Is there a library in C that can calculate the geodesic distance between two points?
It's generally not necessary to compute geodesic distances up to centimeter accuracy as you'd have to model local geography to the same scale.
There are two cases I can think of where such accuracy might be desired:
you're actually interested in the straight-line distance and know exact coordinates, including height above sea level
you're working with a toy model and are interested in general properties, not practical applicability
In the first case, I'd transform from geographical coordinates (φ,λ,h) to geocentric cylindrical coordinates (r,z,λ)
r = (a²/k + h)·cosφ
z = (b²/k + h)·sinφ
λ = λ
where
k = √(a²·cos²φ + b²·sin²φ)
In the second case, I'd use Vincenty's formulae.
You can find (hopefully correct) C implementations of both algorithms under my bitbucket account.
Yes, I've ported the geodesic routines in GeographicLib to C. See
http://geographiclib.sourceforge.net/html/C/index.html
Related
I want to localize node C (refer image below).Here we know the coordinates of A and B and A is in range of B and C so we can calculate the distance AC and AB using the ranging functionality.
I need to calculate the distance between B and C that are not in direct range of each other.I want to use the Law of cosines,
Is there a way to calculate the angle 'γ' represented in the below picture in UnetStack ?Image Ref
If you only know the lengths of 2 sides of a triangle, generally there are an infinite number of triangles that are are compatible with this knowledge. So, there isn't a way for you to figure out BC unless you have some other source of information.
This isn't about UnetStack being able to do it or not, but it is simply mathematically not possible, since your problem is under-constrained as stated above.
Let's say you have 3 inputs: A, B, C. Can an artificial neural network (not necessarily feed forward) learn this pattern?
if C > k
output is A
else
output is B
Are there curtain types of networks, which can or which are well suited for this type of problem?
Yes, that's a relatively easy pattern for a feedforward neural network to learn.
You will need at least 3 layers I think assuming sigmoid functions:
1st layer can test C>k (and possibly also scale A and B down into the linear range of the sigmoid function)
2nd layer can calculate A/0 and 0/B conditional on the 1st layer
3rd (output) layer can perform a weighted sum to give A/B (you may need to make this layer linear rather than sigmoid depending on the scale of values you want)
Having said that, if you genuinely know the structure of you problem and what kind of calculation you want to perform, then Neural Networks are unlikely to be the most effective solution: they are better in situations when you don't know much about the exact calculations required to model the functions / relationships.
If the inputs can be only zeros and ones, then this is the network:
Each neuron has a Heaviside step function as an activation function. The neurons y0 and z have bias = 0.5; the neuron y1 has a bias = 1.5. The weights are shown above the corresponding connections. When s = 0, the output z = d0. When s = 1, the output z = d1.
If the inputs are continuous, then Sigmoid, tanh or ReLU can be used as the activation functions of the neurons, and the network can be trained with the back-propagation algorithm.
What is the most user friendly way to store only the rotation part of an OGL modelview (4x4) matrix?
For example; in a level editor to set the rotation for an object it would be easy to use the XYZ Euler angles. However this seems a very tricky system to use with matrices.
I need to be able to get AND set the rotation from this new representation.
(The alternative is to store the rotation part (4*3 numbers) but it is hard for a user to manipulate these)
I found some code here http://www.google.com/codesearch/p?hl=en#HQY9Wd_snmY/mesh/matrix3.h&q=matrix3&sa=N&cd=1&ct=rc that allows me to set and get rotation from angles (3 floats). This is ideal.
Although they're used regularily, I disregard the use of Euler angles. They're problematic as they only preserve the pointing direction of the object, but not the bitangent to that direction. More important: They're prone to gibal lock http://en.wikipedia.org/wiki/Gimbal_lock
A far superior method for storing rotations are Quarternions. In layman terms a quaternion consists of the rotational axis and the angle of rotation around this axis. It is thus a tuple of 4 scalars a,b,c,d. The quaternion is then Q = a + i*b + j*c + k*d, |Q| = 1, with the special properties of i,j,k that i² = j² = k² = i·j·k = -1 and i·j = k, j·k = i, k·i = j, which implies j·i = -k, k·j = -i, i·k = -j
Quaterions are thus extensions of complex numbers. If you recall complex number theory, you'll remember that the product of two complex numbers a =/= b with |a| = |b| = 1 is a rotation in the complex plane. It is thus easy to assume that rotations in 3D can be described by an extension of complex numbers into a complex hyperplane. This is what quaternions are.
See this article on the details.
http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation
In a standard 3D matrix you only need the top left 3x3 values to give the rotation. To apply the matrix as a 4x4 later on, you need to make the other values 0 apart from on the diagonal.
Here's a rotation only matrix where the values vXY give the rotations.
[v00 v01 v02 0]
[v10 v11 v12 0]
[v20 v21 v22 0]
[ 0 0 0 1]
Interestingly, the values form the bases of the coordinate system you have rotated the object into, so in the new system, the x-axis is along [v00 v01 v02], the y-axis is along [v10 v11 v12] and the z-axis obviously [v20 v21 v22].
You could show these axes beside the object and let the other drag them around to change the rotation, perhaps.
I would say this depends on the user, but to me the most "user friendly" way is to store "roll", "pitch" and "yaw". These are very non-technical terms that an average user can understand and adjust, and it should be easy for you to take these values and compute the matrix.
IMO, the most 'user friendly format' for rotation is storing Euler XYZ angles, this is generally how rotations are exposed in any 3d content creation software.
Euler angles are easy to transform to matrices, see here for the involved matrix product.
But you should not confuse the format given to the GUI/user and the storage format of the data: Euler XYZ angles have problems of their own when doing animation, gimbal lock can introduce unwanted behaviour.
Another candidate for storing/computing rotations is quaternions. They offer mathematical advantages over XYZ angles, essentially when interpolating between two rotations. Of course, you don't want to expose the quaternion values directly to any human user, you'll need to convert them to XYZ angles. You'll find plenty of code to that on the Web.
I would not recommend storing the rotation directly in matrix format. Extracting user friendly values from it is difficult, it does not offer any interesting behaviour for animation/interpolation, it takes for storage. IMO, matrices are to be created when needed to transform the geometry.
To conclude, there are a few options, you should select what suits you most. Do you plan to having animation or not ? etc.
EDIT
Also, you should not make an amalgam with model and view matrices. They are semantically very different, and are combined in OpenGL only for performance reasons. What I had in mind above in the 'model matrix'. The view matrix is generally given by your view system/camera manager, and is combined with you model matrix.
A quaternion is, although the math is "obscure and unintellegible" surprisingly user friendly, as it represents rotation around an axis by a given angle.
The axis of rotation is simply a unit vector pointing in that direction, multiplied by the sine of 1/2 the rotation angle, and the "obscure" 4th component equals the cosine of 1/2 the rotation angle.
It feels kind of "unnatural" at first sight, but once you grasp it... can it be any easier?
Does Postgres' Spatial plugin, or any Spatial package for that manner, factor in the altitude when calculating the distance between 2 points?
I know the Spatial packages factor in the approximate curvature of the earth but if one location is at the top of a mountain and the other location is close to the sea - it seems like the calculated difference between those two points would greatly vary if the difference in altitude was not factored into account.
Also keep in mind that if I have 2 points are at the same ocean altitude but a mountain exists between the 2 points - the distance package should account for this.
Those factors are not being counted at all. Why? The software only knows about the two features (the two points you are getting the distance, the sphere/spheroid and a datum/projection factor).
For that to happen you need to probably use a developed linestring, in which you will connect your point with n vertices, each of them being Z aware.
Imagine this (loose WKT): LINESTRING((0,1,2),(0,2,3),(0,3,4),(0,10,15),(0,11,-1)).
Asking the software to calculate the distance between each vertex and summing it up, will consider the variations of terrain. But without something like that, it is impossible to map for irregularities in terrain.
All GIS softwares cannot tell, by themselves, what are those irregularities in terrain, and therefore, not take them in account.
You can create such linestrings (automatically) with softwares like ArcGIS (and others), using a line (between two points), and a surface file, such as the ones provided freely by NASA (SRTM project). These files come in a raster format, and each pixel has a X Y and Z value, in meters. Traversing the line you want, coupled with that terrain profile, you can achieve the calculation you want to achieve. If you need to have super extra precise calculations, you need a precise surface, and precise Z values in each vertex of this profile line.
That cleared up?
If the distance formula you're using does not take the altitude of the two points as parameters (in addition to the Latitudes and Longitudes of the two points), then it does not factor in altitude to the distance calculation. In any event, altitude difference does not have a very significant effect on calculated distance.
As usual with GPS, the difference in distance calculations that altitude would make is probably smaller than the error in most commercial GPS devices anyway, so in most applications altitude can be safely dispensed with (altitude measurements themselves are pretty inaccurate with commercial GPS devices, although survey data on altitudes is quite accurate).
PostgreSQL does not factor in altitude when calculating distances. It is all done in a planar surface.
Most of database spatial packages will not take this into account, altought, if your point is 3d, i.e., has a Z coordinate that might happend.
I don´t have PostgreSQL in this machine, but try this.
SELECT ST_DISTANCE(ST_POINT(0,0,10),ST_POINT(0,0,0));
It´s fairly easy to know if it is taking into account your Z value, since the return should be > 0; If that turns out to be true, just create Z aware features, and you will be successfull.
What SQL SERVER 2008, for example, takes into account when calculating distances, is the position of a Geography feature in a sphere. Geometry features in SQL SERVER will always use planar calculations.
EDIT: checked this in PostGIS manual
For Z aware points you must use the ST_MakePoint function. It takes up to 4 arguments (X Y Z and M). St_POINT only takes two (X Y)
http://postgis.refractions.net/documentation/manual-1.4/ST_Distance.html
ST_DISTANCE = 2D calculations
ST_DISTANCE_SPHERE documentation (takes in account a fixed sphere for calculations - aka not planar)
http://postgis.refractions.net/documentation/manual-1.4/ST_Distance_Sphere.html
ST_DISTANCE_SPHEROID documentation (takes into account a choosen spheroid for your calculations)
http://postgis.refractions.net/documentation/manual-1.4/ST_Distance_Spheroid.html
ST_POINT documentation
http://postgis.refractions.net/documentation/manual-1.4/ST_Point.html
I search for the fastest or simplest method to compute the outer angle at any point of a convex polygon. That means, always the bigger angle, whereas the two angles in question add up to 360 degrees.
Here is an illustration:
Now I know that I may compute the angles between the two vectors A-B and C-B which involves the dot product, normalization and cosine. Then I would still have to determine which of the two resulting angles (second one is 180 degrees minus first one) I want to take two times added to the other one.
However, I thought there might be a much simpler, less tricky solution, probably using the mighty atan2() function. I got stuck and ask you about this :-)
UPDATE:
I was asked what I need the angle for. I need to calculate the area of this specific circle around B, but only of the polygon that is described by A, B, C, ...
So to calculate the area, I need the angle to use the formula 0.5*angle*r*r.
Use the inner-product (dot product) of the vectors describing the lines to get the inner angle and subtract from 360 degrees?
Works best if you already have the lines in point-vector form, but you can get vectors from point-to-point form pretty easily (i.e. by subtraction).
Taking . to be the dot product we have
v . w = |v| * |w| * cos(theta)
where v and w are vectors and theta is the angle between the lines. And the dot product can be computed from the components of the vectors by
v . w = SUM(v_i * w_i : i=0..3) // 3 for three dimensions. Use more or fewer as needed
here the subscripts indicate components.
Having actually read the question:
The angle returned from inverting the dot-product will always be less than 180 degrees, so it is always the inner angle.
Use this formula:
beta = 360° - arccos(<BA,BC>/|BA||BC|)
Where <,> is the scalar product and BA (BC) are the vectors from B to A (B to C).
I need to calculate the area of the circle outside of the polygon that is described by A, B, C, ...
It sounds like you're taking the wrong approach, then. You need to calculate the area of the circumcircle, then subtract the area of the polygon.
If you need the angle there is no way around normalizing the vectors and do a dot or cross-product. You often have a choice if you want to calculate the angle via asin, acos or atan but in the end that does not make a difference to execution speed.
However, It would be nice if you could tell us what you're trying to archive. If we have a better picture of what you're doing we might be able to give you some hints how to solve the problem without calculating the angle at the first place.
Lots of geometric algorithms can be rewritten to work with cross and dot-products only. Euler-angles are rarely needed.