Orthographic Projections

[qwerty]

How do you determine the precise degree of longitude and latutide, without eyeballing, in an orthgraphic projection (map)? Is there a formula of some kind that you can use??

[Jim Lewandowski]

http://www.progonos.com/furuti/MapProj/ ... rtHow.html

Maybe thats what you're looking for.

[qwerty]

I have found equations for converting known longitude and latitude to 3D cartesian coordinates but not vice versa. I don't know how to reverse the equation so that it converts the cartesian coordinates to degrees in longitude and latitude. Also in some articals it says to rotate the angle of the sphere to find the inverse. what does that mean?

[deleted-71552]

Hello, Qwerty!

I searched for "Orthographic Projection" on Google, and this site was one of the first hits:

Eric W. Weisstein. "Orthographic Projection." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Orthograph ... ction.html

It appears to give you what you are looking for. If not, please repost with more information about the form of the data you have and the form you are trying to convert to.

[Jim Lewandowski]

Hi Qwerty,
Can you explain exactly what you're trying to do? It seems that throwing some equations your way isn't helping. A better understanding of your problem might help us help you.

Jim

[qwerty]

I have images of sunspots from NASA site and I'm trying to determine the longitude and latitude coordinates of the sunspots to track their rotational speed. The pixels coordinates can be found using Microsoft Paint which can be converted into cartesian coordinates. I know that in the http://www.progonos.com/furuti/MapProj/ ... rtHow.html and
http://mathworld.wolfram.com/Orthograph ... ction.html they give you the equations for changing the longitude and latitude into cartesian coordinates but not vice versa. In the math world site, they give you the inverse transformations but I don't know if it applies to what I am doing. Also, I don't quite understand all the symbols.

[Jim Lewandowski]

I think you should have a look at this site...

http://www.meadows3.demon.co.uk/html/location.html

[Jim Lewandowski]

For the previous site look at the bottom of the page. Also check this site out.

http://www.astro.ucla.edu/~obs/150_draw.html

You should try to understand what is behind this stuff, and not just download the software program. In the theory section of your paper you should try to explain the math behind these Stonyhurst disks. And you should probably make sure they are applicable to a photograph.

Hope this helps,

[deleted-71552]

I have images of sunspots from NASA site and I'm trying to determine the longitude and latitude coordinates of the sunspots to track their rotational speed. The pixels coordinates can be found using Microsoft Paint which can be converted into cartesian coordinates. I know that in the http://www.progonos.com/furuti/MapProj/ ... rtHow.html and
http://mathworld.wolfram.com/Orthograph ... ction.html they give you the equations for changing the longitude and latitude into cartesian coordinates but not vice versa. In the math world site, they give you the inverse transformations but I don't know if it applies to what I am doing. Also, I don't quite understand all the symbols.

Hello, Qwerty!

Bravo to you for selecting a challenging project. I agree that the inverse equations with their Greek symbols can seem intimidating at first, but the formulas aren't quite as complicated as they appear at first glance.

If you are trying to calculate rotational speed, I will assume that you are going to be comparing the X,Y location of sunspots on two or more images from the NASA site that are separated in time by some known amount. If I understand your question correctly, you are looking for a way to compute the longitude and latitude of the sunspot in those images and use the differences to calculate rotational speed. Do I have that right?

Please note that the sites Jim Lewandowski has been suggesting talk about a very important characteristic of the Sun as viewed from earth. As we view the Sun, it appears to "wobble". Check out this page off of one of the sites he suggested for an excellent animation of the process:

http://www.meadows3.demon.co.uk/html/sunfromearth.html

In your project, you must either take this "wobble" into account or explain why you believe that you can ignore its effects. One justification for ignoring might be that the images you are using are so close together in time that the effects of this wobble are negligible. If the images were captured weeks or months apart, you probably can't ignore its effects.

Upon further examination, I see that formula 6 on the mathweb site is incorrect. I found the correct set of equations on this site:

http://www.raben.com/articles/MapProjec ... t_3_1.html

This page shows the inverse calculations you are looking for.

I think part of what makes the formulas look complicated is that they call for the use of a "reference" longitude and "reference" latitude. Whenever you measure something, it has to be done referenced to some known quantity or standard. In the case of the planet Earth, the references for lattitude and longitude are the Equator (lattitude 0) and the Prime Meridian (longitude 0). You can see these lines pretty clearly at the following web site:

http://wwp.greenwichmeridian.com/

The equations at the raben.com and meadows3.demon.co.uk sites allow you to take the Sun's "wobble" into account by specifying the lattitude and longitude of your reference lines relative to the center of the image. As the disk wobbles, the intersection of the reference lines moves around. You can use the Stonyhurst disks and the charts from the meadows3.demon.co.uk web site to determine how much the reference lines move from image to image. This is the most accurate way to accomplish your goals.

Don't let the symbols prevent you from making progress. They are just variables. You can rewrite them using a, b and c if doing so makes it easier to understand.

I recommend that you give this some thought and repost with any questions you might have.

[qwerty]

I know that the Sun's axis differ from the Earth's at about 7.25 degrees but since the images were taken by the SOHO satellite and not from the Earth, would that still affect the images? I don't know where SOHO is located now nor if the images are positioned in some distorted way or another. So can you please tell me if there are still things that I need to take into account for?
Here is the page where they explain the techincal stuff to you. http://sohowww.nascom.nasa.gov/data/ancillary/
I don't really understand it so can you explain if there is anything relevent to my experiment? But if the satellite is position in a way that north is north and there was no axis tilt, I don't have to use the Stonyhurst disks, right?
Thanks for your time. The equations are really helpful!

[deleted-71552]

I know that the Sun's axis differ from the Earth's at about 7.25 degrees but since the images were taken by the SOHO satellite and not from the Earth, would that still affect the images? I don't know where SOHO is located now nor if the images are positioned in some distorted way or another. So can you please tell me if there are still things that I need to take into account for?
Here is the page where they explain the techinal stuff to you. http://sohowww.nascom.nasa.gov/data/ancillary/
I don't really understand it so can you explain if there is anything relevent to my experiment? But if the satellite is position in a way that north is north and there was no axis tilt, I don't have to use the Stonyhurst disks, right?
Thanks for your time. The equations are really helpful!

Hello, again, Qwerty!

This project of yours grows more and more interesting the more I learn about it. Thanks for the pointer to the SOHO site. Have you looked at the "best of" image library? Awesome stuff!

The Stonyhurst disks and other Earth-based observational data is not appropriate for images taken from the SOHO satellite. So you were right to question their use in this case. The formulas for calcualting lattitude and longitude are still valid, and your challenge now is to figure out what to do with the reference lattitude and longitude values in the equations.

If I were you, I'd read through the ancillary data linked from the web page you referenced as well as looking at other sources. Your goal is to determine the way the Sun moves or rotates relative to the orbit of the satellite. Keep in mind that the Sun also rotates about an axis, and how you do your calculations will be affected by the tilt of that axis relative to the orbit.

So the ball is in your court on this. Don't let the "jargon" on those data pages scare you. If you don't understand some of it, skip to stuff that you do understand or you think you can look up. One of the keys to a successful web research run is to learn to recognize the data that helps you and that which does not.

Post again when you've made progress or have more questions!

[qwerty]

Thanks for your advice. I am supposing that the coordinate system that they use in the images are aligned to the Sun's equator and the axis are always fixed so that they correspond to the orthographic projection grid. The coordinates of the images are consistent throughout the year. In some other sources, it states that the spacecraft's orbit is centered so that X axis pointing from the center of the Earth to the center of the Sun. Is it relevent to say that the SOHO spacecraft is balanced between the Sun and Earth's graviational pull. But I am still confused if the images are oriented in a strange way. Am I on the right track or is this still kinda off? I really need to understand this..

This is quoted from the data pages.


"The orientation of the SOHO spacecraft is planned to have the spacecraft optical X axis (X) pointing towards the photometric center of the Sun, and the spacecraft optical Z axis (Z) oriented towards the north ecliptic hemisphere such that the (X,Z) plane contains the Sun axis of rotation. As such the Y axis will be parallel to the solar equatorial plane pointing towards the east (opposite to the solar rotation direction). ESA will be responsible for achieving this orientation with the misalignment margins defined in the EID-A.
A standard coordinate system is required for joint observations between instruments on the ground (for test purposes) and in space. This system, designated (X,Y), will be defined as follows: On the ground, the Y axis is parallel to the spacecraft Z axis and the X axis is anti-parallel to the spacecraft Y axis. In space, the (Y,Z) system is however no longer accessible. We will therefore define a virtual system (Y,Z), which is nominally coincident with (Y,Z) and where Y is perfectly aligned with the solar equator and its origin is at the Sun centre, and define (X,Y) in space as above using the virtual system (Y,Z)."



Also, they mention that "the solar rotation axis will be calculated using the Carrington ephemeris elements". What are Carrington ephemeris elements? I googled it and nothing helpful came up.

[deleted-71552]

Thanks for your advice. I am supposing that the coordinate system that they use in the images are aligned to the Sun's equator and the axis are always fixed so that they correspond to the orthographic projection grid. The coordinates of the images are consistent throughout the year. In some other sources, it states that the spacecraft's orbit is centered so that X axis pointing from the center of the Earth to the center of the Sun. Is it relevent to say that the SOHO spacecraft is balanced between the Sun and Earth's graviational pull. But I am still confused if the images are oriented in a strange way. Am I on the right track or is this still kinda off? I really need to understand this..

This is quoted from the data pages.


"The orientation of the SOHO spacecraft is planned to have the spacecraft optical X axis (X) pointing towards the photometric center of the Sun, and the spacecraft optical Z axis (Z) oriented towards the north ecliptic hemisphere such that the (X,Z) plane contains the Sun axis of rotation. As such the Y axis will be parallel to the solar equatorial plane pointing towards the east (opposite to the solar rotation direction). ESA will be responsible for achieving this orientation with the misalignment margins defined in the EID-A.
A standard coordinate system is required for joint observations between instruments on the ground (for test purposes) and in space. This system, designated (X,Y), will be defined as follows: On the ground, the Y axis is parallel to the spacecraft Z axis and the X axis is anti-parallel to the spacecraft Y axis. In space, the (Y,Z) system is however no longer accessible. We will therefore define a virtual system (Y,Z), which is nominally coincident with (Y,Z) and where Y is perfectly aligned with the solar equator and its origin is at the Sun centre, and define (X,Y) in space as above using the virtual system (Y,Z)."



Also, they mention that "the solar rotation axis will be calculated using the Carrington ephemeris elements". What are Carrington ephemeris elements? I googled it and nothing helpful came up.

Hello, again, Qwerty!

Nice work! I think you have identified the information you need to answer your question. You have supposed correctly that the answer lies somewhere in the quoted text you posted. Bravo! You are definitely on the right track.

Now you need to think through the text description. Picture it in your mind or draw it on paper if it helps. The part that is most relevant to your investigation is this:

"The orientation of the SOHO spacecraft is planned to have the spacecraft optical X axis (X) pointing towards the photometric center of the Sun, and the spacecraft optical Z axis (Z) oriented towards the north ecliptic hemisphere such that the (X,Z) plane contains the Sun axis of rotation."

The text is describing the (X,Y,Z) coordinate system of the SOHO spacecraft relative to the Sun. In a previous post, I wrote about how one must always establish a reference from which to take measurements, and this text describes the fundamentals of the reference for the spacecraft. Understanding this is critical to successfully executing your project.

My advice for making forward progress is that you imagine or draw a picture of what these words describe. The X axis is a line that goes out from the spacecraft and goes through where? The Z axis must be perpendicular to the X axis, and it is a line going through where? Think about where the X and Y axes meet. Where is the origin (0,0)? Now, if you imagine what the (X,Z) plane looks like, what does it mean that the Sun's axis, a line, is contained in that plane?

When you can answer these questions and see the (X,Z) plane and the Sun in your mind or on paper, then you can decide how to use the reference lattitude and longitude in the inverse equations.

Once you get this figured out, I strongly suggest that you consider explaining it as part of your project report.

Cheers!

[qwerty]

Am I right in saying that..
The X axis is a line that goes out from the spacecraft and goes through the center of the sun. The y axis is perpendicular and goes through the center also. The Z axis is aligned with the projection of the solar rotation axis. The origin (0,0) is the center of the solar disk.


I have another problem. To calculate the distance between to points of longitude and latitude, I need to use the distance formula for a sphere. There are multiple versions of the great circle distance formula but I don't know which on is the most accurate. Can you help me find one that is relevent to my experiment? One that uses degrees instead of radians.
Thanks.

[deleted-71552]

Am I right in saying that..
The X axis is a line that goes out from the spacecraft and goes through the center of the sun. The y axis is perpendicular and goes through the center also. The Z axis is aligned with the projection of the solar rotation axis. The origin (0,0) is the center of the solar disk.

I think you have this correct. To be sure, let me re-state what I think you wrote. If I were sitting on top of the satellite, the x-axis would be a line that extended out toward the sun and straight through the center of it. The y-axis would shoot out perpendicular to the left and right, also through the center of the Sun, and intersecting with the apparent surface of the Sun at what would be its Equator. The z-axis would be shooting straight up and down, perpendicular to both the x- and y-axes. The z-axis would be going through the center of the Sun as well, with the axis of rotation on the same line as the z-axis. The intersection of these three lines would be the center of the Sun or coordinate (0,0,0) in 3-D space. Now, for all intents and purposes, as I observe the Sun from my perch atop the satellite, I can't see the X-axis. All I can see is the y-axis going left and right and the z-axis going up and down. This defines my 2-D cartesian coordinate system, with the origin, (0,0) at the apparent center of the Sun's circle on the images you are processing.

I have another problem. To calculate the distance between to points of longitude and latitude, I need to use the distance formula for a sphere. There are multiple versions of the great circle distance formula but I don't know which on is the most accurate. Can you help me find one that is relevent to my experiment? One that uses degrees instead of radians.
Thanks.

The inverse formula previously provided (http://www.raben.com/articles/MapProjec ... t_3_1.html) is the formula you need. Since we have determined that the center of the Sun is a good reference position, the reference lattitude and longitude for your equations are zero and zero, dramamtically simplifying the equations.

Remember that lattitude and longitude are nothing more than angles measured in degrees. You have previously stated that you were planning to use the pixel location on the images to record the (y,z) position of sun spots. As long as the position is measured relative to the center of the Sun's disk and you measure the Sun's radius using the same scale as your coordinates, you can use the formulas listed to calculate the lattitude and longitude of the sun spots relative to the center of the Sun's disk. (This is very important to note in your project write up. Your frame of reference must be clearly defined.)

Now to bust your brain. We have established the position of the Sun relative to the satellite. This seems pretty straightforward. But we have not yet discussed one other variable in this experiement: The orbit of the satellite. Is the satellite using a stationary orbit where it is always found above the same spot on the Sun? That is, does the satellite orbit the Sun at a speed that matches the rotation of the Sun? Many of Earth's satellites work this way. That's why all those Dish Network antennae are fixed and pointed in the same direction - their sattelites are orbiting above a fixed point on the globe. They never change position relative to the ground.

The other option is that the satellite orbits faster, slower or in the opposie direction as the Sun's rotation. How the satellite itself moves relative to the sunspots must also be considered. In other words, if you see two images in which a sun spot appears to have moved, how much of the distance is due to the Sun rotating and how much of it is due to the movement of the satellite?

Now there's a puzzle to solve!

[qwerty]

I found that the SOHO satellite orbits the Sun once a year like Earth does. The Sun spins on it's axis every 28 days or so. This project is about the differential rotation of the Sun. So, I guess the satellite does not have a stationary orbit. Its orbit speed most definitely does not match the Sun's. But then I don't really understand how all this connects. How would you be able to calculate in the rotational speed of the sun and the orbiting speed of the Satellite?

[deleted-71552]

I found that the SOHO satellite orbits the Sun once a year like Earth does. The Sun spins on it's axis every 28 days or so. This project is about the differential rotation of the Sun. So, I guess the satellite does not have a stationary orbit. Its orbit speed most definitely does not match the Sun's. But then I don't really understand how all this connects. How would you be able to calculate in the rotational speed of the sun and the orbiting speed of the Satellite?

Hello, Qwerty!

Excellent work. You are quite good at pulling off these little research sub-projects. Bravo! This bodes well for your overall project.

One of the things that engineers and scientists do is to determine when information or variables are important to their experiment. There are at least two decisions you will need to make with regards to this project. As I discuss the second, I will attempt to answer your question about how all this connects.

The first decision is whether to take into account the fact that SOHO's orbit, like that of the earth and other satellites, is not a perfect circle. It's an ellipse. So you have to think about how this egg-shaped orbit affects the differences in the images you are analyzing. I'll give you a hint: It probably only matters if the images you are using are separated by a great deal of time. Over short periods, the arc through which SOHO moves is a close approximation to circular, so if the images you are comparing are taken days or just a few weeks apart you can probably neglect the effects of the ellipse for your experiment. This is certainly an issue that you should discuss in your write up. You should consider noting that to yield more accurate results than what you are presenting would require a precise accounting of the orbit in your work.

For the second issue, I think the best way to describe it would be to make some simplifying assumptions:

1) SOHO's orbit is circular

2) You, the observer, are looking down from high above the solar system

3) From this vantage point, the SOHO's orbit and the Sun look like two concentric wheels. The inner wheel is making one complete rotation every 28 days. The outer wheel is making one complete rotation every 365 days. (We'll use the fact that a circle describes an arc of 360 degrees to simplify SOHO's rotation to be approximately 1 degree per day. That would be 360 degrees, or one full rotation, in a year.)

With that picture in mind, imagine that SOHO is sitting at one point on the wheel. In the time it takes the Sun to make one complete rotation, 28 days, SOHO only rotates approximately 28 degrees. (One degree per day, from above.) Another way to state this is that the Sun rotates approximately 13 times for every single orbital rotation of SOHO.

What does this mean? If you were sitting on top of SOHO gazing at the Sun, how would the fact that the Sun is rotating faster than you are affect your Sun spot observations? How do you think you would take this into account in your project? Or do you think you can neglect its effects?

Curious minds want to know.

[qwerty]

Since the SOHO satellite is balanced between the gravitational pull of the Sun and the Earth, its orbit must be a shape similiar to Earth's orbit. The fact that SOHO is closer to the Sun than the Earth must mean that it orbits at a slower velocity than the Earth. Maybe there is a way to calculate the Earth's orbital angular velocity. This can be used to find out how many degrees the satellite travels in one day. As quoted before, the Sun travels approx. 13 degrees a day. So maybe I can take the number of degrees the satellite travels and find that difference of that to apply to my calculations. This is just a theory so I don't know if it is right or not. I think that this is probably a bit more accurate than just assuming SOHO's orbit is circular. Please give me some feedback on this. THanks!

[deleted-71552]

Since the SOHO satellite is balanced between the gravitational pull of the Sun and the Earth, its orbit must be a shape similiar to Earth's orbit. The fact that SOHO is closer to the Sun than the Earth must mean that it orbits at a slower velocity than the Earth. Maybe there is a way to calculate the Earth's orbital angular velocity. This can be used to find out how many degrees the satellite travels in one day. As quoted before, the Sun travels approx. 13 degrees a day. So maybe I can take the number of degrees the satellite travels and find that difference of that to apply to my calculations. This is just a theory so I don't know if it is right or not. I think that this is probably a bit more accurate than just assuming SOHO's orbit is circular. Please give me some feedback on this. THanks!

In a previous message, you wrote that the SOHO obits the Sun once a year. That's one complete revolution in 365 days. This is so close to 1 degree per day that I think you can use that for your thinking.

If the Sun's "surface" rotates at 13 degrees per day and SOHO orbits at 1 degree per day, how does that affect your observations from the satellite? If you compare two images that are taken 24 hours apart, how does the orbit of the SOHO affect the apparent distance the sun spots have moved? Would they appear to have moved more or less if the SOHO was somehow staionary in space? Since longitude is measured in degrees, do you think it is safe to just add or subtract a degree from your calculated longitudes for every 24 hours that elapse between pictures?