Merge pull request #1445 from daschw/infinite-objects

Infinite objects (fix #1422)
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Daniel Schwabeneder 2018-03-28 21:49:32 +02:00 committed by GitHub
commit bfc26fa407
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12 changed files with 172 additions and 46 deletions

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@ -187,6 +187,8 @@ include("output.jl")
@shorthands sticks @shorthands sticks
@shorthands hline @shorthands hline
@shorthands vline @shorthands vline
@shorthands hspan
@shorthands vspan
@shorthands ohlc @shorthands ohlc
@shorthands contour @shorthands contour
@shorthands contourf @shorthands contourf

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@ -289,8 +289,8 @@ end
function expand_extrema!(ex::Extrema, v::Number) function expand_extrema!(ex::Extrema, v::Number)
ex.emin = NaNMath.min(v, ex.emin) ex.emin = isfinite(v) ? min(v, ex.emin) : ex.emin
ex.emax = NaNMath.max(v, ex.emax) ex.emax = isfinite(v) ? max(v, ex.emax) : ex.emax
ex ex
end end
@ -305,8 +305,8 @@ expand_extrema!(axis::Axis, ::Bool) = axis[:extrema]
function expand_extrema!(axis::Axis, v::Tuple{MIN,MAX}) where {MIN<:Number,MAX<:Number} function expand_extrema!(axis::Axis, v::Tuple{MIN,MAX}) where {MIN<:Number,MAX<:Number}
ex = axis[:extrema] ex = axis[:extrema]
ex.emin = NaNMath.min(v[1], ex.emin) ex.emin = isfinite(v[1]) ? min(v[1], ex.emin) : ex.emin
ex.emax = NaNMath.max(v[2], ex.emax) ex.emax = isfinite(v[2]) ? max(v[2], ex.emax) : ex.emax
ex ex
end end
function expand_extrema!(axis::Axis, v::AVec{N}) where N<:Number function expand_extrema!(axis::Axis, v::AVec{N}) where N<:Number
@ -328,6 +328,9 @@ function expand_extrema!(sp::Subplot, d::KW)
else else
letter == :x ? :y : letter == :y ? :x : :z letter == :x ? :y : letter == :y ? :x : :z
end] end]
if letter != :z && d[:seriestype] == :straightline && any(series[:seriestype] != :straightline for series in series_list(sp)) && data[1] != data[2]
data = [NaN]
end
axis = sp[Symbol(letter, "axis")] axis = sp[Symbol(letter, "axis")]
if isa(data, Volume) if isa(data, Volume)

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@ -51,7 +51,7 @@ const _glvisualize_attr = merge_with_base_supported([
:tick_direction, :tick_direction,
]) ])
const _glvisualize_seriestype = [ const _glvisualize_seriestype = [
:path, :shape, :path, :shape, :straightline,
:scatter, :hexbin, :scatter, :hexbin,
:bar, :boxplot, :bar, :boxplot,
:heatmap, :image, :volume, :heatmap, :image, :volume,
@ -287,7 +287,13 @@ function topoints(::Type{P}, array) where P
end end
function extract_points(d) function extract_points(d)
dim = is3d(d) ? 3 : 2 dim = is3d(d) ? 3 : 2
array = (d[:x], d[:y], d[:z])[1:dim] array = if d[:seriestype] == :straightline
straightline_data(d)
elseif d[:seriestype] == :shape
shape_data(d)
else
(d[:x], d[:y], d[:z])[1:dim]
end
topoints(Point{dim, Float32}, array) topoints(Point{dim, Float32}, array)
end end
function make_gradient(grad::Vector{C}) where C <: Colorant function make_gradient(grad::Vector{C}) where C <: Colorant
@ -1102,7 +1108,7 @@ function _display(plt::Plot{GLVisualizeBackend}, visible = true)
kw_args[:stroke_width] = Float32(d[:linewidth]/100f0) kw_args[:stroke_width] = Float32(d[:linewidth]/100f0)
end end
vis = GL.gl_surface(x, y, z, kw_args) vis = GL.gl_surface(x, y, z, kw_args)
elseif (st in (:path, :path3d)) && d[:linewidth] > 0 elseif (st in (:path, :path3d, :straightline)) && d[:linewidth] > 0
kw = copy(kw_args) kw = copy(kw_args)
points = Plots.extract_points(d) points = Plots.extract_points(d)
extract_linestyle(d, kw) extract_linestyle(d, kw)
@ -1460,7 +1466,7 @@ function make_label(sp, series, i)
d = series.d d = series.d
st = d[:seriestype] st = d[:seriestype]
kw_args = KW() kw_args = KW()
if (st in (:path, :path3d)) && d[:linewidth] > 0 if (st in (:path, :path3d, :straightline)) && d[:linewidth] > 0
points = Point2f0[(0, ho), (w, ho)] points = Point2f0[(0, ho), (w, ho)]
kw = KW() kw = KW()
extract_linestyle(d, kw) extract_linestyle(d, kw)

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@ -48,7 +48,7 @@ const _gr_attr = merge_with_base_supported([
:contour_labels, :contour_labels,
]) ])
const _gr_seriestype = [ const _gr_seriestype = [
:path, :scatter, :path, :scatter, :straightline,
:heatmap, :pie, :image, :heatmap, :pie, :image,
:contour, :path3d, :scatter3d, :surface, :wireframe, :contour, :path3d, :scatter3d, :surface, :wireframe,
:shape :shape
@ -1014,7 +1014,11 @@ function gr_display(sp::Subplot{GRBackend}, w, h, viewport_canvas)
x, y = convert_to_polar(x, y, (rmin, rmax)) x, y = convert_to_polar(x, y, (rmin, rmax))
end end
if st in (:path, :scatter) if st == :straightline
x, y = straightline_data(series)
end
if st in (:path, :scatter, :straightline)
if length(x) > 1 if length(x) > 1
lz = series[:line_z] lz = series[:line_z]
segments_iterator = if lz != nothing && length(lz) > 1 segments_iterator = if lz != nothing && length(lz) > 1
@ -1036,7 +1040,7 @@ function gr_display(sp::Subplot{GRBackend}, w, h, viewport_canvas)
end end
# draw the line(s) # draw the line(s)
if st == :path if st in (:path, :straightline)
for (i, rng) in enumerate(segments_iterator) for (i, rng) in enumerate(segments_iterator)
gr_set_line(series[:linewidth], series[:linestyle], get_linecolor(sp, series, i)) #, series[:linealpha]) gr_set_line(series[:linewidth], series[:linestyle], get_linecolor(sp, series, i)) #, series[:linealpha])
arrowside = isa(series[:arrow], Arrow) ? series[:arrow].side : :none arrowside = isa(series[:arrow], Arrow) ? series[:arrow].side : :none
@ -1182,6 +1186,7 @@ function gr_display(sp::Subplot{GRBackend}, w, h, viewport_canvas)
GR.selntran(1) GR.selntran(1)
elseif st == :shape elseif st == :shape
x, y = shape_data(series)
for (i,rng) in enumerate(iter_segments(x, y)) for (i,rng) in enumerate(iter_segments(x, y))
if length(rng) > 1 if length(rng) > 1
# connect to the beginning # connect to the beginning
@ -1295,7 +1300,7 @@ function gr_display(sp::Subplot{GRBackend}, w, h, viewport_canvas)
st == :shape && gr_polyline(x, y) st == :shape && gr_polyline(x, y)
end end
if st == :path if st in (:path, :straightline)
GR.settransparency(gr_alpha(series[:linealpha])) GR.settransparency(gr_alpha(series[:linealpha]))
if series[:fillrange] == nothing || series[:ribbon] != nothing if series[:fillrange] == nothing || series[:ribbon] != nothing
GR.polyline([xpos - 0.07, xpos - 0.01], [ypos, ypos]) GR.polyline([xpos - 0.07, xpos - 0.01], [ypos, ypos])

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@ -97,7 +97,7 @@ const _hdf5_attr = merge_with_base_supported([
:colorbar_title, :colorbar_title,
]) ])
const _hdf5_seriestype = [ const _hdf5_seriestype = [
:path, :steppre, :steppost, :shape, :path, :steppre, :steppost, :shape, :straightline,
:scatter, :hexbin, #:histogram2d, :histogram, :scatter, :hexbin, #:histogram2d, :histogram,
# :bar, # :bar,
:heatmap, :pie, :image, :heatmap, :pie, :image,

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@ -57,7 +57,7 @@ const _inspectdr_attr = merge_with_base_supported([
]) ])
const _inspectdr_style = [:auto, :solid, :dash, :dot, :dashdot] const _inspectdr_style = [:auto, :solid, :dash, :dot, :dashdot]
const _inspectdr_seriestype = [ const _inspectdr_seriestype = [
:path, :scatter, :shape #, :steppre, :steppost :path, :scatter, :shape, :straightline, #, :steppre, :steppost
] ]
#see: _allMarkers, _shape_keys #see: _allMarkers, _shape_keys
const _inspectdr_marker = Symbol[ const _inspectdr_marker = Symbol[
@ -243,7 +243,11 @@ function _series_added(plt::Plot{InspectDRBackend}, series::Series)
if nothing == plot; return; end if nothing == plot; return; end
_vectorize(v) = isa(v, Vector) ? v : collect(v) #InspectDR only supports vectors _vectorize(v) = isa(v, Vector) ? v : collect(v) #InspectDR only supports vectors
x = _vectorize(series[:x]); y = _vectorize(series[:y]) x, y = if st == :straightline
straightline_data(series)
else
_vectorize(series[:x]), _vectorize(series[:y])
end
#No support for polar grid... but can still perform polar transformation: #No support for polar grid... but can still perform polar transformation:
if ispolar(sp) if ispolar(sp)
@ -268,6 +272,7 @@ For st in :shape:
=# =#
if st in (:shape,) if st in (:shape,)
x, y = shape_data(series)
nmax = 0 nmax = 0
for (i,rng) in enumerate(iter_segments(x, y)) for (i,rng) in enumerate(iter_segments(x, y))
nmax = i nmax = i
@ -299,7 +304,7 @@ For st in :shape:
color = linecolor, fillcolor = fillcolor color = linecolor, fillcolor = fillcolor
) )
end end
elseif st in (:path, :scatter) #, :steppre, :steppost) elseif st in (:path, :scatter, :straightline) #, :steppre, :steppost)
#NOTE: In Plots.jl, :scatter plots have 0-linewidths (I think). #NOTE: In Plots.jl, :scatter plots have 0-linewidths (I think).
linewidth = series[:linewidth] linewidth = series[:linewidth]
#More efficient & allows some support for markerstrokewidth: #More efficient & allows some support for markerstrokewidth:

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@ -39,7 +39,7 @@ const _pgfplots_attr = merge_with_base_supported([
:framestyle, :framestyle,
:camera, :camera,
]) ])
const _pgfplots_seriestype = [:path, :path3d, :scatter, :steppre, :stepmid, :steppost, :histogram2d, :ysticks, :xsticks, :contour, :shape] const _pgfplots_seriestype = [:path, :path3d, :scatter, :steppre, :stepmid, :steppost, :histogram2d, :ysticks, :xsticks, :contour, :shape, :straightline,]
const _pgfplots_style = [:auto, :solid, :dash, :dot, :dashdot, :dashdotdot] const _pgfplots_style = [:auto, :solid, :dash, :dot, :dashdot, :dashdotdot]
const _pgfplots_marker = [:none, :auto, :circle, :rect, :diamond, :utriangle, :dtriangle, :cross, :xcross, :star5, :pentagon, :hline] #vcat(_allMarkers, Shape) const _pgfplots_marker = [:none, :auto, :circle, :rect, :diamond, :utriangle, :dtriangle, :cross, :xcross, :star5, :pentagon, :hline] #vcat(_allMarkers, Shape)
const _pgfplots_scale = [:identity, :ln, :log2, :log10] const _pgfplots_scale = [:identity, :ln, :log2, :log10]
@ -220,6 +220,10 @@ function pgf_series(sp::Subplot, series::Series)
d[:z].surf, d[:x], d[:y] d[:z].surf, d[:x], d[:y]
elseif is3d(st) elseif is3d(st)
d[:x], d[:y], d[:z] d[:x], d[:y], d[:z]
elseif st == :straightline
straightline_data(series)
elseif st == :shape
shape_data(series)
elseif d[:marker_z] != nothing elseif d[:marker_z] != nothing
# If a marker_z is used pass it as third coordinate to a 2D plot. # If a marker_z is used pass it as third coordinate to a 2D plot.
# See "Scatter Plots" in PGFPlots documentation # See "Scatter Plots" in PGFPlots documentation

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@ -51,6 +51,7 @@ const _plotly_attr = merge_with_base_supported([
const _plotly_seriestype = [ const _plotly_seriestype = [
:path, :scatter, :pie, :heatmap, :path, :scatter, :pie, :heatmap,
:contour, :surface, :wireframe, :path3d, :scatter3d, :shape, :scattergl, :contour, :surface, :wireframe, :path3d, :scatter3d, :shape, :scattergl,
:straightline
] ]
const _plotly_style = [:auto, :solid, :dash, :dot, :dashdot] const _plotly_style = [:auto, :solid, :dash, :dot, :dashdot]
const _plotly_marker = [ const _plotly_marker = [
@ -522,12 +523,16 @@ function plotly_series(plt::Plot, series::Series)
plotly_data(series[letter]) plotly_data(series[letter])
end), (:x, :y, :z)) end), (:x, :y, :z))
if st == :straightline
x, y = straightline_data(series)
end
d_out[:name] = series[:label] d_out[:name] = series[:label]
isscatter = st in (:scatter, :scatter3d, :scattergl) isscatter = st in (:scatter, :scatter3d, :scattergl)
hasmarker = isscatter || series[:markershape] != :none hasmarker = isscatter || series[:markershape] != :none
hasline = st in (:path, :path3d) hasline = st in (:path, :path3d, :straightline)
hasfillrange = st in (:path, :scatter, :scattergl) && hasfillrange = st in (:path, :scatter, :scattergl, :straightline) &&
(isa(series[:fillrange], AbstractVector) || isa(series[:fillrange], Tuple)) (isa(series[:fillrange], AbstractVector) || isa(series[:fillrange], Tuple))
d_out[:colorbar] = KW(:title => sp[:colorbar_title]) d_out[:colorbar] = KW(:title => sp[:colorbar_title])
@ -538,7 +543,7 @@ function plotly_series(plt::Plot, series::Series)
end end
# set the "type" # set the "type"
if st in (:path, :scatter, :scattergl) if st in (:path, :scatter, :scattergl, :straightline)
d_out[:type] = st==:scattergl ? "scattergl" : "scatter" d_out[:type] = st==:scattergl ? "scattergl" : "scatter"
d_out[:mode] = if hasmarker d_out[:mode] = if hasmarker
hasline ? "lines+markers" : "markers" hasline ? "lines+markers" : "markers"
@ -708,7 +713,7 @@ function plotly_series_shapes(plt::Plot, series::Series)
base_d[:name] = series[:label] base_d[:name] = series[:label]
# base_d[:legendgroup] = series[:label] # base_d[:legendgroup] = series[:label]
x, y = plotly_data(series[:x]), plotly_data(series[:y]) x, y = shape_data(series)
for (i,rng) in enumerate(iter_segments(x,y)) for (i,rng) in enumerate(iter_segments(x,y))
length(rng) < 2 && continue length(rng) < 2 && continue

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@ -45,7 +45,7 @@ const _pyplot_attr = merge_with_base_supported([
:contour_labels, :contour_labels,
]) ])
const _pyplot_seriestype = [ const _pyplot_seriestype = [
:path, :steppre, :steppost, :shape, :path, :steppre, :steppost, :shape, :straightline,
:scatter, :hexbin, #:histogram2d, :histogram, :scatter, :hexbin, #:histogram2d, :histogram,
# :bar, # :bar,
:heatmap, :pie, :image, :heatmap, :pie, :image,
@ -453,6 +453,11 @@ function py_add_series(plt::Plot{PyPlotBackend}, series::Series)
# ax = getAxis(plt, series) # ax = getAxis(plt, series)
x, y, z = series[:x], series[:y], series[:z] x, y, z = series[:x], series[:y], series[:z]
if st == :straightline
x, y = straightline_data(series)
elseif st == :shape
x, y = shape_data(series)
end
xyargs = (st in _3dTypes ? (x,y,z) : (x,y)) xyargs = (st in _3dTypes ? (x,y,z) : (x,y))
# handle zcolor and get c/cmap # handle zcolor and get c/cmap
@ -486,7 +491,7 @@ function py_add_series(plt::Plot{PyPlotBackend}, series::Series)
# for each plotting command, optionally build and add a series handle to the list # for each plotting command, optionally build and add a series handle to the list
# line plot # line plot
if st in (:path, :path3d, :steppre, :steppost) if st in (:path, :path3d, :steppre, :steppost, :straightline)
if series[:linewidth] > 0 if series[:linewidth] > 0
if series[:line_z] == nothing if series[:line_z] == nothing
handle = ax[:plot](xyargs...; handle = ax[:plot](xyargs...;
@ -1249,7 +1254,7 @@ function py_add_legend(plt::Plot, sp::Subplot, ax)
facecolor = py_color(_cycle(series[:fillcolor],1)), facecolor = py_color(_cycle(series[:fillcolor],1)),
linewidth = py_dpi_scale(plt, clamp(series[:linewidth], 0, 5)), linewidth = py_dpi_scale(plt, clamp(series[:linewidth], 0, 5)),
) )
elseif series[:seriestype] == :path elseif series[:seriestype] in (:path, :straightline)
PyPlot.plt[:Line2D]((0,1),(0,0), PyPlot.plt[:Line2D]((0,1),(0,0),
color = py_color(_cycle(series[:linecolor],1)), color = py_color(_cycle(series[:linecolor],1)),
linewidth = py_dpi_scale(plt, clamp(series[:linewidth], 0, 5)), linewidth = py_dpi_scale(plt, clamp(series[:linewidth], 0, 5)),

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@ -17,7 +17,7 @@ const _unicodeplots_attr = merge_with_base_supported([
:guide, :lims, :guide, :lims,
]) ])
const _unicodeplots_seriestype = [ const _unicodeplots_seriestype = [
:path, :scatter, :path, :scatter, :straightline,
# :bar, # :bar,
:shape, :shape,
:histogram2d, :histogram2d,
@ -142,7 +142,7 @@ function addUnicodeSeries!(o, d::KW, addlegend::Bool, xlim, ylim)
return return
end end
if st == :path if st in (:path, :straightline)
func = UnicodePlots.lineplot! func = UnicodePlots.lineplot!
elseif st == :scatter || d[:markershape] != :none elseif st == :scatter || d[:markershape] != :none
func = UnicodePlots.scatterplot! func = UnicodePlots.scatterplot!
@ -155,7 +155,13 @@ function addUnicodeSeries!(o, d::KW, addlegend::Bool, xlim, ylim)
end end
# get the series data and label # get the series data and label
x, y = [collect(float(d[s])) for s in (:x, :y)] x, y = if st == :straightline
straightline_data(d)
elseif st == :shape
shape_data(series)
else
[collect(float(d[s])) for s in (:x, :y)]
end
label = addlegend ? d[:label] : "" label = addlegend ? d[:label] : ""
# if we happen to pass in allowed color symbols, great... otherwise let UnicodePlots decide # if we happen to pass in allowed color symbols, great... otherwise let UnicodePlots decide

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@ -79,28 +79,50 @@ function hvline_limits(axis::Axis)
end end
@recipe function f(::Type{Val{:hline}}, x, y, z) @recipe function f(::Type{Val{:hline}}, x, y, z)
xmin, xmax = hvline_limits(plotattributes[:subplot][:xaxis])
n = length(y) n = length(y)
newx = repmat(Float64[xmin, xmax, NaN], n) newx = repmat(Float64[-1, 1, NaN], n)
newy = vec(Float64[yi for i=1:3,yi=y]) newy = vec(Float64[yi for i=1:3,yi=y])
x := newx x := newx
y := newy y := newy
seriestype := :path seriestype := :straightline
() ()
end end
@deps hline path @deps hline straightline
@recipe function f(::Type{Val{:vline}}, x, y, z) @recipe function f(::Type{Val{:vline}}, x, y, z)
ymin, ymax = hvline_limits(plotattributes[:subplot][:yaxis])
n = length(y) n = length(y)
newx = vec(Float64[yi for i=1:3,yi=y]) newx = vec(Float64[yi for i=1:3,yi=y])
newy = repmat(Float64[ymin, ymax, NaN], n) newy = repmat(Float64[-1, 1, NaN], n)
x := newx x := newx
y := newy y := newy
seriestype := :path seriestype := :straightline
() ()
end end
@deps vline path @deps vline straightline
@recipe function f(::Type{Val{:hspan}}, x, y, z)
n = div(length(y), 2)
newx = repeat([-Inf, Inf, Inf, -Inf, NaN], outer = n)
newy = vcat([[y[2i-1], y[2i-1], y[2i], y[2i], NaN] for i in 1:n]...)
linewidth --> 0
x := newx
y := newy
seriestype := :shape
()
end
@deps hspan shape
@recipe function f(::Type{Val{:vspan}}, x, y, z)
n = div(length(y), 2)
newx = vcat([[y[2i-1], y[2i-1], y[2i], y[2i], NaN] for i in 1:n]...)
newy = repeat([-Inf, Inf, Inf, -Inf, NaN], outer = n)
linewidth --> 0
x := newx
y := newy
seriestype := :shape
()
end
@deps vspan shape
# --------------------------------------------------------------------------- # ---------------------------------------------------------------------------
# path and scatter # path and scatter
@ -999,15 +1021,7 @@ end
# ------------------------------------------------- # -------------------------------------------------
"Adds a+bx... straight line over the current plot, without changing the axis limits" "Adds a+bx... straight line over the current plot, without changing the axis limits"
function abline!(plt::Plot, a, b; kw...) abline!(plt::Plot, a, b; kw...) = plot!(plt, [0, 1], [b, b+a]; seriestype = :straightline, kw...)
xl, yl = xlims(plt), ylims(plt)
x1, x2 = max(xl[1], (yl[1] - b)/a), min(xl[2], (yl[2] - b)/a)
if x2 > x1
plot!(plt, x -> b + a*x, x1, x2; kw...)
else
nothing
end
end
abline!(args...; kw...) = abline!(current(), args...; kw...) abline!(args...; kw...) = abline!(current(), args...; kw...)

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@ -1079,3 +1079,74 @@ function convert_sci_unicode(label::AbstractString)
end end
label label
end end
function straightline_data(series)
sp = series[:subplot]
xl, yl = isvertical(series) ? (xlims(sp), ylims(sp)) : (ylims(sp), xlims(sp))
x, y = series[:x], series[:y]
n = length(x)
if n == 2
return straightline_data(xl, yl, x, y)
else
k, r = divrem(n, 3)
if r == 0
xdata, ydata = fill(NaN, n), fill(NaN, n)
for i in 1:k
inds = (3 * i - 2):(3 * i - 1)
xdata[inds], ydata[inds] = straightline_data(xl, yl, x[inds], y[inds])
end
return xdata, ydata
else
error("Misformed data. `straightline_data` either accepts vectors of length 2 or 3k. The provided series has length $n")
end
end
end
function straightline_data(xl, yl, x, y)
x_vals, y_vals = if y[1] == y[2]
if x[1] == x[2]
error("Two identical points cannot be used to describe a straight line.")
else
[xl[1], xl[2]], [y[1], y[2]]
end
elseif x[1] == x[2]
[x[1], x[2]], [yl[1], yl[2]]
else
# get a and b from the line y = a * x + b through the points given by
# the coordinates x and x
b = y[1] - (y[1] - y[2]) * x[1] / (x[1] - x[2])
a = (y[1] - y[2]) / (x[1] - x[2])
# get the data values
xdata = [clamp(x[1] + (x[1] - x[2]) * (ylim - y[1]) / (y[1] - y[2]), xl...) for ylim in yl]
xdata, a .* xdata .+ b
end
# expand the data outside the axis limits, by a certain factor too improve
# plotly(js) and interactive behaviour
factor = 100
x_vals = x_vals .+ (x_vals[2] - x_vals[1]) .* factor .* [-1, 1]
y_vals = y_vals .+ (y_vals[2] - y_vals[1]) .* factor .* [-1, 1]
return x_vals, y_vals
end
function shape_data(series)
sp = series[:subplot]
xl, yl = isvertical(series) ? (xlims(sp), ylims(sp)) : (ylims(sp), xlims(sp))
x, y = series[:x], series[:y]
factor = 100
for i in eachindex(x)
if x[i] == -Inf
x[i] = xl[1] - factor * (xl[2] - xl[1])
elseif x[i] == Inf
x[i] = xl[2] + factor * (xl[2] - xl[1])
end
end
for i in eachindex(y)
if y[i] == -Inf
y[i] = yl[1] - factor * (yl[2] - yl[1])
elseif y[i] == Inf
y[i] = yl[2] + factor * (yl[2] - yl[1])
end
end
return x, y
end