Increment
Organized Collection of Points
Mapping
Drive change through a series of closely related values.
Parts may be similar in structure but vary in their inputs. Very
often, input variations are gradual from part to part and parts
in sequences or other arrangements are similar to their
neighbours. Use this pattern when you are making collections of
related parts.
Being able to change one part to another through a gradual
transformation of inputs lends surety and control. As a
form-making strategy, gradual change can provide a background
against which a strong figure can play.
Gradual change has two most simple forms. The first is the
integers, stepping in units of one from low to high, $...-1,0,1,2,3,...$ The
second is the reals, varying continuously (infinitely
divisible). Taken by themselves, the integers and reals can
express only limited kinds of change. Functions transform
sequences of integers and sampled reals into new sequences that
may be dramatically different from the originals.
In turn, the Increment uses the
output of a function to drive change in a variety of ways,
limited only by imagination. Length, size, angle, distance,
orientation, colour, transparency and surface texture can all be
changed in orderly (and disorderly) ways through incrementing
along squences of integers or reals.
The samples in this pattern develop increasingly complex curves\index{curve|(}
\index{point|(}
traced by a single point moving through space. Successive
samples increase both the number of parameters on which an increment
applies and the complexity of the incrementing
functions. Throughout each sample, the structure of the model
remains constant, only the values of the parameters change.
Even a single point can demonstrate the basic structure of an
increment. Start with a point in space, located as it
must be with respect to a coordinate system.
\begin{bodyNote}
\pbOneCol[c]{
\begin{dr}
\drCSOpt{0}{0}{1}{thick}
\node [point, modelPointo] at (2,0) {};
\end{dr}
}
\end{bodyNote}
The point can be thought of having either Cartesian\index{coordinates!Cartesian} $(x,y,z)$ coordinates or cylindrical\index{coordinates!cylindrical} $(r,a,z)$, where $r$ is the
radius, $a$ is the azimuth angle and
$z$ is the height of the point. Using cylindrical
coordinates and incrementing the azimuth angle makes the point
trace out an arc. If the azimuth angle goes from $0\; degrees$ to $360\; degrees$ the arc becomes a circle\index{circle}. Incrementing the radius
turns the arc into a spiral\index{spiral|(}.
\begin{bodyNote}
\pbTwoCol
{\input{\GCFigsPath/Patterns/IncrementCircle0.tex}}
{\currentWidth/2}
{\input{\GCFigsPath/Patterns/IncrementSpiral0.tex}}
{\currentWidth/2}
\end{bodyNote}
Incrementing the height of the point turns the arc into a helix\index{helix|(}
and brings us to the first sample below.
Circular Helix
Move a point uniformly around a centre and upward in space.
As a point moves around the circle, increment its height
by a uniform amount. The result is to trace out a simple
circular helix.
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Conic Helix
Add a reducing radius increment to change a circular to a conic helix.
In addition to the two increments (angle and height) for a
circular helix, reducing the radius incrementally from
an initial value to a minimum value produces a conic helix,
that is a helix whose points lie on a cone\index{conic section!cone}.
Tapered Radius Spiral
Taper the radius of conic helix to produce a spiral.
As a point on a conic helix moves upward its radius reduces
incrementally. The point can be imagined to have a parameter
that is $1$ at the helix base and $0$
at the top. Squaring this parameter will still result in a
series that goes from $1$ to $0$, but
the series will taper across this interval. Mathematically, the curve changes from a helix to a spiral.
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Tapered Height Spiral
Taper the height of a conic helix to produce a spiral.
Instead of tapering the radius, taper the height with the
same device, by squaring the parameter. In this case, the
parameter is $0$ at the helix base and
$1$ at the top. The helix\index{helix|)}, now a spiral, appears to have been
differentially stretched from its base to its top.
Tapered Radius and Height Spiral
Combining increments yields unpredictable forms.
Combining both radius and height tapers can be done
independently in the model. They do not affect each other
computationally, but combine in the geometric result. They
produce a spiral that would be hard to conceive of itself, but
naturally emerges from the parameterization.
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Elliptical Tapered Radius and Height Spiral
Change a circular spiral \index{spiral|)}to an elliptical one.
\index{curve|)}
\index{point|)}
In the prior samples, the radius, angle and height
parameters were independent in the model. In this sample,
the radius becomes a function of the angle, by using a polar
equation for the radius of an ellipse\index{conic section!ellipse}. If an ellipse has
major axis of $r=1$, minor axis of
$s=0.5$, the radius as a function of $a$ is
\[\frac{rs}{\sqrt{{r^2}{\cos^2}\theta + {s^2}{\sin^2}\theta}} = \frac{0.5}{\sqrt{{\cos^2}\theta + {0.25}{\sin^2}\theta}}\]
(*r***s*)/(sqrt(*r*^{2}cos^{2}a + s^{2}sin^{2}a))=(*0.5*)/(sqrt(cos^{2}a + 0.25sin^{2}a)).
Snail Curve
Context of the sample
Details of the sample
Sin Wave
Context of the sample
Details of the sample
Drill
Context of the sample
Details of the sample