Example of how to use presentation, animation & interaction controllers w/ custom segues to create a slide-in modal menu which partially covers presenting view.

hamburger.swift

import UIKit

enum Direction {
    case Left, Right, Up, Down
    
    var pointVector: CGPoint {
        switch self {
        case Left: return CGPoint(x: -1, y: 0)
        case Right: return CGPoint(x: 1, y: 0)
        case Up: return CGPoint(x: 0, y: -1)
        case Down: return CGPoint(x: 0, y: 1)
        }
    }
}

class CoverPartiallyPresentationController: UIPresentationController {
    
    var dismissInteractionController: PanGestureInteractionController? = nil
    var interactiveDismissal: Bool = false
    let coverDirection: Direction
    
    private let margin: CGFloat = 64.0
    
    lazy private var backgroundView: UIView = {
        let view = UIVisualEffectView(effect: UIBlurEffect(style: .Light))
        view.frame = self.containerView?.bounds ?? CGRectZero
        view.backgroundColor = nil
        
        let tapGesture = UITapGestureRecognizer(target: self, action: "backgroundViewTapped")
        view.addGestureRecognizer(tapGesture)
        
        return view
    }()
    
    init(presentedViewController: UIViewController, presentingViewController: UIViewController, coverDirection: Direction) {
        self.coverDirection = coverDirection
        super.init(presentedViewController: presentedViewController, presentingViewController: presentingViewController)
    }
    
    override func presentationTransitionWillBegin() {
        containerView?.addSubview(backgroundView)
        backgroundView.alpha = 0
        presentingViewController.transitionCoordinator()?.animateAlongsideTransition({ [weak self] _ in
            self?.backgroundView.alpha = 1
            }, completion: nil)
    }
    
    override func dismissalTransitionWillBegin() {
        presentingViewController.transitionCoordinator()?.animateAlongsideTransition({ [weak self] _ in
            self?.backgroundView.alpha = 0
            }, completion: nil)
    }
    
    override func presentationTransitionDidEnd(completed: Bool) {
        if !completed {
            backgroundView.removeFromSuperview()
        }
        dismissInteractionController = PanGestureInteractionController(view: containerView!, direction: coverDirection)
        dismissInteractionController?.callbacks.didBeginPanning = { [weak self] in
            self?.interactiveDismissal = true
            self?.presentingViewController.dismissViewControllerAnimated(true, completion: nil)
        }
    }
    
    override func dismissalTransitionDidEnd(completed: Bool) {
        interactiveDismissal = false
        if completed {
            backgroundView.removeFromSuperview()
        }
    }
    
    override func frameOfPresentedViewInContainerView() -> CGRect {
        guard let containerView = containerView else {
            return CGRectZero
        }
        
        switch coverDirection {
        case .Left:
            return CGRect(x: 0, y: 0, width: containerView.bounds.width-margin, height: containerView.bounds.height)
        case .Right:
            return CGRect(x: margin, y: 0, width: containerView.bounds.width-margin, height: containerView.bounds.height)
        case .Up:
            return CGRect(x: 0, y: 0, width: containerView.bounds.width, height: containerView.bounds.height-margin)
        case .Down:
            return CGRect(x: 0, y: margin, width: containerView.bounds.width, height: containerView.bounds.height-margin)
        }
    }
    
    func backgroundViewTapped() {
        presentingViewController.dismissViewControllerAnimated(true, completion: nil)
    }
}

class CoverPartiallySegue: UIStoryboardSegue, UIViewControllerTransitioningDelegate {
    
    var direction: Direction = .Left
    var presentationController: CoverPartiallyPresentationController! = nil
    
    override func perform() {
        destinationViewController.modalPresentationStyle = .Custom
        destinationViewController.transitioningDelegate = self
        super.perform()
    }
    
    func presentationControllerForPresentedViewController(presented: UIViewController, presentingViewController presenting: UIViewController, sourceViewController source: UIViewController) -> UIPresentationController? {
        presentationController = CoverPartiallyPresentationController(presentedViewController: presented, presentingViewController: presenting, coverDirection: direction)
        return presentationController
    }
    
    func interactionControllerForDismissal(animator: UIViewControllerAnimatedTransitioning) -> UIViewControllerInteractiveTransitioning? {
        return presentationController.interactiveDismissal ? presentationController.dismissInteractionController : nil
    }
    
    func animationControllerForPresentedController(presented: UIViewController, presentingController presenting: UIViewController, sourceController source: UIViewController) -> UIViewControllerAnimatedTransitioning? {
        return SlideInTransition(fromDirection: direction)
    }
    
    func animationControllerForDismissedController(dismissed: UIViewController) -> UIViewControllerAnimatedTransitioning? {
        return SlideInTransition(fromDirection: direction, reverse: true, interactive: presentationController.interactiveDismissal)
    }
}

class PanGestureInteractionController: UIPercentDrivenInteractiveTransition {
    struct Callbacks {
        var didBeginPanning: (() -> Void)? = nil
    }
    var callbacks = Callbacks()
    
    let gestureRecognizer: UIPanGestureRecognizer
    
    private let direction: Direction
    
    // MARK: Initialization
    
    init(view: UIView, direction: Direction) {
        self.direction = direction
        gestureRecognizer = UIPanGestureRecognizer()
        view.addGestureRecognizer(gestureRecognizer)
        
        super.init()
        gestureRecognizer.delegate = self
        gestureRecognizer.addTarget(self, action: "viewPanned:")
    }
    
    // MARK: User interaction

    func viewPanned(sender: UIPanGestureRecognizer) {
        switch sender.state {
        case .Began:
            callbacks.didBeginPanning?()
        case .Changed:
            updateInteractiveTransition(percentCompleteForTranslation(sender.translationInView(sender.view)))
        case .Ended:
            if sender.shouldRecognizeForDirection(direction) && percentComplete > 0.25 {
                finishInteractiveTransition()
            } else {
                cancelInteractiveTransition()
            }
        case .Cancelled:
            cancelInteractiveTransition()
        default:
            return
        }
    }
    
    private func percentCompleteForTranslation(translation: CGPoint) -> CGFloat {
        let panDistance = direction.panDistanceForView(gestureRecognizer.view!)
        return (translation*panDistance)/(panDistance.magnitude*panDistance.magnitude)
    }
}

extension PanGestureInteractionController: UIGestureRecognizerDelegate {
    func gestureRecognizerShouldBegin(gestureRecognizer: UIGestureRecognizer) -> Bool {
        guard let panGestureRecognizer = gestureRecognizer as? UIPanGestureRecognizer else {
            return false
        }
        return panGestureRecognizer.shouldRecognizeForDirection(direction)
    }
}

private extension Direction {
    func panDistanceForView(view: UIView) -> CGPoint {
        switch self {
        case .Left: return CGPoint(x: -view.bounds.size.width, y: 0)
        case .Right: return CGPoint(x: view.bounds.size.width, y: 0)
        case .Up: return CGPoint(x: 0, y: -view.bounds.size.height)
        case .Down: return CGPoint(x: 0, y: view.bounds.size.height)
        }
    }
}

class SlideInTransition: NSObject, UIViewControllerAnimatedTransitioning {
    
    let duration: NSTimeInterval = 0.3
    let reverse: Bool
    let interactive: Bool
    let fromDirection: Direction
    
    init(fromDirection: Direction, reverse: Bool = false, interactive: Bool = false) {
        self.reverse = reverse
        self.interactive = interactive
        self.fromDirection = fromDirection
    }
    
    func animateTransition(transitionContext: UIViewControllerContextTransitioning) {
        
        let viewControllerKey = reverse ? UITransitionContextFromViewControllerKey : UITransitionContextToViewControllerKey
        let viewControllerToAnimate = transitionContext.viewControllerForKey(viewControllerKey)!
        let viewToAnimate = viewControllerToAnimate.view
        
        let offsetFrame = fromDirection.offsetFrameForView(viewToAnimate, containerView: transitionContext.containerView()!)
        
        if !reverse {
            transitionContext.containerView()?.addSubview(viewToAnimate)
            viewToAnimate.frame = offsetFrame
        }
        
        let options: UIViewAnimationOptions = interactive ? [.CurveLinear] : []
        
        UIView.animateWithDuration(duration, delay: 0, options: options,
            animations: { [weak self] in
                if self!.reverse {
                    viewToAnimate.frame = offsetFrame
                } else {
                    viewToAnimate.frame = transitionContext.finalFrameForViewController(viewControllerToAnimate)
                }
            }, completion: { _ in
                transitionContext.completeTransition(!transitionContext.transitionWasCancelled())
        })
    }
    
    func transitionDuration(transitionContext: UIViewControllerContextTransitioning?) -> NSTimeInterval {
        return duration
    }
}

private extension Direction {
    func offsetFrameForView(view: UIView, containerView: UIView) -> CGRect {
        var frame = view.bounds
        switch self {
        case .Left:
            frame.origin.x = -frame.width
            frame.origin.y = 0
        case .Right:
            frame.origin.x = containerView.bounds.width
            frame.origin.y = 0
        case .Up:
            frame.origin.x = 0
            frame.origin.y = -frame.height
        case .Down:
            frame.origin.x = 0
            frame.origin.y = containerView.bounds.height
        }
        return frame
    }
}

extension UIPanGestureRecognizer {
    func shouldRecognizeForDirection(direction: Direction) -> Bool {
        guard let view = view else {
            return false
        }
        
        let velocity = velocityInView(view)
        let a = angle(velocity, direction.pointVector)
        return abs(a) < CGFloat(M_PI_4) // Angle should be within 45 degrees
    }
}

Thanks for this. Quick question: Is angle defined in a library? I'm not aware of any vector math functions being defined in Foundation or UIKit...

Improvised solution:

func angle(_ a: CGPoint, _ b: CGPoint) -> CGFloat {
    // TODO | - Not sure if this is correct
    return atan2(a.y, a.x) - atan2(b.y, b.x)
}

@MrAlek, thanks for your sample code. I cannot create a PR, but I have updated your code to work with Swift 4 here: https://gist.github.com/chrisco314/3b58040015ed857498c761a3ea524161