Given four points in the plane, A,B,X,Y, I wish to determine which of the following two angles is smaller ∢ABX or ∢ABY.
The angle ∢ABX is defined as the angle of BX, when AB is translated to lie on the open segment (-∞,0]. Intuitively when saying ∢ABX I mean the angle you get when you turn left after visiting vertex B.
I'd rather not use cos or sqrt, in order to preserve accuracy, and to minimize performance (the code would run on an embedded system).
In the case where A=(-1,0),B=(0,0), I can compare the two angles ∢ABX and ∢ABY, by calculating the dot product of the vectors X,Y, and watch its sign.
What I can do in this case is:
Determine whether or not ABX turns right or left
If ABX turns left check whether or not Y and A are on the same side of the line on segment BX. If they are - ∢ABX is a smaller than ABY.
If ABX turns right, then Y and A on the same side of BX means that ∢ABX is larger than ∢ABY.
But this seems too complicated to me.
Any simpler approach?
Here's some pseudocode. Doesn't detect the case when both angles are the same. Also doesn't deal with angle orientation, e.g. assumes all angles are <= 180 degrees.
v0 = A-B
v1 = X-B
v2 = Y-B
dot1 = dot(v0, v1)
dot2 = dot(v0, v2)
if(dot1 > 0)
if(dot2 < 0)
// ABX is smaller
if(dot1 * dot1 / dot(v1,v1) > dot2 * dot2 / dot(v2, v2) )
// ABX is smaller
// ABY is smaller
if(dot2 > 0)
// ABY is smaller
if(dot1 * dot1 / dot(v1,v1) > dot2 * dot2 / dot(v2,v2) )
// ABY is smaller
// ABX is smaller
Note that much of this agonizing pain goes away if you allow taking two square roots.
Center the origin on B by doing
X = X - B
Y = Y - B
A = A - B
EDIT: you also need to normalise the 3 vectors
A = A / |A|
X = X / |X|
Y = Y / |Y|
Find the two angles by doing
acos(A dot X)
acos(A dot Y)
===
I don't understand the point of the loss of precision. You are just comparing, not modifying in any way the coordinates of the points...
You might want to check out Rational Trigonometry. The ideas of distance and angle are replaced by quadrance and spread, which don't involve sqrt and cos. See the bottom of that webpage to see how spread between two lines is calculated. The subject has its own website and even a youtube channel.
I'd rather not use cos or sqrt, in order to preserve accuracy.
This makes no sense whatsoever.
But this seems too complicated to me.
This seems utterly wrong headed to me.
Take the difference between two vectors and look at the signs of the components.
The thing you'll have to be careful about is what "smaller" means. That idea isn't very precise as stated. For example, if one point A is in quadrant 4 (x-component > 0 and y-component < 0) and the other point B is in quadrant 1 (x-component > 0 and y-component > 0), what does "smaller" mean? The angle of the vector from the origin to A is between zero and π/2; the angle of the vector from the origin to B is between 3π/4 and 2π. Which one is "smaller"?
I am not sure if you can get away without using sqrt.
Simple:
AB = A-B/|A-B|
XB = X-B/|X-B|
YB = Y-B/|Y-B|
if(dot(XB,AB) > dot (YB,AB)){
//<ABY is grater
}
else
{
...
}
Use the law of cosines: a**2 + b**2 - 2*a*b*cos(phi) = c**2
where a = |ax|, b =|bx| (|by|), c=|ab| (|ay|) and phi is your angle ABX (ABY)
Related
I have a simple maths/physics problem here: In a Cartesian coordinate system, I have a point that moves in time with a known velocity. The point is inside a box, and bounces orthognally on its walls.
Here is a quick example I did on paint:
What we know: The red point position, and its velocity which is defined by an angle θ and a speed. Of course we know the dimensions of the green box.
On the example, I've drawn in yellow its approximate trajectory, and let's say that after a determined period of time which is known, the red point is on the blue point. What would be the most efficient way to compute the blue point position?
I've tought about computing every "bounce point" with trigonometry and vector projection, but I feel like it's a waste of resources because trigonometric functions are usually very processor hungry. I'll have more than a thousand points to compute like that so I really need to find a more efficient way to do it.
If anyone has any idea, I'd be very grateful.
Apart from programming considerations, it has an interesting solution from geometric point of view. You can find the position of the point at a specific time T without considering its temporal trajectory during 0<t<T
For one minute, forget the size and the boundaries of the box; and assume that the point can move on a straight line for ever. Then the point has constant velocity components vx = v*cos(θ), vy = v*sin(θ) and at time T its virtual porition will be x' = x0 + vx * T, y' = y0 + vy * T
Now you need to map the virtual position (x',y') into the actual position (x,y). See image below
You can recursively reflect the virtual point w.r.t the borders until the point comes back into the reference (initial) box. And this is the actual point. Now the question is how to do these mathematics? and how to find (x,y) knowing (x',y')?
Denote by a and b the size of the box along x and y respectively. Then nx = floor(x'/a) and ny = floor(y'/b) indicates how far is the point from the reference box in terms of the number of boxes. Also dx = x'-nx*a and dy = y'-ny*b introduces the relative position of the virtual point inside its virtual box.
Now you can find the true position (x,y): if nx is even, then x = dx else x = a-dx; similarly if ny is even, then y = dy else y = b-dy. In other words, even number of reflections in each axis x and y, puts the true point and the virtual point in the same relative positions, while odd number of reflections make them different and complementary.
You don't need to use trigonometric function all the time. Instead get normalized direction vector as (dx, dy) = (cos(θ), sin(θ))
After bouncing from vertical wall x-component changes it's sign dx = -dx, after bouncing from horizontal wall y-component changes it's sign dy = -dy. You can see that calculations are blazingly simple.
If you (by strange reason) prefer to use angles, use angle transformations from here (for ball with non-zero radius)
if ((ball.x + ball.radius) >= window.width || (ball.x - ball.radius) <= 0)
ball.theta = M_PI - ball.theta;
else
if ((ball.y + ball.radius) >= window.height || (ball.y - ball.radius) <= 0)
ball.theta = - ball.theta;
To get point of bouncing:
Starting point (X0, Y0)
Ray angle Theta, c = Cos(Theta), s = Sin(Theta);
Rectangle coordinates: bottom left (X1,Y1), top right (X2,Y2)
if c >= 0 then //up
XX = X2
else
XX = X1
if s >= 0 then //right
YY = Y2
else
YY = Y1
if c = 0 then //vertical ray
return Intersection = (X0, YY)
if s = 0 then //horizontal ray
return Intersection = (XX, Y0)
tx = (XX - X0) / c //parameter when vertical edge is met
ty = (YY - Y0) / s //parameter when horizontal edge is met
if tx <= ty then //vertical first
return Intersection = (XX, Y0 + tx * s)
else //horizontal first
return Intersection = (X0 + ty * c, YY)
Making a game using Golang since it seems to work quite well for games. I made the player face the mouse always, but wanted a turn rate to make certain characters turn slower than others. Here is how it calculates the turn circle:
func (p *player) handleTurn(win pixelgl.Window, dt float64) {
mouseRad := math.Atan2(p.pos.Y-win.MousePosition().Y, win.MousePosition().X-p.pos.X) // the angle the player needs to turn to face the mouse
if mouseRad > p.rotateRad-(p.turnSpeed*dt) {
p.rotateRad += p.turnSpeed * dt
} else if mouseRad < p.rotateRad+(p.turnSpeed*dt) {
p.rotateRad -= p.turnSpeed * dt
}
}
The mouseRad being the radians for the turn to face the mouse, and I'm just adding the turn rate [in this case, 2].
What's happening is when the mouse reaches the left side and crosses the center y axis, the radian angle goes from -pi to pi or vice-versa. This causes the player to do a full 360.
What is a proper way to fix this? I've tried making the angle an absolute value and it only made it occur at pi and 0 [left and right side of the square at the center y axis].
I have attached a gif of the problem to give better visualization.
Basic summarization:
Player slowly rotates to follow mouse, but when the angle reaches pi, it changes polarity which causes the player to do a 360 [counts all the back to the opposite polarity angle].
Edit:
dt is delta time, just for proper frame-decoupled changes in movement obviously
p.rotateRad starts at 0 and is a float64.
Github repo temporarily: here
You need this library to build it! [go get it]
Note beforehand: I downloaded your example repo and applied my change on it, and it worked flawlessly. Here's a recording of it:
(for reference, GIF recorded with byzanz)
An easy and simple solution would be to not compare the angles (mouseRad and the changed p.rotateRad), but rather calculate and "normalize" the difference so it's in the range of -Pi..Pi. And then you can decide which way to turn based on the sign of the difference (negative or positive).
"Normalizing" an angle can be achieved by adding / subtracting 2*Pi until it falls in the -Pi..Pi range. Adding / subtracting 2*Pi won't change the angle, as 2*Pi is exactly a full circle.
This is a simple normalizer function:
func normalize(x float64) float64 {
for ; x < -math.Pi; x += 2 * math.Pi {
}
for ; x > math.Pi; x -= 2 * math.Pi {
}
return x
}
And use it in your handleTurn() like this:
func (p *player) handleTurn(win pixelglWindow, dt float64) {
// the angle the player needs to turn to face the mouse:
mouseRad := math.Atan2(p.pos.Y-win.MousePosition().Y,
win.MousePosition().X-p.pos.X)
if normalize(mouseRad-p.rotateRad-(p.turnSpeed*dt)) > 0 {
p.rotateRad += p.turnSpeed * dt
} else if normalize(mouseRad-p.rotateRad+(p.turnSpeed*dt)) < 0 {
p.rotateRad -= p.turnSpeed * dt
}
}
You can play with it in this working Go Playground demo.
Note that if you store your angles normalized (being in the range -Pi..Pi), the loops in the normalize() function will have at most 1 iteration, so that's gonna be really fast. Obviously you don't want to store angles like 100*Pi + 0.1 as that is identical to 0.1. normalize() would produce correct result with both of these input angles, while the loops in case of the former would have 50 iterations, in the case of the latter would have 0 iterations.
Also note that normalize() could be optimized for "big" angles by using floating operations analogue to integer division and remainder, but if you stick to normalized or "small" angles, this version is actually faster.
Preface: this answer assumes some knowledge of linear algebra, trigonometry, and rotations/transformations.
Your problem stems from the usage of rotation angles. Due to the discontinuous nature of the inverse trigonometric functions, it is quite difficult (if not outright impossible) to eliminate "jumps" in the value of the functions for relatively close inputs. Specifically, when x < 0, atan2(+0, x) = +pi (where +0 is a positive number very close to zero), but atan2(-0, x) = -pi. This is exactly why you experience the difference of 2 * pi which causes your problem.
Because of this, it is often better to work directly with vectors, rotation matrices and/or quaternions. They use angles as arguments to trigonometric functions, which are continuous and eliminate any discontinuities. In our case, spherical linear interpolation (slerp) should do the trick.
Since your code measures the angle formed by the relative position of the mouse to the absolute rotation of the object, our goal boils down to rotating the object such that the local axis (1, 0) (= (cos rotateRad, sin rotateRad) in world space) points towards the mouse. In effect, we have to rotate the object such that (cos p.rotateRad, sin p.rotateRad) equals (win.MousePosition().Y - p.pos.Y, win.MousePosition().X - p.pos.X).normalized.
How does slerp come into play here? Considering the above statement, we simply have to slerp geometrically from (cos p.rotateRad, sin p.rotateRad) (represented by current) to (win.MousePosition().Y - p.pos.Y, win.MousePosition().X - p.pos.X).normalized (represented by target) by an appropriate parameter which will be determined by the rotation speed.
Now that we have laid out the groundwork, we can move on to actually calculating the new rotation. According to the slerp formula,
slerp(p0, p1; t) = p0 * sin(A * (1-t)) / sin A + p1 * sin (A * t) / sin A
Where A is the angle between unit vectors p0 and p1, or cos A = dot(p0, p1).
In our case, p0 == current and p1 == target. The only thing that remains is the calculation of the parameter t, which can also be considered as the fraction of the angle to slerp through. Since we know that we are going to rotate by an angle p.turnSpeed * dt at every time step, t = p.turnSpeed * dt / A. After substituting the value of t, our slerp formula becomes
p0 * sin(A - p.turnSpeed * dt) / sin A + p1 * sin (p.turnSpeed * dt) / sin A
To avoid having to calculate A using acos, we can use the compound angle formula for sin to simplify this further. Note that the result of the slerp operation is stored in result.
result = p0 * (cos(p.turnSpeed * dt) - sin(p.turnSpeed * dt) * cos A / sin A) + p1 * sin(p.turnSpeed * dt) / sin A
We now have everything we need to calculate result. As noted before, cos A = dot(p0, p1). Similarly, sin A = abs(cross(p0, p1)), where cross(a, b) = a.X * b.Y - a.Y * b.X.
Now comes the problem of actually finding the rotation from result. Note that result = (cos newRotation, sin newRotation). There are two possibilities:
Directly calculate rotateRad by p.rotateRad = atan2(result.Y, result.X), or
If you have access to the 2D rotation matrix, simply replace the rotation matrix with the matrix
|result.X -result.Y|
|result.Y result.X|
I am breaking my head trying to find an appropriate formula to calculate a what sounds to be an easy task but in practice is a big mathematical headache.
I want to find out the offset it needs to turn my vector's angle (X, Y, Angle) to face a coord ( X, Y )
My vector won't always be facing 360 degrees, so i need that as a variable as well..
Hoping an answer before i'm breaking my pc screen.
Thank you.
input
p1 = (x1,y1) point1 (vector origin)
p2 = (x2,y2) point2
a1 = 360 deg direction of vector
assuming your coodinate system is: X+ is right Y+ is up ang+ is CCW
your image suggest that you have X,Y mixed up (angle usually start from X axis not Y)
da=? change of a1 to match direction of p2-p1
solution 1:
da=a1-a2=a1-atanxy(x2-x1,y1-y1)
atanxy(dx,dy) is also called atan2 on some libs just make sure the order of operands is the right one
you can also use mine atanxy in C++
it is 4 quadrant arctangens
solution 2:
v1=(cos(a1),sin(a1))
v2=(x2-x1,y2-y1)
da=acos(dot(v1,v2)/(|v1|*|v2|))
or the same slightly different
v1=(cos(a1),sin(a1))
v2=(x2-x1,y2-y1)
v2/=|v2| // makes v2 unit vector, v1 is already unit
da=acos(dot(v1,v2))
so:
da=acos((cos(a1)*(x2-x1)+sin(a1)*(y2-y1)/sqrt((x2-x1)*(x2-x1)+(y2-y1)*(y2-y1)));
[notes]
just change it to match your coordinate system (which you did not specify)
use radians or degrees according to your sin,cos,atan dependencies ...
The difference between the vectors is also a vector.
Then calculate the tangens (y part / x part) and invert it to an angle.
Of course use the sign of y if x = 0.
if the coord to face is (x2 ,y2)
deltaY = y2 - y1
deltaX = x2 - x1
You have the angle in degrees between the two points using this formula...
angleInDegrees = arctan(deltaY / deltaX) * 180 / PI
subtract the original angle of your vector and you will get the correct offset!
I am looking for a fast and effective way to determine if vector B is Between the small angle of vector A and vector C. Normally I would use the perpendicular dot product to determine which sides of each line B lies on but in this case is not so simple because of the following:
None of the vectors can be assumed to be normalized and normalizing them is an extra step I would prefer to avoid.
I have no clear notion as to which side is the smallest angle so it is hard to say which side of the line is good or not.
It is possible for A and B to be co-linear or exactly 180 degrees apart in which case I want to return false.
While I am working in a 3D enviroment it is easy for me to simplify this to 2D if that makes things easier and more importantly faster. This test will be used in an algorithm that needs to run as fast as possible.
If there is some easy and efficient method to determine which direction my perpendicular vectors should both point I could use the two dot products for my test.
Another approach I have been considering without much success so far is using a matrix. In theory from what I understand of matrix transforms I should be able to use A and C as basis vectors. Then multiplying B by the matrix I should be able to test what quadrant B then lies in by whether X and Y are both positive. If I could get this approach to work it would likely be the best since one matrix multiplication should be faster than two dot products and I should not have to worry about which side has the smallest angle on it.
The problem is from my tests I cannot simply use A and C as bases and multiply it normally and get correct behavior. I am really not sure what i am doing wrong here. I have run across the term "Vector spaces" a few times which as near as I can figure seems to be a very similar concept to matrix transforms without any requirements for orthogonal bases or orthonormal bases. Is it the same thing as matrix? If not, might there be a better approach and how would I use that?
Just to give a more visual explanation of what I am talking about:
#Aki Suihkonen
I can't seem to get it working. Coded up a mock case I could run through and see if I can't figure somthing out
For this case using
Ax 2.9579773 Ay 3.315979
Cx 2.5879822 Cy 5.1630249
For B I rotated around the four quadrants the vectors divide the space up into.
The signs I got:
- For Q1 --
- For Q2 +-
- For Q3 +-
- For Q4 --
Assuming I rotated around in the enviroment the same direction as the image I am fairly sure I did.
I think Aki's solution is close, but there are cases where it doesn't work:
From his solution:
return (ay * bx - ax * by) * (ay * cx - ax * cy) < 0;
This is equivalent to checking whether the cross product between B and A has the same sign as the cross product between C and A.
The sign of the cross product (U x V) tells you whether V lies on one side of U or the other (out of the board, into the board). In most coordinate systems, if U needs to rotate counter-clockwise (out of the board), then the sign will be positive.
So Aki's solution checks to see if B needs to rotate in one direction to get to A, while C needs to rotate in the other direction. If this is the case, B is not within A and C. This solution doesn't work when you don't know the 'order' of A and C, as follows:
To know for certain whether B is within A and C you need to check both ways. That is, the rotation direction from A to B should be the same as from A to C, and the rotation direction from C to B should be the same as from C to A.
This reduces to:
if (AxB * AxC >= 0 && CxB * CxA >= 0)
// then B is definitely inside A and C
One method to think about this is to regard all these vectors A, B, C as complex numbers.
Multiplying A, C all with B*, which is the complex conjugate of B, both the resulting vectors will be rotated in complex plane so that the reference axis (B*Conj(B)) is now the real axis (or y = 0) -- and that axis doesn't need to be calculated. In this case one only has to check if the sign of 'y' or imaginary component differ. Also in this case both resulting vectors have been scaled by the same length |B|.
`return sign(Imag(A * Conj(B))) != sign(Imag(C * Conj(B)));`
A = ax + i * ay; B = bx + i * by; C = cx + i * cy;
Conj(B) = bx - i * by;
A * B = (ax * bx - ay * by) + i * (ax * by + ay * bx);
I think this equation leads to even better performance, as only the Imaginary component of the multiplication is needed.
As a full solution, this converts to:
return (ay * bx - ax * by) * (ay * cx - ax * cy) < 0;
The middle multiplication is a short cut for:
return Sign(ay * bx - ax * by) != Sign(ay * cx - ax * cy);
Without complex numbers, the problem can be also seen as vector B being
{ Rcos beta, Rsin beta }, which can be represented as a rotation matrix.
R*[ cb -sb ] [ bx -by ], cb = cos(beta), sb = sin(beta)
[ sb cb ] = [ by bx ] cos(-beta) = cos(beta), sin(-beta) = -sin(beta)
Multiplying [ax,ay], [cx,cy] with the transpose of the scaled matrix [bx by, -by bx] affects the lengths of [ax, ay] * rotMatrix(-beta), [cx, cy] * rotMatrix(-beta) in exactly the same way.
In polar coordinates, you would just be asking if θA < θB < θC. So transform to polar first:
a_theta = ax ? atan(ay / ax) : sign(ay) * pi
Hello all math masters, I got a problem for you:
I have a 2D game (top down), and I would like to make the character escape from a shot, but not just walk away from the shot (I mean, don't be pushed by the shot), I want it to have a good dodging skills.
The variables are:
shotX - shot x position
shotY - shot y position
shotSpeedX - shot x speed
shotSpeedY - shot x speed
charX - character x position
charY - character y position
keyLeft - Set to true to make the character press the to left key
keyRight - Set to true to make the character press the to right key
keyUp - Set to true to make the character press the to up key
keyDown - Set to true to make the character press the down key
I can understand the following languages:
C/C++
Java
Actionscript 2/3
Javascript
I got this code (Actionscript 3), but sometimes it doesn't work:
var escapeToLeft:Boolean = false;
var r:Number = Math.atan2(0 - shotSpeedY, 0 - shotSpeedX)
var angle:Number = Math.atan2(charY - (shotY + shotSpeedY), charX - (shotX + shotSpeedX));
var b:Number = diepix.fixRotation(r-angle); // This function make the number between -180 and 180
if(b<0) {
escapeToLeft = true;
}
r += (escapeToLeft?1:0 - 1) * Math.PI / 2;
var cx:Number = Math.cos(r);
var cy:Number = Math.sin(r);
if(cx < 0.0) {
keyLeft = true;
}else {
keyRight = true;
}
if(cy < 0.0) {
keyUp = true;
}else {
keyDown = true;
}
Some observations:
Optimal dodging probably involves moving at a 90 degree angle from the bullets direction. That way, you get out of harms way the quickest.
If you do err, you want to err in the direction of the bullet, as that buys you time.
you can calculate 90 degrees to bullet direction with the scalar product
find the closest compass direction to the calculated optimal angle (4 possible answers)
are you allowed to go up and left at the same time? Now you have 8 possible answers to a bullet
bonus points for dodging in optimal direction according to second point
The scalar product of two vectors (ax, ay) and (bx, by) is ax * bx + ay * by. This is 0 if they are orthogonal (90 degrees). So, given the bullet (ax, ay), find a direction (bx, by) to run that has a scalar product of 0:
ax * bx must equal ay * by, so (bx, by) = (-ax, -ay)
Now to find the closest point on the compass for (bx, by), the direction you would like to run to. You can probably figure out the technique from the answer to a question of mine here on SO: How to "snap" a directional (2D) vector to a compass (N, NE, E, SE, S, SW, W, NW)? (note, thow, that I was using a wonky coordinate system there...)
If you have only 4 compass directions, your easiest path is to take:
max(abs(bx), abs(by))
The bigger vector component will show you the general direction to go - for
bx positive: right
bx negative: left
by positive: up (unless (0, 0) is top left with y positive in bottom left...)
by negative: down
I guess you should be able to come up with the rest on your own - otherwise, good luck on writing your own game!
I am not following what the line
var angle:Number = Math.atan2(charY - (shotY + shotSpeedY), charX - (shotX + shotSpeedX));
is supposed to be doing. The vector ( charY - shotY, charX - shotX ) would be the radius vector pointing from the location of the shot to the location of the character. But what do you have when you subtract a speed vector from that, as you are doing in this line?
It seems to me that what you need to do is:
Calculate the radius vector (rY, rX) where rY = shotY - charY; rX = xhotX - charX
Calculate the optimal direction of jump, if the character weren't constrained to a compass point.
Start with a vector rotated 90 degrees from the shot-character radius vector. Say vJump = ( rX, -rY ). (I think Daren has this calculation slightly wrong--you are transposing the two coordinates, and reversing one of their signs.)
The character should either wants to jump in the direction of vJump or the direction of -vJump. To know which, take the scalar product of vJump with (shotSpeedY, shotSpeedX). If this is positive, then the character is jumping towards the bullet, which you don't want, obviously, so reverse the sign of both components of vJump in this case.
Jump in the permissible direction that is closest to vJump. In the code you listed, you are constrained to jump in one of the diagonal directions--you will never jump in one of the cardinal directions. This may in fact be the mathematically optimal solution, since the diagonal jumps are probably longer than the cardinal jumps by a factor of 1.414.
If your jumps are actually equal distance, however, or if you just don't like how it looks if the character always jumps diagonally, you can test each of the eight cardinal and intermediate directions by calculating the scalar product between vJump and each of the eight direction vectors (0,1), (0.7071,0.7071), (1,0), (0.7071,-0.7071), etc. Take the direction that gives you the biggest positive scalar product. Given the patterns present, with some clever programming you can do this in fewer than eight tests.
Note that this algorithm avoids any math more complicated than addition and multiplication, so will likely have much better performance than something that requires trig functions.