, a translation moves every point on a figure exactly the same distance and in exactly the same direction. In your figure, points 321 and 322 move in different directions. Also, 322 has to move further than 321 if both are rotated through the same angle. So, no translation occurred.
When I first looked at the problem, I translated the line 321-322 so that 321 coincided with 14 and then rotated the 321-322 line until it was collinear with the 14-13 line. The difference in the coordinates of 13 and 322 was the distance difference that would have to be minimized. But that did not meet the condition specified for a solution.
As you pointed out, the lines have different directions (slopes to mathematicians). Because of that, I'm not sure that a single rotation can be devised that keeps the lines from crossing. In other words, the best fit from a single rotation would, I think, show the lines intersecting somewhere.
As to the worth of the exercise, that, like beauty, is in the eye of the beholder. The problem poses a legitimate question, but the answer may indeed be of no practical value.
MathTeacher
Interesting and thank you. I think it is a matter of semantics. In my mathematics courses, approximately 45 years ago, our instructor was adamant about translations in that they were a matter of movement independent of path. However, if the current meaning for translation is as you say, I can agree so long as we each know each other's definitions.
I think the situation is similar to that in one of your posts in which you stated:
"Saying 'But if there's a geometric solution, then there's likely an algebraic one, too.' would come close to squaring you with the mathematicians. An exception would be those who would say that using a trig function disqualifies the operation as an algebraic one. But that goes from being picky to being picky, picky."
As for the lines intersecting, they do because they are coincident.
MLSchumann, post: 329468, member: 471 wrote: MathTeacher
As for the lines intersecting, they do because they are coincident.
I thought two intersecting lines cross at a point, or share a point. I looked up the geometry definition though and it indeed says that two intersecting lines share one OR MORE points. Is that the basis of your statement?
On a lighter note: they don't actually intersect, nor are they coincident, because 322-321 are at their actual elevation, and 13-14 are at zero. Of course, no one would know that, looking at the 2D sketch I provided. 😀
I see what you're saying. Let me offer this. If you tell a geometry student to translate a line, he will move it so that every part of the line moves exactly the same distance and in exactly the same direction. If you tell him to rotate the line about, say, a point, he will move the line so that the distance between every point on the line and the point you specified will stay the same. The student will do precisely what the definitions of those operations tell him to do. So will a game programmer or a geodetic software programmer.
If you tell a precalculus student to solve a purely trigonometric equation, the student will likely know that the equation is transcendental, not algebraic. However, many of the equations that arise in surveying and other applications involve both algebraic and trigonometric components. Examples are problems involving the Law of Sines or the Law of Cosines. They are transcendental, but a lot of algebra is involved in their solution.
I see some difference in the need for strictness of the terminology in the geometric versus the algebraic-transcendental applications, but, I have to tell you, many of my colleagues would not permit looseness in either place.
Like yours, my college days ended in the sixties, and things have changed. In geometry curricula today, much more emphasis is placed on transformations than in my day. In fact, it starts in elementary school. For surveying and engineering, that has to be a good thing. You guys rotate and translate coordinates systems fairly often and good geometry students know today what you're talking about (but not how to apply it.)
Like RFC, I think this has been a great thread. His problem has opened doors to a lot of stuff.
If you are running this travers through Star*net, then I suggest the following inline commands
.DATA OFF
. DATA ON
Pick some place to start, I suggest the first few observations lines after which you will enter the .DATA OFF inline command
Run the dat file
if the file runs to complete adjustment and pass the Chi test
Then move the .DATA OFF the file until you get to a set of measurements that will tank the adjustment. When that happens you will have to identify which of the lines need to be edited.
Make those edits and process again, and keep doing this till you reach the end of the file...
you will identify your error and correct it no muss no fuss really
Bill93, post: 329214, member: 87 wrote: If the goal were only to rotate one line to become co-linear with the other, I think there is a continuous locus of pivot points available, not just your A and B. Each choice of pivot point on that locus results in the rotated endpoints being a different distance along that target line.
This is far off the topic of the original post, but I had to work it out ...
The locus of possible rotation points that will move one line to be co-linear with another line is a straight line. That locus passes through the intersection of the given lines and is perpendicular to the bisector of the angle formed by the given lines.
Each rotation point results in a different distance between a chosen point on the rotated line and that intersection. Thus there is only one rotation point which solves the originally posted problem.
Bill93, post: 329684, member: 87 wrote: This is far off the topic of the original post, but I had to work it out ...
The locus of possible rotation points that will move one line to be co-linear with another line is a straight line. That locus passes through the intersection of the given lines and is perpendicular to the bisector of the angle formed by the given lines.
Each rotation point results in a different distance between a chosen point on the rotated line and that intersection. Thus there is only one rotation point which solves the originally posted problem.
Could, or would, you upload a drawing. I'm having a head-warp trying to figure out what your saying???
AlanG, post: 329692, member: 7306 wrote: Could, or would, you upload a drawing. I'm having a head-warp trying to figure out what your saying???
It sure seems to me that MLSCHUMANN, in post #40 did exactly that.
The line is drawn perpendicular to the line between the midpoints of the two lines we're trying to match up.
Am I wrong, Bill93?
In MathTeacher's picture, the locus of rotation points that would make the lines co-linear is a line through points A and B. Using any rotation point on that locus and the delta angle would superimpose the lines, but would give various displacements along the target line.
In Schumann's picture, the locus is a straight line through the upper blue vertex labeled "rotation point" and through the lower left-hand dashed-line vertex.
Double head warp
So far, there have been 1543 views and 48 posts indicating significant interest when compared to most other recent posts. Only "Question For Javad Users" appears to have more replies. Bill93's most recent posts set me to examine the situation all the way from post #1.
first -
Bearings from post #1[pre]
321 - 322 N73-17-19E
14 - 13 N76-38-08E[/pre]
second -
in the drawing, lines appear to be northwest to southeast
third -
assuming coordinates are north and east in post #6
Azimuths and distances computed from coordinates[pre]
321 - 322 115å¡14'38" (S64å¡45'22"E) 150.118
14 - 13 121å¡02'38" (S58å¡57'22"E) 149.984[/pre]
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Question: which are correct, the bearings or the coordinates?
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Solutions or answers presented
post #3 and #8 Tom Adams, bearings from post #1, coordinates for rotation point not shown
post #4 Jumbomotive, a "belief" in "the intersection of the perpendicular bisectors of the two error vectors"
post #5 MLSchumann, a generalized least squares solutions, no actual numbers shown
post #11 Conrad, "tom has the answer i was going to give."
post #23 Bill93, "The pivot is near N427763.70, E1618061.68"
post #29 MathTeacher, "you can rotate one of the lines so that it aligns exactly with the other. And, one of the two pivot points can be found easily with algebra." - note reference to "two" pivot points, coordinates are shown
post #40 MLSchumann, solution with diagram, coordinates shown, residuals shown
post #46 Bill93 "The locus of possible rotation points that will move one line to be co-linear with another line is a straight line. That locus passes through the intersection of the given lines and is perpendicular to the bisector of the angle formed by the given lines." - no coordinates or residuals are shown
post#48 Bill93, reveals a difference between MathTeacher's and MLSchumann's rotation point coordinates[pre]
427763.797 1618058.086 MathTeacher
427764.396 1618061.593 MLSchumann
From MathTeacher to MLSchumann
Az 80å¡18'39" 3.558[/pre]
========================================================================
Question: which post, or posts, gives the correct solution? Are any of the posts right?
========================================================================
Well, my picture is not the solution.
Question: which are correct, the bearings or the coordinates?
I used the coordinates, as I think most other replies did.
Note that one of the given bearings N73-17-19E nearly matches the reverse of the line of travel 13-322, rather than the points it was labeled with. I can't find the other angle given in the first post anywhere between the point coordinates.
post #3 and #8 Tom Adams, bearings from post #1, coordinates for rotation point not shown
post #4 Jumbomotive, a "belief" in "the intersection of the perpendicular bisectors of the two error vectors"
I think these comments are on track, except for disregarding the small difference in the lengths of 14-13 versus 321-322.
post #5 MLSchumann, a generalized least squares solutions, no actual numbers shown
I didn't analyze that post.
post #11 Conrad, "tom has the answer i was going to give."
He goes on to give the correction to the above nearly correct approach, that the optimum solution requires you to consider the motion of the midpoint of the lines.
post #23 Bill93, "The pivot is near N427763.70, E1618061.68
I was off a tenth sideways and 2.6 ft short. As I said, LS convergence isn't great, so the algebraic method is easier to get a precise answer.
post #29 MathTeacher, "you can rotate one of the lines so that it aligns exactly with the other. And, one of the two pivot points can be found easily with algebra." - note reference to "two" pivot points, coordinates are shown
He is right that both his points will overlay the lines, but only one rotates the endpoints close to the other endpoints. He has ignored the difference in the length of the lines, so didn't get the optimum answer.
post #40 MLSchumann, solution with diagram, coordinates shown, residuals shown
This is the first totally correct post. I had worked out the exact same coordinates but was still puzzling over some details so he beat me to it.
post #46 Bill93 "The locus of possible rotation points that will move one line to be co-linear with another line is a straight line. That locus passes through the intersection of the given lines and is perpendicular to the bisector of the angle formed by the given lines." - no coordinates or residuals are shown
This is considering an offshoot of the posted problem.
post#48 Bill93, reveals a difference between MathTeacher's and MLSchumann's rotation point coordinates
Yes, they are different, as explained above. Schumann's are the optimum for placing the rotated endpoints as close as possible to the other endpoints. Post 48 was again on that offshoot of the problem.
=======
It's been an interesting exercise.
AlanG, post: 329826, member: 7306 wrote: Double head warp
So far, there have been 1543 views and 48 posts indicating significant interest when compared to most other recent posts.
Ya; I guess it takes a Grasshopper to ask the really hard questions.:-S:-D
rfc, post: 329859, member: 8882 wrote: Ya; I guess it takes a Grasshopper to ask the really hard questions.:-S:-D
Well -- I guess that's all relative isn't it? Hard questions for grasshoppers, easy answers for surveyors!
