Unless you were a Math Geek or an Ancient Greek, Geometry was probably not your favorite subject in high school. No, chances are that you were that kid in class who dutifully programmed all of the necessary formulæ into your TI8X calculator.
So for those of you who spent more time learning TIBASIC than Euclidean geometry, here’s the cheatsheet for how geometry works in Quartz 2D, the drawing system used in iOS and OS X:

A
CGPoint
is a struct that represents a point in a twodimensional coordinate system. For iOS, the origin is at the topleft, so points move right and down as theirx
andy
values, respectively, increase. OS X, by contrast, is oriented with(0, 0)
in the bottom left, withy
moving up as it increases. 
A
CGSize
is a struct that represents the dimensions ofwidth
andheight
. 
A
CGRect
is a struct with both aCGPoint
(origin
) and aCGSize
(size
), representing a rectangle drawn from itsorigin
point with thewidth
andheight
of itssize
.
Because CGRect
is used to represent the frame
of every view drawn on screen, a programmer’s success in graphical programming is contingent on their ability to effectively manipulate rectangle geometry.
Fortunately for us, Quartz comes with a slew of useful functions to reduce the amount of floating point math we have to do ourselves. As central as view programming is to Cocoa, and as useful as these functions are, however, they remain relatively unknown to most iOS developers.
This will not stand! Let’s shine some light on the most useful functions and save y’all some typing!
In Swift, the CoreGraphics framework augments
CGRect
by adding all this functionality as static properties, instance properties, and both mutating and nonmutating functions where appropriate. For idiomatic Swift code, prefer properties and nonmutating instance methods whenever possible.
Transformations
First on our list are the geometric transformations. These functions return a CGRect
, which is the result of performing a particular set of operations on the passed rectangle.
CGRectOffset
offsetBy
/CGRectOffset
: Returns a rectangle with an origin that is offset from that of the source rectangle.
// methods:
extension CGRect {
func offsetBy(dx: CGFloat, dy: CGFloat) > CGRect
mutating func offsetInPlace(dx: CGFloat, dy: CGFloat)
}
// function:
func CGRectOffset(rect: CGRect, dx: CGFloat, dy: CGFloat) > CGRect
CGRect CGRectOffset(
CGRect rect,
CGFloat dx,
CGFloat dy
)
Consider using this anytime you’re changing the origin of a rectangle. Not only can it save a line of code when changing both the horizontal and vertical position, but more importantly, it represents the translation more semantically than manipulating the origin values individually.
CGRectInset
rectByInsetting
/CGRectInset
: Returns a rectangle that is smaller or larger than the source rectangle, with the same center point.
// methods:
extension CGRect {
func insetBy(dx: CGFloat, dy: CGFloat) > CGRect
mutating func insetInPlace(dx: CGFloat, dy: CGFloat)
}
// function:
func CGRectInset(rect: CGRect, dx: CGFloat, dy: CGFloat) > CGRect
CGRect CGRectInset(
CGRect rect,
CGFloat dx,
CGFloat dy
)
Want to make a viewwithinaview look good? Give it a nice 10pt padding with CGRectInset
. Keep in mind that the rectangle will be resized around its center by ± dx
on its left and right edge (for a total of 2 × dx
), and ± dy
on its top and bottom edge (for a total of 2 × dy
).
If you’re using CGRectInset
as a convenience function for resizing a rectangle, it is common to chain this with CGRectOffset
by passing the result of CGRectInset
as the rect
argument in CGRectOffset
.
CGRectIntegral
integral
/CGRectIntegral
: Returns the smallest rectangle that results from converting the source rectangle values to integers.
// methods:
extension CGRect {
var integral: CGRect { get }
mutating func makeIntegralInPlace()
}
// function:
func CGRectIntegral(rect: CGRect) > CGRect
CGRect CGRectIntegral (
CGRect rect
)
It’s important that CGRect
values all are rounded to the nearest whole point. Fractional values cause the frame to be drawn on a pixel boundary. Because pixels are atomic units (cannot be subdivided†) a fractional value will cause the drawing to be averaged over the neighboring pixels, which looks blurry.
CGRectIntegral
will floor
each origin value, and ceil
each size value, which will ensure that your drawing code will crisply align on pixel boundaries.
As a rule of thumb, if you are performing any operations that could result in fractional point values (e.g. division, CGRectGetMid[XY]
, or CGRectDivide
), use CGRectIntegral
to normalize rectangles to be set as a view frame.
† Technically, since the coordinate system operates in terms of points, Retina screens, which have 4 pixels for every point, can draw
± 0.5f
point values on odd pixels without blurriness.
Value Helper Functions
These functions provide a shorthand way to calculate interesting dimensional values about a particular CGRect
.
CGRectGet[MinMidMax][XY]
CGRectGetMinX
CGRectGetMinY
CGRectGetMidX
CGRectGetMidY
CGRectGetMaxX
CGRectGetMaxY
These six functions return the minimum, middle, or maximum x
or y
value for a rectangle, taking the form:
func CGRectGet[MinMidMax][XY](rect: CGRect) > CGFloat
CGFloat CGRectGet[MinMidMax][XY] (
CGRect rect
)
These functions will replace code like frame.origin.x + frame.size.width
with cleaner, more semantically expressive equivalents (especially with the mid and max functions).
CGRectGet[WidthHeight]
// Returns the height of a rectangle.
func CGRectGetHeight(rect: CGRect) > CGFloat
// Returns the width of a rectangle.
func CGRectGetWidth(rect: CGRect) > CGFloat
// Returns the height of a rectangle.
CGFloat CGRectGetHeight (
CGRect rect
)
// Returns the width of a rectangle.
CGFloat CGRectGetWidth (
CGRect rect
)
Much like the previous functions, CGRectGetWidth
& CGRectGetHeight
are often preferable to returning the corresponding member of a CGRect
’s size
. While it’s not extremely competitive in terms of character savings, remember that semantic clarity trumps brevity every time.
Swift Additions
extension CGRect {
var minX: CGFloat { get }
var minY: CGFloat { get }
var midX: CGFloat { get }
var midY: CGFloat { get }
var maxX: CGFloat { get }
var maxY: CGFloat { get }
var width: CGFloat { get }
var height: CGFloat { get }
}
The CGRect
Swift extensions for dimensional values make a huge difference in code readability, as each of the value helper functions is mapped to individual computed properties.
// instead of calling the function
let rightEdge = CGRectMaxX(view.frame)
// simply access the property
let leftEdge = view.frame.minX
Identities
There are three special rectangle values, each of which have unique properties that are important to know about:
CGRectZero
, CGRectNull
, & CGRectInfinite
CGRect.zero
/const CGRect CGRectZero
: A rectangle constant with location (0,0), and width and height of 0. The zero rectangle is equivalent to CGRectMake(0.0f, 0.0f, 0.0f, 0.0f).CGRect.null
/const CGRect CGRectNull
: The null rectangle. This is the rectangle returned when, for example, you intersect two disjoint rectangles. Note that the null rectangle is not the same as the zero rectangle.CGRect.infinite
/const CGRect CGRectInfinite
: A rectangle that has infinite extent.
CGRectZero
is perhaps the most useful of all of the special rectangle values. When initializing subviews, their frames are often initialized to CGRectZero
, deferring their layout to layoutSubviews
.
CGRectNull
is distinct from CGRectZero
, despite any implied correspondence to NULL
== 0
. This value is conceptually similar to NSNotFound
, in that it represents the absence of an expected value. Be aware of what functions can return CGRectNull
, and be prepared to handle it accordingly, by testing with CGRectIsNull
.
CGRectInfinite
is the most exotic of all, and has some of the most interesting properties. It intersects with all points and rectangles, contains all rectangles, and its union with any rectangle is itself. Use CGRectIsInfinite
to check to see if a rectangle is infinite.
Relationships
Moving from one rectangle to two, a pair of rectangles can be either intersected or combined to create a new CGRect
:
CGRectIntersection
intersect
/CGRectIntersection
: Returns the intersection of two rectangles.
// methods:
extension CGRect {
func intersect(withRect: CGRect) > CGRect
mutating func intersectInPlace(withRect: CGRect)
}
// function:
func CGRectIntersection(rect1: CGRect, rect2: CGRect) > CGRect
CGRect CGRectIntersection (
CGRect rect1,
CGRect rect2
)
CGRectIntersection
is a fast way to find the overlapping region between two views. The intersection of two nonoverlapping rectangles is a null rect, but if you need to simply check for intersection or containment, use intersects
/ CGRectIntersectsRect
or contains
/ CGRectContainsRect
instead.
CGRectUnion
union
/CGRectUnion
: Returns the smallest rectangle that contains the two source rectangles.
// methods:
extension CGRect {
func union(rect: CGRect) > CGRect
mutating func unionInPlace(rect: CGRect)
}
// function:
func CGRectUnion(rect1: CGRect, rect2: CGRect) > CGRect
CGRect CGRectUnion (
CGRect rect1,
CGRect rect2
)
Need a rectangle that can wrap two separate regions in your view? Remember that you can chain together different methods to produce the rectangle you need. Use CGRectUnion
and negative values with CGRectInset
to find a padded rectangle around two items:
let combinedRect = imageRect.union(textRect).insetBy(dx: 10, dy: 10)
CGRect combinedRect = CGRectInset(CGRectUnion(imageRect, textRect), 10, 10);
And Finally…
Behold, the most obscure, misunderstood, and useful of the CGGeometry
functions: CGRectDivide
.
CGRectDivide
CGRectDivide
: Divides a source rectangle into two component rectangles.
// method:
extension CGRect {
func divide(atDistance: CGFloat, fromEdge: CGRectEdge) > (slice: CGRect, remainder: CGRect)
}
// function:
CGRectDivide(rect: CGRect,
slice: UnsafeMutablePointer<CGRect>,
remainder: UnsafeMutablePointer<CGRect>,
amount: CGFloat,
edge: CGRectEdge)
void CGRectDivide(
CGRect rect,
CGRect *slice,
CGRect *remainder,
CGFloat amount,
CGRectEdge edge
)
CGRectDivide
divides a rectangle into two components in the following way:
 Take a rectangle and choose an
edge
(left, right, top, or bottom).  Measure out an
amount
from that edge.  Everything from the
edge
to the measuredamount
is stored in the rectangle referenced in theslice
argument.  The rest of the original rectangle is stored in the
remainder
out argument.
Don’t fret about the
UnsafeMutablePointer<CGRect>
in the Swift version; those pointers act just likeinout
properties in this case. Create your slice and remainder instances upfront, and prefix with an&
in the call. Or better yet, use the instance method on an existingCGRect
:
let (slice, remainder) = frame.divide(120, fromEdge: .MinXEdge)
That edge
argument takes a value from the CGRectEdge
enum:
enum CGRectEdge {
case MinXEdge
case MinYEdge
case MaxXEdge
case MaxYEdge
}
enum CGRectEdge {
CGRectMinXEdge,
CGRectMinYEdge,
CGRectMaxXEdge,
CGRectMaxYEdge
}
CGRectDivide
is perfect for dividing up available space among several views (call it on subsequent remainder
amounts to accommodate more than two views). Give it a try next time you’re manually layingout a UITableViewCell
.
So what if you didn’t pay attention in Geometry class—this is the real world, and in the real world, you have CGGeometry.h
.
Know it well, and you’ll be on your way to discovering great new user interfaces in your apps. Do good enough of a job with that, and you may run into the greatest arithmetic problem of all: adding up all of the money you’ll make with your awesome new app. Mathematical!