plot::Waterman – Waterman polyhedra

plot::Waterman(r) creates the Waterman polyhedron of radius .

Call:

plot::Waterman(r, <a = amin .. amax>, Options)

Parameters:

r:	an arithmetical expression: the radius of the polyhedron (see below for details). r is equivalent to the attribute Radius.

Attributes for plot::Waterman

AffectViewingBox = TRUE	influence of objects on the ViewingBox of a scene
Center = [0, 0, 0]	center of objects, rotation center
CenterX = 0	center of objects, rotation center, x-component
CenterY = 0	center of objects, rotation center, y-component
CenterZ = 0	center of objects, rotation center, z-component
Color = RGB::SafetyOrange	the main color
FillColor = RGB::SafetyOrange	color of areas and surfaces
FillColor2 = RGB::CornflowerBlue	second color of areas and surfaces for color blends
FillColorDirection = [0, 0, 1]	the direction of color transitions on surfaces
FillColorDirectionX = 0	x-component of the direction of color transitions on surfaces
FillColorDirectionY = 0	y-component of the direction of color transitions on surfaces
FillColorDirectionZ = 1	z-component of the direction of color transitions on surfaces
FillColorFunction	functional area/surface coloring
FillColorType = Flat	surface filling types
Filled = TRUE	filled or transparent areas and surfaces
Frames = 50	the number of frames in an animation
Legend	makes a legend entry
LegendEntry = FALSE	add this object to the legend?
LegendText	short explanatory text for legend
LineColor = RGB::Grey40.[0.4]	color of lines
LineColor2 = RGB::DeepPink	color of lines
LineColorDirection = [0, 1, 1]	the direction of color transitions on lines
LineColorDirectionX = 0	x-component of the direction of color transitions on lines
LineColorDirectionY = 1	y-component of the direction of color transitions on lines
LineColorDirectionZ = 1	z-component of the direction of color transitions on lines
LineColorFunction	functional line coloring
LineColorType = Flat	line coloring types
LineStyle = Solid	solid, dashed or dotted lines?
LinesVisible = TRUE	visibility of lines
LineWidth = 0.25	width of lines
Name	the name of a plot object (for browser and legend)
ParameterBegin	initial value of the animation parameter
ParameterEnd	end value of the animation parameter
ParameterName	name of the animation parameter
ParameterRange	range of the animation parameter
PointSize = 1.5	the size of points
PointStyle = FilledCircles	the presentation style of points
PointsVisible = FALSE	visibility of mesh points
Radius	radius of circles, spheres etc.
Shading = Smooth	smooth color blend of surfaces
TimeBegin = 0.0	start time of the animation
TimeEnd = 10.0	end time of the animation
TimeRange = 0.0 .. 10.0	the real time span of an animation
Title	object title
TitleAlignment = Center	horizontal alignment of titles w.r.t. their coordinates
TitleFont = ["sans-serif", 11]	font of object titles
TitlePosition	position of object titles
TitlePositionX	position of object titles, x component
TitlePositionY	position of object titles, y component
TitlePositionZ	position of object titles, z component
Visible = TRUE	visibility
VisibleAfter	object visible after this time value
VisibleAfterEnd = TRUE	object visible after its animation time ended?
VisibleBefore	object visible until this time value
VisibleBeforeBegin = TRUE	object visible before its animation time starts?
VisibleFromTo	object visible during this time range

Example 1

With increasing radius, Waterman polyhedra get ever closer to spheres:

plot(plot::Waterman(r, r=0..10,
                    PointsVisible, PointSize=1,
                    LineColor=RGB::Black,
                    Color=RGB::Red.[0.75]),
     plot::Sphere(r, [0,0,0], r=0..10,
                  Color=RGB::Yellow.[0.3]),
     CameraDirection=[2,10,1])

MuPAD graphics

Example 2

Waterman polyhedra have a rather general definition and can be made from spheres centered anywhere:

plot(plot::Waterman(5, Center=[0,0,0]))

MuPAD graphics

plot(plot::Waterman(5, Center=[0,0,1]))

MuPAD graphics

To translate or scale a Waterman polyhedron, use plot::Translate3d and plot::Scale3d:

n := 3:
r := i -> 3/2+sqrt(i+1):
plot(plot::Translate3d([i mod n, i div n, 0],
               plot::Scale3d([1/(3*r(i)) $ 3],
                  plot::Waterman(r(i), Color=RGB::random())))
     $ i = 0..n^2-1, Axes=None)

MuPAD graphics

Example 3

As usual, many attributes can be animated, although by the nature of Waterman polyhedra, the resulting animation will not be smooth:

plot(plot::Waterman(5, Center=[a/PI, cos(a), 0],
a=0..2*PI),
AnimationStyle=BackAndForth, CameraDirection=[0,0.1,1])

MuPAD graphics

Example 4

The LineColorFunction and FillColorFunction attributes can be set to functions which get indices of the currently painted surfacepolygon and its current vertex as fourth and fifth argument, repectively. This allows to color the polygons individually:

colors := [RGB::random() $ i = 1..42]:
plot(plot::Waterman(5, FillColorFunction=((x,y,z,i) -> colors[i])))

MuPAD graphics

Another way of getting random colors which remain constant for each polygon is to use a procedure with option remember:

col :=
proc(n)
option remember;
begin
RGB::fromHSV([360*frandom(), 1, 1]);
end:
plot(plot::Waterman(7, FillColorFunction=((x,y,z,i) -> col(i))))

MuPAD graphics

Background:

plot::Waterman uses plot::hull (and therefore, the Geometry Centre's qhull code) to compute the convex hull of the coordinates. Most of the remaining code has been contributed by Mirek Majewski.

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