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. \renewcommand{\sampleNewPage}[0]{false} 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. \renewcommand{\sampleNewPage}[0]{false} 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. \renewcommand{\sampleNewPage}[0]{false} 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(r²cos²a + s²sin²a))=(0.5)/(sqrt(cos²a + 0.25sin²a)). 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