Translate rotate translate?

When working with transforms and wanting to rotate around some other point than center or origo you hear that you need to translate (move), then rotate and then translate back. Let explain what that means.

Behind the scenes

When you apply a transform to a view the coordinates of that views corners gets multiplied by a transformation matrix to calculate their new positions. When you apply multiple transforms to a view the matrixes for the different transforms are multiplied with the corners of your view in the order that they should be applied1. The important thing to take away from this is that the order is important. Translating first and then rotating does not give the same result as rotating first and then translating. The reason for this is that matrixes are not commutative, which means that Mtrans × Mrotate ≠ Mrotate × Mtrans.

Don’t worry! You don’t need to understand matrixes to apply transforms as long as you know that order matters. The animations below illustrate how the order of translating and rotating makes a difference.

◀◀
begin
end
view
  1. translate
  2. rotate
  3. translate
A view being translated, rotated and translated back.

Creating such a transform in code

Both Core Animation and Core Graphics provide nice abstractions for working with transforms. In Core Graphics we have 2D affine transforms called CGAffineTransform and in Core Animation we have 3D transforms called CATransform3D. In the rest of the post I’m going to use the 3D transforms but it all applies to affine transforms as well.

You can create a new rotation transform by calling CATransform3DMakeRotation(…) or even rotate an existing transform using CATransform3DRotate(…). Translating and scaling has similar methods. That’s means that our three step transform is three lines of code (excluding two variables for the angle and the distance):

CGFloat thirtyDegrees = 30.0 * M_PI / 180.0;
CGFloat distanceToRotationPoint = 100.0;
CATransform3D rotation = 
    CATransform3DMakeTranslation(-distanceToRotationPoint, 0.0, 0.0);
rotation = 
    CATransform3DRotate(rotation, -thirtyDegrees, 0.0, 0.0, 1.0);
rotation = 
    CATransform3DTranslate(rotation, distanceToRotationPoint, 0.0, 0.0);

If we set that as the transform of our view we will see that it has moved to the correct position, just as if it was rotated around a pointer other than its center.

Animating a rotation

Being able to set a rotation transform is cool and all but we want to animate the rotation as well. So we create a basic animation from the identity matrix (meaning no transformation) to our translate-rotate-translate transform and add it to a layer.

CABasicAnimation *rotate = 
    [CABasicAnimation animationWithKeyPath:@"transform"];
rotate.fromValue = [NSValue valueWithCATransform3D:CATransform3DIdentity];
rotate.toValue = [NSValue valueWithCATransform3D:externalRotation];
rotate.duration = 1.0;    

[rotatingLayer addAnimation:rotate forKey:@"myRotationAnimation"];
◀◀
begin
end
view
Animating to the translate-rotate-translate transform.

Something is not right. It may not look very strange for small rotation like this one but if we slow down the animation a little and make a rotation of more than π/2 (90º) it will be obvious what is happening. The view moves from the start position to the end position in a straight line! It doesn’t rotate around that point, it’s just strange.

◀◀
begin
end
view
Animating a bigger rotation to see what is wrong.

Making it work

Lets take a step back and see how this can be solved. If you start thinking that it worked fine for small angles so let’s just do a key frame for every angle. Please just stop and don’t do that.

Let’s look at alternatives. There is a property on the view layer called the anchor point that can change which point the transform is being applied relative to2. If we can change the anchor point to the point we want to rotate around we won’t have to deal with translating back and forth, we just rotate. There are some caveats though. Changing the anchor point changes how the frame is drawn relative the position which causes the view to move on screen.

To counter the frame moving we can change the position of the layer to the point we are rotating around. That way the view/layer will appear in the same frame as before.

CGPoint rotationPoint = // The point we are rotating around

CGFloat minX   = CGRectGetMinX(view.frame);
CGFloat minY   = CGRectGetMinY(view.frame);
CGFloat width  = CGRectGetWidth(view.frame);
CGFloat height = CGRectGetHeight(view.frame);

CGPoint anchorPoint =  CGPointMake((rotationPoint.x-minX)/width,
                                   (rotationPoint.y-minY)/height);

view.layer.anchorPoint = anchorPoint;
view.layer.position = rotationPoint; 

If we are using Auto Layout it becomes a bit more complicated. The answers to this question on Stack Overflow show some of the many ways that it can be done. The rest of the code for this post will assume you are not using Auto Layout. If you are, I refer you to the those answers.

Now that have changed the point we are transforming relative to, we don’t have to translate back and forth so the animation code becomes very simple.

CABasicAnimation *rotate = 
    [CABasicAnimation animationWithKeyPath:@"transform.rotation.z"];
rotate.toValue = @(-M_PI_2); // The angle we are rotating to
rotate.duration = 1.0;

[view.layer addAnimation:rotate forKey:@"myRotationAnimation"];

And the view finally rotates around the point we were expecting it to.

◀◀
begin
end
view
Finally rotating around the point we wanted.
  1. In reality the different transformation matrixes are being multiplied in the order they should be applied to get a single transformation matrix that reflects all the transformations in total.

  2. You can read more about the anchor point in my article for iDeveloper.tv