Calculating new coordinates based on current position, bearing and speed
 

Hi

I'm writing a program that needs to calculate a new set of coordinates based on the current position, the directional bearing and the speed of movement. The map is simply a flat 2D representation and therefore the Earth's curving axis does not need to be taken into account.

I've searched through Google, but I can only find information on achieving this using lat/long on the Earth's 3D surface (which is a different representation and thus problem).

Does anybody have any idea on what the equation is to achieve this?

Thanks

Cotun

Check out Mr Pythagoras' famous idea

Start from entering

cartesian coordinates
polar coordinates

into your favorite WEB search engine and reading the Wiki articles.

Define bearing a in radians,

a = (angle in degrees) × Pi / 180

Normally we define zero angle as towards the positive x axis, increasing towards the positive y axis.

If at time t=0 you are at (x(0), y(0)), bearing a and velocity v, then you can calculate the position as a function of time t using

x(t) = x(0) + t v cos a
y(t) = y(0) + t v sin a

As long as your units for x, y, v and t match ([x] = [y] = [t v]), and the units are uniform over the entire map, you can use any units you wish.

Thanks to all of you, particularly Nominal Animal for providing the easy-to-understand reference on the equation. It took me only minutes to implement this into my program and I do appreciate the effort you took.

How do you know the provided equation is correct ? I.e. if a number of different posters provide different equations, how will you know which of them to choose ?

Because I implemented and tested the provided equation on a spreadsheet first. It provides exactly what I needed and works within expected boundaries.

How do you know the results returned by the search engines are correct? That the equations in Wiki articles are correct? When you see the different equations, how will you know which of them to choose?

Sergei, do you realize how much math you need to actually know and understand to derive the equation? Cotun is obviously unfamiliar with this. Just because you (and I, I admit) think that this is something every programmer should know, does not make it so.

Most of the links you show point to this familiar equation of motion, using vector notation:

x(t) = x(0) + v t + 1/2 a t^2

First, you need to understand vector notation, and that the above is actually equivalent to

x(t) = x(0) + v_x t + 1/2 a_x t^2
y(t) = y(0) + v_y t + 1/2 a_y t^2

Then you need to realize that because the motion in this case is at constant velocity, the acceleration is zero:

x(t) = x(0) + v_x t
y(t) = y(0) + v_y t

Finally, you must be able to convert the bearing from polar coordinates to cartesian coordinates, i.e.

v_x = v cos b
v_y = v sin b

where "total" velocity is v, and bearing b is in radians counterclockwise from positive x axis.

Then, and only then, you can arrive at the final equation,

x(t) = x(0) + v t cos b
y(t) = y(0) + v t sin b

Math is unlike programming in that the end result is almost always easier to understand and apply than the method of arriving at it. In fact, usually the method of finding a solution requires not only experience, but creative leaps (or tedious hours of trial and error). Therefore, I believe, we should point those with math problems towards the equations themselves, and perhaps explain (with references) why and how they work. After all, this is not a theoretical math forum; all math problems we're likely to see are "applied" math.

Don't you agree?

I don't really agree that every programmer needs to know this kind of thing. I have more than 15 years of experience as a programmer, but have largely ended up working on the business end of the market rather than any computer-science or mathematical modelling in code. Until today, I have never needed this equation to make anything work. Only a few days ago, I came across a need to calculate bearing for the first time. It is quite possible to be a programmer and not be heavily involved in applied mathematics, a field which I fully admit I'm neither experienced or knowledgeable with at present.

No, I don't agree. I was taught at school (yes, a special kind of high school, before the university) trigonometry and kinematics. And when necessary, in simple cases like this one I can really prove (mathematically) formulas from Wiki.

If a programmer is a domain programmer, he/she should know the domain. "Know" means being able to prove what is provable

FWIW, looking up info on Digital Fourier Transform in Wiki I disagreed ((slightly) with their formula for Hanning window (the disagreement is N <-> N - 1 and for large Ns is insignificant, but still).

I was once looking into somebody else's FFT code and did find a significant bug in windowing - significant because quite audible. And mathematically it was a bug - much more significant than the above N <-> N - 1.


More on DFT - other pitfalls are scaled/unscaled and complex exponent imaginary part sign.

This is a lovely nonsense.

I.e. suppose I tell you that linear function is

y(x) = k * x + b + sin(x * pi)

and tell you to check it for x = 0, 1, 2, 3 ...

- though the above function is wrong, it will work for the suggested set of test points exactly as the correct

y(x) = k * x + b

one.

I think in a Niklaus Wirth's book I once read: "Testing never proves absence of bugs, it can only prove their presence" - I don't claim it's the exact quote, but the sense was definitely like this.

Here I agree. Prove yes, derive from first principles no. That is why I showed the final formula, not just point at the basic principles.

That is quite common in Wikipedia (although I believe you mean Discrete Fourier Transform). I did something similar with trilateration (under Simplified solution without extra assumptions), but lost interest when even such an easy simplification raised a controversy. The original article used a number of coordinate system transforms with trigonometric functions. I think the original author was unfamiliar with vector algebra, but familiar with matrices and coordinate transforms with trigonometric functions.

I concede that this is of course possible. However, the equation given would have to give results that were good enough to fool my quite comprehensive test cases and yet still be wrong for my application. If this was done, it is much more likely by design than by accident and therefore Nominal Animal would likely have been deceiving me deliberately. I didn't think he was and can't see why he would in all honesty.

If I was working on any job of a safety-critical nature, then of course I should be able to prove all the mathematics or find somebody who works in the same organisation who could for the good of the public. However, I was merely writing a computer game which will be later released as open-source. If there is some minor obscure error that develops due to me using the wrong calculations, the most likely event is that nobody will notice. It would not be a good use of time for me to spend weeks learning about mathematics just to come to what is a relatively simple equation to resolve a problem I have for a basic, unimportant task.

We all depend on things done by others on a daily basis and often build upon those too. You should be able to prove that the water you drink is safe before you actually do, but few people would do that. Most people wouldn't know how to start doing that, just as I don't have presently have the mathematical basis to do what you expected me to do.

I realise of course that you were talking generically about being able to understand what you are implementing, but I think you are being too theoretical and thus harsh when you look at the real-world application.

This thread reminded me of a joke I read on facebook:

But seriously, I doubt most non-trivial programs can be "proven". If we did that, there would be nothing like for example this forum.

The OP's need/formulas he needs are trivial.