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# Transformation and Projection

Hello,

I am a Civil Engineer with limited survey experience and have been reviewing Transformation and Projection concepts.

I noticed the idea of scale factor as it relates to transformations refers to the idea of moving points from ground to grid.  This gets confusing because it is not considered part of the Projection concepts, but yet involves the grid.  So I have to ask, the Grid to Ground scale factors appear to be a kind of "vertical" Transformation by moving points vertically, and shifting a datum would be the "horizontal" transformation.  Is my understanding reasonably correct ?

This Content Originally Published by a land surveyor to Land Surveyors United Network

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### Replies to This Discussion

when is a meter not a meter? The scale factor.

I refer to transformation as converting the position (x,y,z coordinates) of a point from one datum to another and scale factor as a multiplier used to project or represent a curved or spherical distance on a flat/plane surface.

https://www.ngs.noaa.gov/corbin/class_description/SPC_Doyle_2010.ppt

This link should help. Transformations are not for the uninitiated and should be left to those who know how to do it (surveyors). Based on liability alone I would hire a surveyor do define your transformations.

Amen.

I think you'll find this webinar helpful...

http://infogeospatial.trimble.com/2017-7-26TBCPH_Take2_RecordedWebi...

Let me know if that link doesn't work, or you can google

## "TBC Power Hour - Defining and Working with Grid and Ground Coordinates"

Chuck,

I took a look at your profile and it states that you are a student?.  Now, I ask this without trying to sound smug, but what surveying thery courses have you taken?   Your questions could have been or should have been covered by the completion of a 200 level (2nd year) course.

The basis of this scale factor concept starts with the "earth centered - earth fixed" or ECEF anchor of lats and longs.  Lats and Longs have no vertical component - they are strictly division lines on a sphere(oid).  When 18th century sailors got the sextant out - sea level was truly sea level.  As we surveyors got into more and more precise methods and techninques, we found we had to make adjustments to lat-longs between in the coastal areas and lat-longs in the mountians for distances to work out with any accuracy at all.

Draw two lines about 18 to 20 inches long on some paper with about a 15 degree angle between them. Now draw an arc with a radius of 12 inches from the intersection.  This first arc is your ellipsiod, the WGS 84, from which most all modern lat-longs are based.  Now draw a second arc with a radius of about 15 inches.  This second arc is the Geoid, it is a more localized model of the earths surface and it is this model that grid coordinates are calculated to.  Now draw a freehand squigglly arc near the ends of your first two lines - this is the Ground surface.  It this the difference in heights or elevations from the Geoid to the Ground that we must adjust for.

Image a tennis ball and a beach ball with the tennis ball suspended in the center of the beach ball.  With two straight wires, push each thru the beach ball and into the tennis ball (assume they are true vertical lines that will intersect the ECEF point inside the tennis ball).

At the point were each wire passes thru the surface of each ball you have a lat-long.  The tennis ball is the Geoid and the beach ball is the ground surface.  As you move from the tennis ball to the beach ball, the distance btween the two wires increases, but the lat-long values remain the same.

Now within a limited area this "projection" from the geoid to the ground can be simplyfied down to considering the difference in heights to be a change in radius from ECEF.

So, if you a scale between the two points on the "Geoid" arc you drew, that would be your "grid distance"

Then measure between the lines where they intersect your "squiggle arc" and that will be your ground distance.

Now, if any other members want to chime in on this feel free. haha.  I AM NOT A PROFESSOR!!

Hope this very simple ( maybe overly simple ) explaination helps.

Kevin

Thank you Kevin for your effort and time to explain.  Your perception is correct, I am a non surveyor but have limited exposure to the science.  As a Civil Engineer / Designer I am always looking for more understanding of the coordinate system subject due to its complexity.  If you know of any good survey reference texts (200 level course texts) I am definitely open to learning more.  Thanks again for your help and time.

Dear Mr. Turlington,

The concept is rendered overly complex by the way it is usually presented, either by a hopeful mentor or by sequence of acquaintance with the topics. If we separate them, the AHA! moment often happens.

1) Lets first deal with projection. It may be thought of as the means of projecting things on an irregular or one a round(ish) surface on to a plane surface. The various formulae we may are myriad and some quite complex, but it is best to think of a transparent globe with a spot light shining through it, for grasping the concept.

2) Next, let's consider the mapped plane that we have created. Its geometry is very simple compared to the spherical(ish) stuff above. parallel lines don't intersect triangles interior angles add to 180°, &c. Keep in mind for later that you and I may create projections with different parameters resulting in different maps with different coordinates.

3) Last, we have the slightly irrational notion of ground coordinates. Supposedly, plane coordinates pasted somehow on to the generally spherical and irregular surface of the earth. It is a compromised notion of making a projection to a convenient compromise elevation within the range of elevations of our project. If we are working on a dry salt lake it probably works well. Otherwise, it depends.

On the ground, actual distances taped and angles turned are the best data on which to rely, as does the PLSS.

Originally, the data was collected on the ground and a local plane was assumed and little else was needed. Celestial observations could disclose polar azimuth and latitude, longitude.

The next problem was the reverse of what most people ask about today. With the NAD27 state plane coordinates, the surface of the datum was the MSL and coordinates were developed at ground level but often the need was to express them in state plane coordinates. THE GROUND COORDINATES WERE KNOWN! The scale factor was a product of the elevation and position within the projection. The coordinates controlled by a known monument.

Along came NAD83 and office computers and GPS. Now the native coordinates ECEF or Lat, Lon on the ellipsoid (grid). The ellipsoid has no natural relationship to the local ground, being a simple 2 parameter smooth construct. The NAD83 state plane are likewise projected to the ellipsoid instead the assumed geoid (MSL). The problem is now how to convert ellipsoid based grid coordinates to the often more practical, ground coordinates for inversing and the like.

THIS is a tripping point. The resulting scaled coordinates depend upon what point one holds fixed when applying the scale factor. This was not an issue when going from ground to grid (27) but for each point one might hold a different result will occur. Hold a point near the centroid of the project and they will tend to change the least. A frequent choice is to hold the false origin of 0,0,0 that results in gross changes in the coordinate values.

Variables include what point is held, what point was held by the designer, where one is within the projection and the elevations involved.

I hope this helps. After noticing this and similar questions about transformations, projections, geoids &c. I wonder if there would be an interest in a seminar. What say you?

JAC

Yes, I appreciate your thoughts and time invested to reply to all my questions.  A seminar from the folks that practice this stuff would be great.  As a practicing designer/non surveyor, the subject of practical understanding is lacking and even the textbook explanation does not always add-up without knowledge of practical workflows and experience.  So yes, any webinar on this subject would be Great, I encourage it !

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