Mike Swanson is a developer who writes about Apple design details, immersive video, and his own apps and projects.
Commenters call his analysis of the iOS 7 icon shape the best available and cite his posts on Apple's fisheye video projection.
On iOS 11, CALayer has a `continuousCorners` private property [0].
I did a spartan reverse engineering by recreating the UIBezierPath equivalent of the two types of radii found in the Apple Design Resources [1], then adding a convenience initializer on UIBezierPath that recreates the shape with arbitrary radius of the chosen kind (either UISwitch-like, or App Icon-like).
Earlier attempts include the fascinating Unleashing Genetic Algorithms on the iOS 7 Icon [2].
[0] https://mobile.twitter.com/argentumko/status/955773459463790592
[1] https://developer.apple.com/design/resources/
[2] http://blog.mikeswanson.com/post/62341902567/unleashing-genetic-algorithms-on-the-ios-7-icon
Very cool.
For those who want to follow along at home: http://blog.mikeswanson.com/iosroundedrect
I modified this myself to produce better background colors for testing different icon designs with an angular gradient and some layer cropping. This script and a script to save icons at different sizes really helped when making 30 iOS apps for iOS 7 recently. (I don't have time now to share my changes but if anyone's interested, I'll check this comment later for replies.)
Apple used straight segments and circular arcs on its icons upto iOS 6, and switched to 2nd-order continuous shapes on version 7. Here is a side-by-side comparison :
https://blog.mikeswanson.com/photoshop-script-for-ios-7-rounded-rectangles/
Your visual system can actually see those discontinuities.
This is probably the best analysis that I've seen of the iOS 7 icon shape: http://blog.mikeswanson.com/post/62341902567/unleashing-genetic-algorithms-on-the-ios-7-icon
(I am slightly biased, though.)
This outlines how Apple approaches the immersive video format to put the most pixels on the horizon line:
https://blog.mikeswanson.com/apples-mysterious-fisheye-projection/
Not too many steps away from this: https://blog.mikeswanson.com/spatial/