I'm having a bit of trouble with Trilateration using iBeacon's. I found a formula using vector math in this thread, and I implemented it in my project. However it seems to be giving me odd numbers.
It seems like it works most of the time, however I've noticed that sometimes it will give me extremely similar answers when using distances that are vastly different, which seems odd to me.
CGPoint pointA = [[curBlufiPoints objectAtIndex:0] CGPointValue];
CGPoint pointB = [[curBlufiPoints objectAtIndex:1] CGPointValue];
CGPoint pointC = [[curBlufiPoints objectAtIndex:2] CGPointValue];
/*Point A = (40, 612)
Point B = (379, 788)
Point C = (0, 352)*/
float distanceA = [[distances objectAtIndex:0] floatValue];
float distanceB = [[distances objectAtIndex:1] floatValue];
float distanceC = [[distances objectAtIndex:2] floatValue];
/*distanceA = 13.535637
distanceB = 46.931522
distanceC = 51.585461
----OR----
distanceA = 349.9057;
distanceB = 352.84134;
distanceC = 353.37515;*/
CGPoint P2MinusP1 = CGPointMake(pointB.x - pointA.x, pointB.y - pointA.y);
CGPoint P3MinusP1 = CGPointMake(pointC.x - pointA.x, pointC.y - pointA.y);
CGFloat magP2MinusP1 = sqrt(pow((P2MinusP1.x), 2) + pow((P2MinusP1.y), 2));
CGPoint eX = CGPointMake(P2MinusP1.x / magP2MinusP1, P2MinusP1.y / magP2MinusP1);
CGFloat i = eX.x * P3MinusP1.x + eX.y * P3MinusP1.y;
CGPoint eXTimesI = CGPointMake(eX.x * i, eX.y * i);
CGPoint P3MinusP1MinusITimesEX = CGPointMake(P3MinusP1.x - eXTimesI.x, P3MinusP1.y - eXTimesI.y);
CGFloat magP3MinusP1MinusITimesEX = sqrt(pow(P3MinusP1MinusITimesEX.x, 2) + pow(P3MinusP1MinusITimesEX.y, 2));
CGPoint eY = CGPointMake(P3MinusP1MinusITimesEX.x / magP3MinusP1MinusITimesEX, P3MinusP1MinusITimesEX.y / magP3MinusP1MinusITimesEX);
CGFloat j = eY.x * P3MinusP1.x + eY.y * P3MinusP1.y;
CGFloat x = (pow(distanceA, 2) - pow(distanceB, 2) + pow(magP2MinusP1, 2)) / (2 * magP2MinusP1);
CGFloat y = (pow(distanceA, 2) - pow(distanceC, 2) + pow(i, 2) + pow(j, 2)) / (2 * j) - (i * x) / j;
CGPoint finalPoint = CGPointMake(pointA.x + x * eX.x + y * eY.x, pointA.y + x * eX.y + y * eY.y);
NSLog(#"%f %f %f %f", finalPoint.x, finalPoint.y);
//finalPoint.x = ~343
//finalPoint.y = ~437
As you can see from the values I've commented in the code above, when I am using the same points, but different distances I end up with the same result. It doesn't make sense to me how they can both have around the same answer. If the point (343, 437), (the answer I get for both sets of input) is 13.5 units away from point (40, 612), how can the same point also be 349.9 units away?
I'm not sure where my math has gone wrong but I think that something about calculating x and y is where the problems comes in. I've discovered that the lines "pow(distanceA, 2) - pow(distanceB, 2)" and "pow(distanceA, 2) - pow(distanceC, 2)" give me approximately the same answer for both sets of numbers, which is why x and y end up being the same regardless of which set of numbers I use.
I don't think that this should be the case, any help would be greatly appreciated, thank you!
I tried your code with a sample (and bad looking code test).
The testing code is here.
To debug, I used the "3D" new feature of XCode 6, to see the points and circle that were out of the screen bounds, because I didn't want to do translation (recentering the plane, etc.)
The main issue (with your first sample test (#[#(13.535637), #(46.931522), #(51.585461)];) is that the circles aren't overlapping. There is NO intersection. So you can't guess the position.
If you draw a circle with center pointA, and radius distanceA, and do it for point B and C, you'll need to find the intersection to geolocalize, which is clearly illustrated on the wiki-related article on the linked question.
I added a "check" method to see if they overlap correctly, with a "allowed mistake", because we're doing some maths with float and there could be a few rounding issue.
The check method consist on recalculating the distance between the finalPoint (coordinate found) and each point. The question is now if they differs (a lot?) from the original ones.
Screenshot with first set of distances
Screenshot with second set of distances
Related
I created a "ship" node to move along the circular path as follows:
self.orbit = [OrbitManager getCirclePathWithCenter:centerRealPt radius:radius startingAngle:angelFromCenter isClockwise:self.isClockwise];
SKAction* orbitAction = [SKAction followPath:self.orbit asOffset:NO orientToPath:YES speed:300];
[self.ship.node runAction:orbitAction];
and I have a cannon which shoots a bullet by applying a velocity to it as follows:
bullet.node.physicsBody.velocity = [ActionHelper getVelocityFrom:bullet.node toNodeB:self.target speed:bullet.speed];
as the ship is moving along the path. But the bullet will miss every time. How can I calculate the position which the cannon should aim at, with a given speed?
This is my Objective-C (it is actually a C function) Solution to fire a projectile in to a moving target.
You can look at the derivations In this SO topic
This will give you a hit point and an angle to shoot,
you can simply translate it to velocity because you know the angle and a projectile speed, it will be something like:
`CGVector Velocity = CGVectorMake(speed * cos(theta), speed * sin(theta));`
BOOL calculateAngleToShoot(CGVector targetVelocity, CGPoint targetPosition, CGPoint selfPos,CGFloat projectileSpeed,CGFloat *theta, CGPoint * hitPoint)
{
CGFloat dx = targetPosition.x - selfPos.x;
CGFloat dy = targetPosition.y - selfPos.y;
CGVector v = targetVelocity;
CGFloat a = v.dx * v.dx + v.dy * v.dy - projectileSpeed * projectileSpeed;
CGFloat b = 2 * (v.dx * dx + v.dy * dy);
CGFloat c = v.dx * v.dx + v.dy * v.dy;
CGFloat q = b * b - 4 * a * c;
if (q < 0)
{
//Dead End;
return NO;
}
CGFloat t = ((a < 0 ? -1 : 1) * sqrt(q) - b) / (2 * a);
// Aim for where the target will be after time t
dx += t * v.dx;
dy += t * v.dy;
*theta = atan2(dy, dx);
*hitPoint = CGPointMake(targetPosition.x + v.dx * t, targetPosition.y + v.dy * t);
return YES;
}
After some investigation I got how to get the answer
first I need to get the distance(d) between the target and the center
and the time for the bullet from center to the target.
since the ship is moving along the circle, so the radius is also equals to distance(d)
CGFloat timeToArriveTarget = bullet.speed/distance;
CGFloat angularSpeed = bullet.speed/distance;
Find the angle moved within this period of time
CGFloat angle = angularSpeed * timeToArriveTarget;
CGFloat x = self.target.position.x;
CGFloat y = self.target.position.y;
CGFloat a = bullet.node.position.x;
CGFloat b = bullet.node.position.y;
and finally using this formula:
details are give by this link https://math.stackexchange.com/a/266837
CGPoint targetPt = CGPointMake((x - a) * cos(angle) - (y - b) * sin(angle) + a, (x - a) * sin(angle) + (y - b) * cos(angle) + b);
bullet.node.physicsBody.velocity = [ActionHelper getVelocityFrom:bullet.node.position toPtB:targetPt speed:bullet.speed];
the getVelocity function is given by
+(CGVector)getVelocityFrom:(CGPoint)ptA toPtB:(CGPoint)ptB speed:(CGFloat)speed{
CGPoint targetPosition = ptB;
CGPoint currentPosition = ptA;
double angle = [MathHelper getRotateAngleFrom:currentPosition toTargetPosition:targetPosition];
double velocityX = speed * cos(angle);
double velocityY = speed * sin(angle);
CGVector newVelocty = CGVectorMake(velocityX, velocityY);
return newVelocty;
}
Like many that post about this topic, I too am busy trying to write myself an accurate simulator for the movement of objects in a 2D gravitation field.
I decided early on that I would settle on Velocity Verlet Integration, as I want my objects to maintain stable orbits and conserve energy even if the timestep is rather large. So, what might the problem be?
Well, so far, everything seems to behave correctly, except for one component. When I try to calculate the correct velocity for a stable orbit at a certain distance, the resulting velocity sends them into odd elliptical orbits that quickly increase in magnitude each time.
So, to begin, here are the following methods that determine an objects next position, velocity, and acceleration in scene: (Objective C)
Acceleration:
-(CGVector)determineAccelerationFor:(SKObject *)object
{ // Ok, let's find Acceleration!
CGVector forceVector = (CGVector){0,0}; // Blank vector that we will add forces to
for (SKObject *i in self.sceneObjects)
{
if (![i isEqual:object]) // Just make sure we're not counting ourselves here
{
CGPoint distance = [self getDistanceBetween:i.position And:object.position];
float hypotenuse = sqrtf(powf(distance.x, 2)+ powf(distance.y, 2));
float force = ((self.gravity * object.mass * i.mass)/powf(hypotenuse, 3));
float xMagnitude = (force * distance.x);
float yMagnitude = (force * distance.y);
forceVector.dx += xMagnitude;
forceVector.dy += yMagnitude;
}
}
CGVector acceleration = (CGVector){forceVector.dx/object.mass, forceVector.dy/object.mass};
return acceleration;
}
Cool, so basically, I just take an object, add all the other forces that each other object imposes on it together then divide the X & Y factor by the mass of the current object to get the acceleration!
Next up is Velocity. Here I use the following equation:
The method for it is pretty straightforward too:
-(CGVector)determineVelocityWithCurrentVelocity:(CGVector)v OldAcceleration:(CGVector)ao NewAcceleration:(CGVector)a
{
float xVelocity = (v.dx + ((ao.dx + a.dx)/2) * self.timeStep);
float yVelocity = (v.dy + ((ao.dy + a.dy)/2) * self.timeStep);
CGVector velocity = (CGVector){xVelocity,yVelocity};
return velocity;
}
And finally, position! The equation for this is:
And it is determined with the following method!
-(CGPoint)determinePositionWithCurrentPosition:(CGPoint)x CurrentVelocity:(CGVector)v OldAcceleration:(CGVector)ao
{
float xPosition = (x.x + v.dx * self.timeStep + ((ao.dx * powf(self.timeStep, 2))/2));
float yPosition = (x.y + v.dy * self.timeStep + ((ao.dy * powf(self.timeStep, 2))/2));
CGPoint position = (CGPoint){xPosition,yPosition};
return position;
}
This is all called from the below method!!
-(void)refreshPhysics:(SKObject *)object
{
CGPoint position = [self determinePositionWithCurrentPosition:object.position CurrentVelocity:object.velocity OldAcceleration:object.acceleration]; // Determine new Position
SKAction *moveTo = [SKAction moveTo:position duration:0.0];
[object runAction:moveTo]; // Move to new position
CGVector acceleration = [self determineAccelerationFor:object]; // Determine acceleration off new position
CGVector velocity = [self determineVelocityWithCurrentVelocity:object.velocity OldAcceleration:object.acceleration NewAcceleration:acceleration];
NSLog(#"%# Old Velocity: %f, %f",object.name,object.velocity.dx,object.velocity.dy);
NSLog(#"%# New Velocity: %f, %f\n\n",object.name,velocity.dx,velocity.dy);
[object setAcceleration:acceleration];
[object setVelocity:velocity];
}
Okay, so those methods above dictate how objects are moved in scene. Now onto the initial issue, the ever present problem of achieving a stable orbit!
In order to determine what velocity an object should have to maintain an orbit, I use the following equation:
And I implement that as follows:
-(void)setObject:(SKObject *)object ToOrbit:(SKObject *)parent
{
float defaultSeparation = 200;
// Move Object to Position at right of parent
CGPoint defaultOrbitPosition = (CGPoint){parent.position.x + (parent.size.width/2)+ defaultSeparation,parent.position.y};
[object setPosition:defaultOrbitPosition];
// Determine Orbital Velocity
float velocity = sqrtf((self.gravity * parent.mass)/(parent.size.width/2+defaultSeparation));
CGVector vector = (CGVector){0,velocity};
[object setVelocity:vector];
}
And for some reason, despite this, I get abysmal results. Here is some of the output:
Information:
Gravity(constant) = 1000 (For test purposes)
Mass(Parent) = 5000 units
Mass(Satellite) = 1 units
Separation = 224 pixels
It determines that in order for the Satellite to Orbit the Parent, a velocity of:
149.403580 pixels/timeStep
is required. And that checks out on my calculator.
So this has left me a little confused as to what could be going wrong. I log all the output concerning new velocities and positions, and it does use the velocity I set it to, but that just doesn't seem to make a difference. If anyone could possible help spot what's going wrong here I would be immensely grateful.
If anyone believes I have left something out, tell me and I will edit this right away. Thanks!
So I'm trying to generate a NSBezierPath that looks like an arrow between two points, which can lie anywhere on the view, so the startPoint can be larger or smaller than the endpoint.
The arrow is updated while a user drags the mouse like in a drawing app.
I already figured out, that I probably have to use transformations and some math to do trigonometric and have come up with this implementation:
+(NSBezierPath *)arrowWithStart:(NSPoint)startPoint andEnd:(NSPoint)endPoint{
NSBezierPath* path = [NSBezierPath bezierPath];
CGFloat width = endPoint.x - startPoint.x;
CGFloat height = endPoint.y - startPoint.y;
CGFloat angle = atan2(width, height);
NSAffineTransform *tr = [NSAffineTransform transform];
[tr translateXBy:startPoint.x yBy:startPoint.y];
[tr scaleXBy:width yBy:height];
[tr rotateByDegrees:angle];
[path moveToPoint:CGPointZero];
[path lineToPoint:CGPointMake(0.75, 0.7)];
[path lineToPoint:CGPointMake(0.8, 0.65)];
[path lineToPoint:CGPointMake(1, 1)];
[path lineToPoint:CGPointMake(0.65, 0.8)];
[path lineToPoint:CGPointMake(0.7, 0.75)];
[path closePath];
[path transformUsingAffineTransform:tr];
return path;
}
This Code generates pretty nice arrows, when the points are some kind of diagonal,
like
(0,0)
(-2,-2)
to each other, but when the points are getting nearer to a horizontal or vertical line, like
(2,3)
(5,3)
the result becomes a straight line without an arrowhead.
So I think I'm doing something wrong in the transformation Matrix.
If somebody knows where I'm making a mistake it would be great.
What you need is an affine transformation that transforms the point (0, 0) to startPoint and (1, 1) to endPoint. This transformation can be computed directly:
CGFloat tx = startPoint.x;
CGFloat ty = startPoint.y;
CGFloat a = ((endPoint.x - startPoint.x) + (endPoint.y - startPoint.y))/2.;
CGFloat b = (-(endPoint.x - startPoint.x) + (endPoint.y - startPoint.y))/2.;
NSAffineTransformStruct transformStruct = { a, b, -b, a, tx, ty };
NSAffineTransform *tr = [NSAffineTransform transform];
[tr setTransformStruct:transformStruct];
Explanation:
NSAffineTransformStruct transformStruct = { a, b, -b, a, tx, ty };
describes a general combination of translation, scaling and rotation, i.e. an affine transformation without shearing. The requirements (0, 0) -> startPoint, (1, 1) -> endPoint give the equations
startPoint.x = 0 * a + 0 * (-b) + tx
startPoint.y = 0 * b + 0 * a + ty
endPoint.x = 1 * a + 1 * (-b) + tx
endPoint.y = 1 * b + 1 * a + tx
and solving these equations for a, b, tx, ty gives above solution. (See Transform Mathematics in the "Cocoa Drawing Guide" for more information.)
The problem with your original code is that
atan2 takes y as first argument, so atan2(height, width) would compute the angle.
For horizontal or vertical lines, width or height and therefore one scaling factor is zero, this causes the straight lines without arrowhead.
I asked a quick question Bezier curve algorithm in objective-c that seemed to solve my issues. I'm asking this new question as I think its different enough rather than repurposing the old one.
I have what seems like a working Bezier Curve algorithm but when compared to built in NSBezierPath version there are some major issues. It looks as though certain types of curves are very much distorted.
You can see from the above image the differences, the red line is my function and the lighter color is the built in version. I am not expecting and exat match, pixel for pixel, but as you can see the red lines go way off course at times.
The first method I'm listing is what calls the 2 Bezier methods, it shows that the inputs are the same to both versions.
- (void)MakeBezier
{
int x1 = [self getMegaNumber:2];
int y1 = self.frame.size.height - [self getMegaNumber:2];
int x2 = [self getMegaNumber:2];
int y2 = self.frame.size.height - [self getMegaNumber:2];
int x3 = [self getMegaNumber:2];
int y3 = self.frame.size.height - [self getMegaNumber:2];
int x4 = [self getMegaNumber:2];
int y4 = self.frame.size.height - [self getMegaNumber:2];
int cnt = [self getMegaNumber:2];
NSBezierPath *bezierPath = [[NSBezierPath alloc] init];
[bezierPath setLineWidth:1.0f];
[bezierPath moveToPoint:NSMakePoint(x1, y1)];
[bezierPath curveToPoint:NSMakePoint(x4, y4) controlPoint1:NSMakePoint(x2, y2) controlPoint2:NSMakePoint(x3, y3)];
// Draw path to image with build in NSBezierPath
[self drawPath:bezierPath fill:NO];
// Draw path with custom algorithm
[self drawBezierFrom:NSMakePoint(x1, y1) to:NSMakePoint(x4, y4) controlA:NSMakePoint(x2, y2) controlB:NSMakePoint(x3, y3) sections:cnt color:4];
}
This next method is the custom algorithm thats used to draw the red lines in the sample image.
- (void)drawBezierFrom:(NSPoint)from to:(NSPoint)to controlA:(NSPoint)a controlB:(NSPoint)b sections:(NSUInteger)cnt color:(NSUInteger)color
{
float qx, qy;
float q1, q2, q3, q4;
int lastx = - 1, lasty;
int plotx, ploty;
float t = 0.0;
while (t <= 1)
{
q1 = t*t*t*-1 + t*t*3 + t*-3 + 1;
q2 = t*t*t*3 + t*t*-6 + t*3;
q3 = t*t*t*-3 + t*t*3;
q4 = t*t*t;
qx = q1*from.x + q2*a.x + q3*to.x + q4*b.x;
qy = q1*from.y + q2*a.y + q3*to.y + q4*b.y;
plotx = round(qx);
ploty = round(qy);
if (lastx != -1)
[self drawLineFrom:NSMakePoint(lastx, lasty) to:NSMakePoint(plotx, ploty) color:color];
else
[self drawLineFrom:NSMakePoint(from.x, from.y) to:NSMakePoint(plotx, ploty) color:color];
lastx = plotx;
lasty = ploty;
t = t + (1.0/(cnt + 0.0f));
}
[self drawLineFrom:NSMakePoint(lastx, lasty) to:NSMakePoint(to.x, to.y) color:color];
}
So my question is; is the custom algorithm way off or is it just missing an edge case for specific kinds of lines, or something else? Either way any help in fixing the algorithm would be very appreciated. To reiterate, I am not looking for a pixel perfect match, but I am expecting the curves to lineup together.
Looking at the Wikipedia page here it seems that your coefficients for the q1-q4 are incorrect. Shouldn't they be symmetric?
It also seems that to.x and b.x should be swapped:
qx = q1*from.x + q2*a.x + q3*to.x + q4*b.x;
qy = ...
Given:
point clickWheelCenter
point startPoint (first touch)
point point (actual touch)
I want to determine the angle between these three point, and I want to know, in which quadrant the last touch was found.
The calculation of the angle is working well, while the quadrant is always 1 or 2, never 3 or 4.
Where am I wrong?
CGFloat DistanceBetweenTwoPoints(CGPoint point1,CGPoint point2)
{
CGFloat dx = point2.x - point1.x;
CGFloat dy = point2.y - point1.y;
return sqrt(dx*dx + dy*dy );
};
NSInteger GetQuadrant(double angle)
{
double sinAngle = sin(angle);
double cosAngle = cos(angle);
double tanAngle = tan(angle);
double cotAngle = 1.0/tanAngle;
NSLog(#"%f %f %f %f", sinAngle, cosAngle, tanAngle, cosAngle);
if(sinAngle > 0 && cosAngle > 0 && tanAngle > 0 && cotAngle > 0) return 1;
if(sinAngle > 0 && cosAngle < 0 && tanAngle < 0 && cotAngle < 0) return 2;
if(sinAngle < 0 && cosAngle < 0 && tanAngle > 0 && cotAngle > 0) return 3;
if(sinAngle < 0 && cosAngle > 0 && tanAngle < 0 && cotAngle < 0) return 4;
return 0;
}
double AngleBetweenThreePoints(CGPoint point1,CGPoint point2, CGPoint point3)
{
CGPoint point_a = point1;
CGPoint point_b = point2;
CGPoint point_c = point3;
CGFloat a, b, c;
a = DistanceBetweenTwoPoints(point_b, point_c);
b = DistanceBetweenTwoPoints(point_a, point_c);
c = DistanceBetweenTwoPoints(point_a, point_b);
double result = acos((b*b+c*c-a*a)/(2*b*c));
NSLog(#"%d", GetQuadrant(result));
return result/M_PI * 180.0;
}
-(void)touchesBegan:(NSSet *)touches withEvent:(UIEvent *)event
{
startPoint = [[touches anyObject] locationInView:self];
}
-(void)touchesMoved:(NSSet *)touches withEvent:(UIEvent *)event
{
CGPoint point = [[touches anyObject] locationInView:self];
double angle = AngleBetweenThreePoints(self.clickWheelCenter, startPoint, point);
NSLog(#"%f", angle);
}
This can all be done much easier.
With {x1,y1} = first point - center point, and {x2,y2} = third point - center point, the signed angle (-pi,+pi) is given by:
atan2(x2*y1 - x1*y2,x1*x2 + y1*y2)
Depending on your definition of the sign of the angle you may or may not want to add a minus in front.
You can easily derive the quadrant from sign and absolute size.
size <= pi/2, sign +
size > pi/2, sign +
size > pi/2, sign -
size <= pi/2, sign -
As others have mentioned, what you are doing gets the inner angle which is always less than 180 degrees. I am assuming you want the quadrant as measured by the counter clockwise angle from one vector to the other.
An easy way to get this quadrant is to do dot product and perp dot product and check the signs of each result.
This works because dot product equals |A|*|B|*cos(theta) where || means magnitude and theta is the angle between A and B. Since magnitude is always positive, by checking the sign of the result you are actually checking sign of cos(theta).
Perp dot product is actually |A|*|B|*sin(theta) so by checking the sign of perp dot you are actually checking the sign of sin(theta).
Lets say we are measuring theta as the counter clockwise angle from A to B. The quadrant is also measured by theta. We can compute the dot and perpdot by the following
First you will need to get the vectors A and B. A and B will be the vectors from clickWheelCenter to startpoint and from clickWheelCenter to what you called 'point'
dot = A.x*B.x + A.y*B.y
perpdot = A.x*B.y - A.y*B.x
then check the signs, noting the below table.
quadrant dot perpdot
1 positive positive
2 negative positive
3 positive negative
4 negative negative
Again, by checking the sign of dot product you are actually checking the sign of cos(theta) and by checking perpdot you are checking the sign of sin(theta). All you need is a set of if statements to compare if these are > 0 or < 0.
(note: I wrote the above quickly and haven't checked everything well for accuracy, but the general idea is correct)
BBitmaster
Because law of cosine is only going to return an interior angle of 180 degrees or less. I'm not quite sure what you are trying to do here, but this approach doesn't make sense in terms of quadrents. Also, you need to error check if this is production level code. Law of Cosines blows up when your triangle in degenerate.
It seems like your function AngleBetweenThreePoints is only ever returning an angle between 0 and Pi which is then being passed on.
Your problem is that acos is not a one to one function. For any given input there are multiple values so without any further knowledge it will just choose one in a given range. An easy example is that 270degrees and 90degrees both have cos equal to 0. So if you acos(0) then it can clearly only return one of the two.
I'm a little unclear on what you mean by which quadrant it is in though. Usually you can just tell by looking at the sign of the x and y co-ordinates and use that to work out what your quadrant is. Looking at the angle between two lines will give a strange value for quadrant (quadrant relative to the first line) - is this what you want?
You are calculating the inner angle which is always less than 180 degrees.
Here is a quick and correct way of calculating the right angle value:
double AngleBetweenThreePoints(CGPoint pointA, CGPoint pointB, CGPoint pointC)
{
CGFloat a = pointB.x - pointA.x;
CGFloat b = pointB.y - pointA.y;
CGFloat c = pointB.x - pointC.x;
CGFloat d = pointB.y - pointC.y;
CGFloat atanA = atan2(a, b);
CGFloat atanB = atan2(c, d);
return atanB - atanA;
}