8 Plot Utilities

▼ |  | PLo T: Graph Plotting

1 | Introduction
2 | 2D Plot Procedures
3 | 2D Renderers
4 | 3D Plot Procedures
5 | 3D Renderers
6 | Nonrenderers
7 | Axis Transforms and Ticks
8 | Plot Utilities
9 | Plot and Renderer Parameters
10 | Plot Contracts
11 | Porting From PLo T <= 5.1.3
12 | Compatibility Module

► | 8 | Plot Utilities

8.1 | Formatting
8.2 | Sampling
8.3 | Plot Colors and Styles
8.4 | Plot-Specific Math
8.5 | Dates and Times

On this page:

8.1`` Formatting
digits-for-range
real->plot-label
ivl->plot-label
->plot-label
real->string/ trunc
real->decimal-string*
integer->superscript
8.2`` Sampling
linear-seq
linear-seq*
nonlinear-seq
mapped-function
kde
8.3`` Plot Colors and Styles
color-seq
color-seq*
->color
->pen-color
->brush-color
->pen-style
->brush-style
8.4`` Plot-Specific Math
8.4.1`` Real Functions
polar->cartesian
3d-polar->3d-cartesian
ceiling-log/ base
floor-log/ base
maybe-inexact->exact
8.4.2`` Vector Functions
v+
v-
vneg
v*
v/
v=
vcross
vcross2
vdot
vmag^2
vmag
vnormalize
vcenter
vrational?
8.4.3`` Intervals and Interval Functions
ivl
rational-ivl?
bounds->intervals
clamp-real
8.5`` Dates and Times
datetime->real
plot-time
plot-time->seconds
seconds->plot-time

5.3.6
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### 8`` Plot Utilities

(require plot/utils)

Typed Racket users should use

(require plot/typed/utils)

#### 8.1`` Formatting

> > procedure
> > (digits-for-range |  | x-min |  |  |  |
> > |  | x-max |  |  |  |
> > | [ | base |  |  |  |
> > |  | extra-digits]) |  | → |  | exact-integer?
> x-min : real?
> x-max : real?
> base : (and/c exact-integer? (>=/c 2)) = 10
> extra-digits : exact-integer? = 3

Given a range, returns the number of decimal places necessary to distinguish numbers in the range. This may return negative numbers for large ranges.

Examples:
> > (digits-for-range 0.01 0.02)
> 5
> > (digits-for-range 0 100000)
> -2

> procedure (real->plot-label x digits [scientific?]) → any
> x : real?
> digits : exact-integer?
> scientific? : boolean? = #t

Converts a real number to a plot label. Used to format axis tick labels, point-labels, and numbers in legend entries.

Examples:
> > (let ([d (digits-for-range 0.01 0.03)])
> (real->plot-label 0.02555555 d))
> ".02556"
> > (real->plot-label 2352343 -2)
> "2352300"
> > (real->plot-label 1000000000.0 4)
> "1×10⁹"
> > (real->plot-label 1000000000.1234 4)
> "(1×10⁹)+.1234"

> procedure (ivl->plot-label i [extra-digits]) → string?
> i : ivl?
> extra-digits : exact-integer? = 3

Converts an interval to a plot label.

If i = (ivl x-min x-max), the number of digits used is (digits-for-range x-min x-max 10 extra-digits) when both endpoints are rational?. Otherwise, it is unspecified—but will probably remain 15.

Examples:
> > (ivl->plot-label (ivl -10.52312 10.99232))
> "[-10.52,10.99]"
> > (ivl->plot-label (ivl -inf.0 pi))
> "[-inf.0,3.141592653589793]"

> procedure (->plot-label a [digits]) → string?
> a : any/c
> digits : exact-integer? = 7

Converts a Racket value to a label. Used by discrete-histogram and discrete-histogram3d.

> procedure (real->string/trunc x e) → string?
> x : real?
> e : exact-integer?

Like real->decimal-string, but removes any trailing zeros and any trailing decimal point.

> > procedure
> > (real->decimal-string* |  | x |  |  |  |
> > |  | min-digits |  |  |  |
> > | [ | max-digits]) |  | → |  | string?
> x : real?
> min-digits : exact-nonnegative-integer?
> max-digits : exact-nonnegative-integer? = min-digits

Like real->decimal-string, but accepts both a maximum and minimum number of digits.

Examples:
> > (real->decimal-string* 1 5 10)
> "1.00000"
> > (real->decimal-string* 1.123456 5 10)
> "1.123456"
> > (real->decimal-string* 1.123456789123456 5 10)
> "1.1234567891"

Applying (real->decimal-string* x min-digits) yields the same value as (real->decimal-string x min-digits).

> procedure (integer->superscript x) → string?
> x : exact-integer?

Converts an integer into a string of superscript Unicode characters.

Example:
> > (integer->superscript -1234567890)
> "⁻¹²³⁴⁵⁶⁷⁸⁹⁰"

Systems running some out-of-date versions of Windows XP have difficulty with Unicode superscripts for 4 and up. Because integer->superscript is used by every number formatting function to format exponents, if you have such a system, PLoT will apparently not format all numbers with exponents correctly (until you update it).

#### 8.2`` Sampling

> > procedure
> > (linear-seq |  | start |  |  |  |
> > |  | end |  |  |  |
> > |  | num |  |  |  |
> > | [ | #:start? start? |  |  |  |
> > |  | #:end? end?]) |  | → |  | (listof real?)
> start : real?
> end : real?
> num : exact-nonnegative-integer?
> start? : boolean? = #t
> end? : boolean? = #t

Returns a list of uniformly spaced real numbers between start and end. If start? is #t, the list includes start. If end? is #t, the list includes end.

This function is used internally to generate sample points.

Examples:
> > (linear-seq 0 1 5)
> '(0 1/4 1/2 3/4 1)
> > (linear-seq 0 1 5 #:start? #f)
> '(1/9 1/3 5/9 7/9 1)
> > (linear-seq 0 1 5 #:end? #f)
> '(0 2/9 4/9 2/3 8/9)
> > (linear-seq 0 1 5 #:start? #f #:end? #f)
> '(1/10 3/10 1/2 7/10 9/10)
> > (define xs (linear-seq -1 1 11))
> > (plot (lines (map vector xs (map sqr xs))))

> > procedure
> > (linear-seq* |  | points |  |  |  |
> > |  | num |  |  |  |
> > | [ | #:start? start? |  |  |  |
> > |  | #:end? end?]) |  | → |  | (listof real?)
> points : (listof real?)
> num : exact-nonnegative-integer?
> start? : boolean? = #t
> end? : boolean? = #t

Like linear-seq, but accepts a list of reals instead of a start and end. The #:start? and #:end? keyword arguments work as in linear-seq. This function does not guarantee that each inner value will be in the returned list.

Examples:
> > (linear-seq* '(0 1 2) 5)
> '(0 1/2 1 3/2 2)
> > (linear-seq* '(0 1 2) 6)
> '(0 2/5 4/5 6/5 8/5 2)
> > (linear-seq* '(0 1 0) 5)
> '(0 1/2 1 1/2 0)

> > procedure
> > (nonlinear-seq |  | start |  |  |  |
> > |  | end |  |  |  |
> > |  | num |  |  |  |
> > |  | transform |  |  |  |
> > | [ | #:start? start? |  |  |  |
> > |  | #:end? end?]) |  | → |  | (listof real?)
> start : real?
> end : real?
> num : exact-nonnegative-integer?
> transform : axis-transform/c
> start? : boolean? = #t
> end? : boolean? = #t

Generates a list of reals that, if transformed using transform, would be uniformly spaced. This is used to generate samples for transformed axes.

Examples:
> > (linear-seq 1 10 4)
> '(1 4 7 10)
> > (nonlinear-seq 1 10 4 log-transform)
> '(1.0 2.154434690031884 4.641588833612779 10.000000000000002)
> > (parameterize ([plot-x-transform log-transform])
> (plot (area-histogram sqr (nonlinear-seq 1 10 4 log-transform))))

> > struct
> > (struct |  | mapped-function (f fmap)
> > |  | #:extra-constructor-name make-mapped-function)
> f : (any/c . -> . any/c)
> fmap : ((listof any/c) . -> . (listof any/c))

Represents a function that maps over lists differently than (map f xs).

With some functions, mapping over a list can be done much more quickly if done specially. (An example is a piecewise function with many pieces that first must decide which interval its input belongs to. Deciding that for many inputs can be done more efficiently by sorting all the inputs first.) Renderer-producing functions that accept a (-> real? real?) also accept a mapped-function, and use its fmap to sample more efficiently.

> > procedure
> > (kde xs h) |  | → |  |
> > mapped-function?
> > (or/c rational? #f)
> > (or/c rational? #f)
> xs : (listof real?)
> h : real?

Given samples and a kernel bandwidth, returns a mapped-function representing a kernel density estimate, and bounds, outside of which the density estimate is zero. Used by density.

#### 8.3`` Plot Colors and Styles

> > procedure
> > (color-seq |  | c1 |
> > |  | c2 |
> > |  | num |
> > | [ | #:start? start? |
> > |  | #:end? end?]) |
> → (listof (list/c real? real? real?))
> c1 : color/c
> c2 : color/c
> num : exact-nonnegative-integer?
> start? : boolean? = #t
> end? : boolean? = #t

Interpolates between colors—red, green and blue components separately—using linear-seq. The #:start? and #:end? keyword arguments work as in linear-seq.

Example:
> > (plot (contour-intervals (λ (x y) (+ x y)) -2 2 -2 2
> #:levels 4 #:contour-styles '(transparent)
> #:colors (color-seq "red" "blue" 5)))

> > procedure
> > (color-seq* |  | colors |
> > |  | num |
> > | [ | #:start? start? |
> > |  | #:end? end?]) |
> → (listof (list/c real? real? real?))
> colors : (listof color/c)
> num : exact-nonnegative-integer?
> start? : boolean? = #t
> end? : boolean? = #t

Interpolates between colors—red, green and blue components separately—using linear-seq*. The #:start? and #:end? keyword arguments work as in linear-seq.

Example:
> > (plot (contour-intervals (λ (x y) (+ x y)) -2 2 -2 2
> #:levels 4 #:contour-styles '(transparent)
> #:colors (color-seq* '(red white blue) 5)))

> procedure (->color c) → (list/c real? real? real?)
> c : color/c

Converts a non-integer plot color to an RGB triplet.

Symbols are converted to strings, and strings are looked up in a color-database<%>. Lists are unchanged, and color% objects are converted straightforwardly.

Examples:
> > (->color 'navy)
> '(36 36 140)
> > (->color "navy")
> '(36 36 140)
> > (->color '(36 36 140))
> '(36 36 140)
> > (->color (make-object color% 36 36 140))
> '(36 36 140)

This function does not convert integers to RGB triplets, because there is no way for it to know whether the color will be used for a pen or for a brush. Use ->pen-color and ->brush-color to convert integers.

> procedure (->pen-color c) → (list/c real? real? real?)
> c : plot-color/c

Converts a line color to an RGB triplet. This function interprets integer colors as darker and more saturated than ->brush-color does.

Non-integer colors are converted using ->color. Integer colors are chosen for good pairwise contrast, especially between neighbors. Integer colors repeat starting with 128.

Examples:
> > (equal? (->pen-color 0) (->pen-color 8))
> #f
> > (plot (contour-intervals
> (λ (x y) (+ x y)) -2 2 -2 2
> #:levels 7 #:contour-styles '(transparent)
> #:colors (map ->pen-color (build-list 8 values))))

> procedure (->brush-color c) → (list/c real? real? real?)
> c : plot-color/c

Converts a fill color to an RGB triplet. This function interprets integer colors as lighter and less saturated than ->pen-color does.

Non-integer colors are converted using ->color. Integer colors are chosen for good pairwise contrast, especially between neighbors. Integer colors repeat starting with 128.

Examples:
> > (equal? (->brush-color 0) (->brush-color 8))
> #f
> > (plot (contour-intervals
> (λ (x y) (+ x y)) -2 2 -2 2
> #:levels 7 #:contour-styles '(transparent)
> #:colors (map ->brush-color (build-list 8 values))))

In the above example, mapping ->brush-color over the list is actually unnecessary, because contour-intervals uses ->brush-color internally to convert fill colors.

The function-interval function generally plots areas using a fill color and lines using a line color. Both kinds of color have the default value 3. The following example reverses the default behavior; i.e it draws areas using line color 3 and lines using fill color 3:

> > (plot (function-interval sin (λ (x) 0) -4 4
> #:color (->pen-color 3)
> #:line1-color (->brush-color 3)
> #:line2-color (->brush-color 3)
> #:line1-width 4 #:line2-width 4))

> procedure (->pen-style s) → symbol?
> s : plot-pen-style/c

Converts a symbolic pen style or a number to a symbolic pen style. Symbols are unchanged. Integer pen styles repeat starting at 5.

Examples:
> > (eq? (->pen-style 0) (->pen-style 5))
> #t
> > (map ->pen-style '(0 1 2 3 4))
> '(solid dot long-dash short-dash dot-dash)

> procedure (->brush-style s) → symbol?
> s : plot-brush-style/c

Converts a symbolic brush style or a number to a symbolic brush style. Symbols are unchanged. Integer brush styles repeat starting at 7.

Examples:
> > (eq? (->brush-style 0) (->brush-style 7))
> #t
> > (map ->brush-style '(0 1 2 3))
> '(solid bdiagonal-hatch fdiagonal-hatch crossdiag-hatch)
> > (map ->brush-style '(4 5 6))
> '(horizontal-hatch vertical-hatch cross-hatch)

#### 8.4`` Plot-Specific Math

##### 8.4.1`` Real Functions

> procedure (polar->cartesian θ r) → (vector/c real? real?)
> θ : real?
> r : real?

Converts 2D polar coordinates to 3D cartesian coordinates.

> procedure (3d-polar->3d-cartesian θ ρ r) → (vector/c real? real? real?)
> θ : real?
> ρ : real?
> r : real?

Converts 3D polar coordinates to 3D cartesian coordinates. See parametric3d for an example of use.

> procedure (ceiling-log/base b x) → exact-integer?
> b : (and/c exact-integer? (>=/c 2))
> x : (>/c 0)

Like (ceiling (/ (log x) (log b))), but ceiling-log/base is not susceptible to floating-point error.

Examples:
> > (ceiling (/ (log 100) (log 10)))
> 2.0
> > (ceiling-log/base 10 100)
> 2
> > (ceiling (/ (log 1/1000) (log 10)))
> -2.0
> > (ceiling-log/base 10 1/1000)
> -3

Various number-formatting functions use this.

> procedure (floor-log/base b x) → exact-integer?
> b : (and/c exact-integer? (>=/c 2))
> x : (>/c 0)

Like (floor (/ (log x) (log b))), but floor-log/base is not susceptible to floating-point error.

Examples:
> > (floor (/ (log 100) (log 10)))
> 2.0
> > (floor-log/base 10 100)
> 2
> > (floor (/ (log 1000) (log 10)))
> 2.0
> > (floor-log/base 10 1000)
> 3

This is a generalization of order-of-magnitude.

> procedure (maybe-inexact->exact x) → (or/c rational? #f)
> x : (or/c rational? #f)

Returns #f if x is #f; otherwise (inexact->exact x). Use this to convert interval endpoints, which may be #f, to exact numbers.

##### 8.4.2`` Vector Functions

> procedure (v+ v1 v2) → (vectorof real?)
> v1 : (vectorof real?)
> v2 : (vectorof real?)

> procedure (v- v1 v2) → (vectorof real?)
> v1 : (vectorof real?)
> v2 : (vectorof real?)

> procedure (vneg v) → (vectorof real?)
> v : (vectorof real?)

> procedure (v* v c) → (vectorof real?)
> v : (vectorof real?)
> c : real?

> procedure (v/ v c) → (vectorof real?)
> v : (vectorof real?)
> c : real?

Vector arithmetic. Equivalent to vector-mapp-ing arithmetic operators over vectors, but specialized so that 2- and 3-vector operations are much faster.

Examples:
> > (v+ #(1 2) #(3 4))
> '#(4 6)
> > (v- #(1 2) #(3 4))
> '#(-2 -2)
> > (vneg #(1 2))
> '#(-1 -2)
> > (v* #(1 2 3) 2)
> '#(2 4 6)
> > (v/ #(1 2 3) 2)
> '#(1/2 1 3/2)

> procedure (v= v1 v2) → boolean?
> v1 : (vectorof real?)
> v2 : (vectorof real?)

Like equal? specialized to numeric vectors, but compares elements using =.

Examples:
> > (equal? #(1 2) #(1 2))
> #t
> > (equal? #(1 2) #(1.0 2.0))
> #f
> > (v= #(1 2) #(1.0 2.0))
> #t

> procedure (vcross v1 v2) → (vector/c real? real? real?)
> v1 : (vector/c real? real? real?)
> v2 : (vector/c real? real? real?)

Returns the right-hand vector cross product of v1 and v2.

Examples:
> > (vcross #(1 0 0) #(0 1 0))
> '#(0 0 1)
> > (vcross #(0 1 0) #(1 0 0))
> '#(0 0 -1)
> > (vcross #(0 0 1) #(0 0 1))
> '#(0 0 0)

> procedure (vcross2 v1 v2) → real?
> v1 : (vector/c real? real?)
> v2 : (vector/c real? real?)

Returns the signed area of the 2D parallelogram with sides v1 and v2. Equivalent to (vector-ref (vcross (vector-append v1 #(0)) (vector-append v2 #(0))) 2) but faster.

Examples:
> > (vcross2 #(1 0) #(0 1))
> 1
> > (vcross2 #(0 1) #(1 0))
> -1

> procedure (vdot v1 v2) → real?
> v1 : (vectorof real?)
> v2 : (vectorof real?)

Returns the dot product of v1 and v2.

> procedure (vmag^2 v) → real?
> v : (vectorof real?)

Returns the squared magnitude of v. Equivalent to (vdot v v).

> procedure (vmag v) → real?
> v : (vectorof real?)

Returns the magnitude of v. Equivalent to (sqrt (vmag^2 v)).

> procedure (vnormalize v) → (vectorof real?)
> v : (vectorof real?)

Returns a normal vector in the same direction as v. If v is a zero vector, returns v.

Examples:
> > (vnormalize #(1 1 0))
> '#(0.7071067811865475 0.7071067811865475 0)
> > (vnormalize #(1 1 1))
> '#(0.5773502691896258 0.5773502691896258 0.5773502691896258)
> > (vnormalize #(0 0 0.0))
> '#(0 0 0.0)

> procedure (vcenter vs) → (vectorof real?)
> vs : (listof (vectorof real?))

Returns the center of the smallest bounding box that contains vs.

Example:
> > (vcenter '(#(1 1) #(2 2)))
> '#(3/2 3/2)

> procedure (vrational? v) → boolean?
> v : (vectorof real?)

Returns #t if every element of v is rational?.

Examples:
> > (vrational? #(1 2))
> #t
> > (vrational? #(+inf.0 2))
> #f
> > (vrational? #(#f 1))
> #f

##### 8.4.3`` Intervals and Interval Functions

> > struct
> > (struct |  | ivl (min max)
> > |  | #:extra-constructor-name make-ivl)
> min : real?
> max : real?

Represents a closed interval.

An interval with two real-valued endpoints always contains the endpoints in order:

> > (ivl 0 1)
> (ivl 0 1)
> > (ivl 1 0)
> (ivl 0 1)

An interval can have infinite endpoints:

> > (ivl -inf.0 0)
> (ivl -inf.0 0)
> > (ivl 0 +inf.0)
> (ivl 0 +inf.0)
> > (ivl -inf.0 +inf.0)
> (ivl -inf.0 +inf.0)

Functions that return rectangle renderers, such as rectangles and discrete-histogram3d, accept vectors of ivls as arguments. The ivl struct type is also provided by plot so users of such renderers do not have to require plot/utils.

> procedure (rational-ivl? i) → boolean?
> i : any/c

Returns #t if i is an interval and each of its endpoints is rational?.

Example:
> > (map rational-ivl? (list (ivl -1 1) (ivl -inf.0 2) 'bob))
> '(#t #f #f)

> procedure (bounds->intervals xs) → (listof ivl?)
> xs : (listof real?)

Given a list of points, returns intervals between each pair.

Use this to construct inputs for rectangles and rectangles3d.

Example:
> > (bounds->intervals (linear-seq 0 1 5))
> (list (ivl 0 1/4) (ivl 1/4 1/2) (ivl 1/2 3/4) (ivl 3/4 1))

> procedure (clamp-real x i) → real?
> x : real?
> i : ivl?

#### 8.5`` Dates and Times

> procedure (datetime->real x) → real?
> x : (or/c plot-time? date? date*? sql-date? sql-time? sql-timestamp?)

Converts various date/time representations into UTC seconds, respecting time zone offsets.

For dates, the value returned is the number of seconds since a system-dependent UTC epoch. See date-ticks for more information.

To plot a time series using dates pulled from an SQL database, simply set the relevant axis ticks (probably plot-x-ticks) to date-ticks, and convert the dates to seconds using datetime->real before passing them to lines. To keep time zone offsets from influencing the plot, set them to 0 first.

> > struct
> > (struct |  | plot-time (second minute hour day)
> > |  | #:extra-constructor-name make-plot-time)
> second : (and/c (>=/c 0) (</c 60))
> minute : (integer-in 0 59)
> hour : (integer-in 0 23)
> day : exact-integer?

A time representation that accounts for days, negative times (using negative days), and fractional seconds.

PLoT (specifically time-ticks) uses plot-time internally to format times, but because renderer-producing functions require only real values, user code should not need it. It is provided just in case.

> procedure (plot-time->seconds t) → real?
> t : plot-time?

> procedure (seconds->plot-time s) → plot-time?
> s : real?

Convert plot-times to real seconds, and vice-versa.

Examples:
> > (define (plot-time+ t1 t2)
> (seconds->plot-time (+ (plot-time->seconds t1)
> (plot-time->seconds t2))))
> > (plot-time+ (plot-time 32 0 12 1)
> (plot-time 32 0 14 1))
> (plot-time 4 1 2 3)

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