UIViewControllerAnimatedTransitioning simple usage. Source : http://gist.tolpp.com/2015/09/swift-uiviewcontrolleranimatedtransitio.html

FirstViewController.swift

import UIKit

class FirstViewController: UIViewController {
    
    var secondVc : SecondViewController!
    
    override func viewDidLoad() {
        super.viewDidLoad()
    }
    
    @IBAction func toSecondTapped(sender: UIButton) {
        
        let sb = UIStoryboard(name: "Main", bundle: nil)
        secondVc = sb.instantiateViewControllerWithIdentifier("SecondVC") as! SecondViewController
        secondVc.transitioningDelegate = self
        
        self.presentViewController(secondVc, animated: true, completion: nil)
    }
}

extension FirstViewController : UIViewControllerTransitioningDelegate {
    func animationControllerForPresentedController(presented: UIViewController, presentingController presenting: UIViewController, sourceController source: UIViewController) -> UIViewControllerAnimatedTransitioning? {
        return TransitionAnimator()
    }
    
    func animationControllerForDismissedController(dismissed: UIViewController) -> UIViewControllerAnimatedTransitioning? {
        return TransitionAnimator()
    }
}

SecondViewController.swift

import UIKit

class SecondViewController: UIViewController {

    override func viewDidLoad() {
        super.viewDidLoad()
        self.modalPresentationStyle = .Custom
    }

    @IBAction func toFirstTapped(sender: UIButton) {
        self.dismissViewControllerAnimated(true, completion:  nil)
    }
}

TransitionAnimator.swift

// Little Updated version of 
// http://www.raywenderlich.com/86521/how-to-make-a-view-controller-transition-animation-like-in-the-ping-app
import UIKit

class TransitionAnimator : NSObject, UIViewControllerAnimatedTransitioning {
    
    weak var transitionContext: UIViewControllerContextTransitioning?
    
    func transitionDuration(transitionContext: UIViewControllerContextTransitioning?) -> NSTimeInterval {
        return 0.5
    }
    
    func animateTransition(transitionContext: UIViewControllerContextTransitioning) {
        
        //1
        self.transitionContext = transitionContext
        
        //2
        let containerView = transitionContext.containerView()
        let fromViewController = transitionContext.viewControllerForKey(UITransitionContextFromViewControllerKey) as UIViewController!
        let toViewController = transitionContext.viewControllerForKey(UITransitionContextToViewControllerKey) as UIViewController!
        
        //3
        containerView!.addSubview(toViewController.view)
        
        //4
        let circleMaskPathInitial = UIBezierPath(ovalInRect: CGRect())
        let extremePoint = CGPoint(x: fromViewController.view.center.x - 0, y: fromViewController.view.center.y - CGRectGetHeight(toViewController.view.bounds))
        let radius = sqrt((extremePoint.x*extremePoint.x) + (extremePoint.y*extremePoint.y))
        let circleMaskPathFinal = UIBezierPath(ovalInRect: CGRectInset(fromViewController.view.frame, -radius, -radius))
        
        //5
        let maskLayer = CAShapeLayer()
        maskLayer.path = circleMaskPathFinal.CGPath
        toViewController.view.layer.mask = maskLayer
        
        //6
        let maskLayerAnimation = CABasicAnimation(keyPath: "path")
        maskLayerAnimation.fromValue = circleMaskPathInitial.CGPath
        maskLayerAnimation.toValue = circleMaskPathFinal.CGPath
        maskLayerAnimation.duration = self.transitionDuration(transitionContext)
        maskLayerAnimation.delegate = self
        maskLayer.addAnimation(maskLayerAnimation, forKey: "path")
    }
    
    override func animationDidStop(anim: CAAnimation, finished flag: Bool) {
        self.transitionContext?.completeTransition(!self.transitionContext!.transitionWasCancelled())
        self.transitionContext?.viewControllerForKey(UITransitionContextFromViewControllerKey)?.view.layer.mask = nil
    }
}