January 5th, 2009
object 1 is at x1,y1,z1 and object 2 is at x2,y2,z2.I need to find
what angles object 1 is to object 2.I making simple free program I
making.I need it to help me with the program for stuff like what angle
would object 1 need to go toward object 2Hi there,
There are a several ways to answer this question, so I will give you
one of the simplest.
I actually had to research problems like this (in 2D though) for the
programming contests at http://www.topcoder.com not long ago.
We are given two points P1 and P2 in Euclidean three-dimensional space
at coordinates (x1,y1,z1) and (x2,y2,z2) and we want to find the angle
between the rays O-P1 and O-P2 where O is the origin.
Let theta be this angle.
Then it is a theorem that the inner product of P1 and P2, see
http://mathworld.wolfram.com/DotProduct.html, that
P1 dot P2
has value
P1 P2 cos theta
The inner product of P1 and P2 is:
x1 x2 + y1 y2 + z1 z2
And the norm of a vector is the square root of the sum of the squares
of its coordinates (this works also in any number of dimensions)
Thus, here is how one might solve this in pseudocode
double angle (double P1, double P2){ //compute angle between P1
and P2
double P1norm=0; // P1
double P2norm=0; // P2
for (int i=0;i<3;i++)P1norm+=P1[i]*P1[i];
P1norm=sqrt(P1norm); // Get P1norm
for (int i=0;i<3;i++) P2norm+=P2[i]*P2[i];
P2norm=sqrt(P2norm); //Get P2norm
double epsilon=.000000000000001 //some very small positive value,
depending on machine precision and application details
if (P1norm
I will try to clarify what I am looking for better.I make will
startrek like.
object 1 is enterprise d.it is located at x1, y1, z1.
object 2 is Romulan warbird that just decloaked inside federation
space. it is located at x2,y2,z2
The enterprise d wants to move toward the romulan warbird.The helmsman
needs to know the following angles:
1 the angle that how left or right
I will try to clear up what I am looking for.I will startrek like.
object 1 is enterprise d. it located at x1, y1, z2
object 2 is romulan warbird in federation space.it is located at
x2,y2,z3
of course the enterprise d want to go to the romulan warbird.The
helmsman needs the following info.
1 the angle saying how left or right enterprise d needs to be go to
strait toward the romulan warbird.
2 the angle saying how up or down the enterprise d needs to go strait
toward the romulan warbird.
Ps the weapons officer may need the info also.
Since I am darkbasic for the programming language it does not appear
to have inverse sin or inverse tan.They have the following math
commands.
SQRT() will return the square root of an expression
ABS() will return the positive equivalent of an expression
INT() will return the largest integer before the decimal point
RND() will return a random number within a given range
EXP() will return a number, raised to the power of a value
COS() will return the cosine of a value
SIN() will return the sine of a value
TAN() will return the tangent of the a value
ACOS() will return the arccosine of a value
ASIN() will return the arcsine of a value
ATAN() will return the tangent of a value
ATANFULL() will return the angle between points
HCOS() will return the hyperbolic cosine of a value
HSIN() will return the hyperbolic sine of a value
HTAN() will return the hyperbolic tangent of a value
First, the arc cosine, ACOS, is the same function as "inverse cosine".
These are just different terms for the same thing.
Second, in all these computations I am going to assume that the
objects reside in Euclidean 3-dimensional space. In fact the topology
of the universe may be different from Euclidean three-space (it might
be a Riemannian manifold) in which case the computations become more
complex because the meaning of "straight line" changes (by analogy,
think of a ship on the surface of the Earth trying to go from point A
to point B: the ship cannot simply "point towards" A and B, since it
must travel along geodesics, great circles.)
Third, I am going to assume that both objects are at rest relative to
one another. If Object 2 is moving at some rate then the Enterprise
may wish to aim at a location such that the Enterprise arrives to
where the object is at the time the Enterprise arrives; but I will
neglect this. Similarly, if the Enterprise is moving at some rate then
I will neglect the time it takes the Enterprise to change its
orientation and momentum vector.
Fourth, there is not quite enough information given to solve the
problem. Specifically, you will need the orientation of the
Enterprise.
Let me explain. Suppose the Enterprise is at (0,0,0) and the other
ship is at (1,0,0). Then if the Enterprise is already pointing in the
direction of (1,0,0) then clearly the Enterprise does not need to turn
at all. On the other hand, if the Enterprise is facing opposite to
(1,0,0) then the Enterprise must perform a 180-degree rotation about,
for example, the z-axis.
Now suppose we idealize the Enterprise is a small directed line
segment from endpoint P1_a to P1_b representing the stern and the bow
of the Enterprise. One might think that this would be sufficient to
solve the problem, since one could determine how much to rotate the
line segment about lines parallel to the coordinate axes and throught
the center of mass of the Enterprise. However, remember that
(presumably) the coordinate system used by the helmsman of the
Enterprise will be the natural coordinate system given by the
Enterprise itself; that is, the helmsman can rotate about axes
perpindicular to the line segment through Captain Picard, if Picard
were standing straight up and facing forward.
There are several ways one can use to denote the orientation of the
Enterprise. If you like, you can tell me how you plan to represent
this orientation, or I can choose some one and then give you the
answer from that representation. Which would you prefer?
Well think I got it right.Well I plugged some numbers in it looked
about what I think it should be.For example x1 ,y1, z1 where all 1 and
x2,y2,z2 where all 2 and both angles showed up as 45 which I believe
is correct.Below is source code I made for it.Does it look right to
you?
xdif = x2 -x1
ydif = y2 - y1
zdif = z2 - z1
if ydif > 0 then xyangle = atan(xdif/ydif)
if ydif < 0 then xyangle = atan(xdif/ydif) + 180
h = sqrt( (xdif)^2 + (ydif)^2 )
if h > 0 then zangle = atan(zdif/h)
if h < 0 then zangle = atan(zdif/h) + 180
We are actually discussing three separate issues here:
1. How does one usefully mathematically model the angular navigation
commands of the Enterprise?
2. Given a mathematical definition of the problem in 1, how does one
solve for the angles required?
3. Given a mathematical solution in 2, how do we actually code it?
Regarding 1, one resource I used is:
Star Trek: The Next Generation Technical Manual by Rick Sternbach and
Michael Okuda (Pocket Books, 1991).
Section 3.4 of that book discusses flight control. Pages 36-37 discuss
navigation methods specifically. Six input modes for Conn are given:
1. Destination planet or star system.
2. Destination sector.
3. Spacecraft intercept (only when the target spacecraft has a
tactical sensor lock).
4. Relative bearing. The flight vector is specified as
azimuth/elevation relative to the current orientation of the
Enterprise.
5. Absolute heading. Flight vector is specified as azimuth/elevation
relative to the center of the galaxy
6. Galactic coordinates (XYZ).
Presumably you are most interested in input modes 4 and 5 for the
purposes of this question.
As I pointed out in my previous answer clarification, however, in
order to implement mode 4 you will need to keep track of the spatial
orientation of the Enterprise: which way it is pointed and how much it
is twisted about its axis.
6. This implementation would require a lot of additional information
and would require some linear algebra to do, so you may not want to
implement input mode 4.
Input mode 5 does suggest that you can use absolute angles and so you
do not need to keep track of the orientation of the Enterprise.
My recommendation if you choose to use input mode 5 would be to
convert both the Enterprise P1 and the target P2 into spherical
coordinates. (see http://mathworld.wolfram.com/SphericalCoordinates.html
) .
The spherical coordinates of a point [x,y,z], assuming the sphere has
origin at the galactic center, is given by:
r = sqrt(x^2 + y^2 + z^2)
theta = ATAN(y/x)
phi = ACOS (z/r)
Here, theta is the "azimuthal angle", the angle in the xy-plane from
the x axis, and phi is the "polar angle", the angle from the z-axis,
formed by the ray O-[x,y,z].
The picture at the above URL is clearer than I can draw here as well.
Anyway, get the azimuth and polar angles for the Enterprise at P1, say
phi_1 and theta_1
Next, get the azimuth and polar angles for the target, say the Romulan
warbird, at
phi_2 and theta_2
using the above formulas.
Now, the angles you need to rotate the enterprise are given by the XY
rotation, the azimuthal rotation, of:
theta_2 - theta_1 = A
and the elevation angle, given by:
phi_2 - phi_1 = E
Thus, the instruction would be A mark E, where A and E are defined
above.
Finally, I will answer your specific request for clarification:
You write:
"For example x1 ,y1, z1 where all 1 and
x2,y2,z2 where all 2 and both angles showed up as 45"
I'm afraid I do not understand why in this case, in which P1, P2, and
the origin are all collinear, that "both angles should show up as 45
degrees".
Either you are using relative bearing or absolute heading (input modes
4 and 5 above).
If you are using relative bearing, then the amount to rotate the
Enterprise depends not only on x1,x2,x3 but also on the orientation of
the Enterprise. Since you do not specify the orientation of the
Enterprise, there is no way to know what its orientation is, hence,
there is no way to know how much to rotate it. Your answer would be
correct, however, if you are using relative bearing AND if the natural
coordinate system defined by the Enterprise was parallel to the
galactic coordinate system, but this is not necessarily the case (and
doesn't make sense, since the Enterprise is going to turn around as
soon as it executes the mark order anyway; that is to say, the
Enterprise is going to be turning all the time).
On the other hand, if you are using absolute bearing, then the
Enterpise need not change its mark at all: it does not need to perform
any azimuthal or elevation rotation, since its polar and azimuthal
angles are identical to that of the target.
Thus, since it remains uspecified what it is that your code intends to
compute so I cannot comment on it (well, of course, h is never
negative and you do not need to check for that case)
I just wanted to add a couple of other comments.
You can find additional information on Star Trek cartography at
http://www.stdimension.de/int/Cartography/IntroTools.htm . This pretty
much agrees with my suggestion to use spherical coordinates with the
origin at the galaxy center, which was taken from the Star Trek Next
Generation Technical manual (op cit.) . This site has some additional
detail on Star Trek cartography, including coordinates for numerous
stars.
For implementation, you might consider first writing a routine to
convert galactic coordinates to spherical coordinates. For example,
the point (1,1,1) should have spherical coordinates of r=sqrt(3), and
azimuthal and polar angles each equal to 45 degrees, using the
formulae in my previous clarification.
Finally, one suggestion that you might consider, if you have not
already done so, is first to build your system working for simplified
2-D Star Trek, and only then to add in the third dimension. I
personally find it useful when developing programs to get some simple
prototypes up first before tackling the full problem; I don't really
have a reference for you on that, maybe
http:///www.extremeprogramming.org .
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