Mapping
Controller
Jig
Use a function in a new domain and range.
\index{function|(}
\index{function!domain|(}
\index{function!range|(}
\index{transformation!mapping|(}
A function accepts inputs and produces a value. Geometric functions such
as , , and are extremely common in parametric modeling. In fact, they
form an indispensable base for much modeling work. These functions all
naturally defined over their own domains and ranges. Use this pattern when
you want to use a function in the domain and range specific to a model.
The terms domain and range need to be explained. The
domain of a function is the set of input values over which it is defined
or used. The range of a function is the set of output values it
generates. For example, we might chose to use a domain of to degrees for the function. This corresponds to its natural
repetitive cycle. The range generated by over this domain is to .
\begin{tikzpicture}[domain=0:6.2832,
dimension/.style={|-|,black,thin}]
\draw[very thin,color=gray, xstep = 0.7854, ystep = 1] (-0.1,-1.1) grid (6.4,1.1);
\draw[->] (-0.2,0) -- (6.5,0) node[right] {$x$};
\draw[->] (0,-1.2) -- (0,1.2) node[above] {$f(x)$};
\draw[color=red, thick] plot (\x,{sin(\x r)}) node[below=6mm] {$f(x) = \sin(x)$};
\drDim{(0,-1)}{(6.2832,-1)}{-8mm}{Domain $= 0$ : $360$}{0.5}{3mm}
\drDim{(6.2832,-1)}{(6.2832,1)}{-12mm}{Range $= -1$ : $1$}{0.5}{5mm}
\end{tikzpicture}
Much form-making in design comes directly from relatively simple
functions, for many reasons. The repetitive use of simple functions can
unify a design across its parts and across design scales. If the ease and
cost of fabrication is a concern, simple functions can help control
complexity and/or cost (but do not necessarily do so). In contrast, the
so-called free-form curves and surfaces provide interfaces to
more complex functions, but cede some control of the form-making process
to the underyling algorithms.
Simple functions come with natural domains and ranges. It is much, much
easier to think about a function in its natural domain and range than in
some transformed version.
To use a function in a model requires reframing it so that it
makes sense in the model. Reframing turns out to be a
suprisingly difficult and error-prone process for many
designers. Yet, there is a universal method of precisely seven
parameters that works for almost all reframing tasks. Further
this method is based on one simple equation---essentially the
same equation that defines a one-dimensional affine map or,
equivalently, an affine function.
The term "affine map" is extremely common in linear algebra and
its dependent fields (computer graphics and geometric
modeling). It has a precise mathematical meaning, and we use
this meaning here---precision of language is important! An
affine map is a linear transformation followed by a
translation. In turn, a linear transformation preserves the
operations of vector
addition\index{vector!addition} and scalar
multiplication\index{vector!scalar
multiplication} --- you can add or multiply before or
after the transformation and the result is the same. In one
dimension (for instance, the real number line), the only linear
transformation is scaling. A 1D affine map changes a function's
scale and moves it along the real number line. The function
becomes under the affine map.
\begin{tikzpicture}[domain=0:6.2832,
dimension/.style={|-|,black,thin}]
\draw[very thin,color=gray, xstep = 0.7854, ystep = 1] (-0.1,-1.1) grid (6.4,1.1);
\draw[->] (-0.2,0) -- (6.5,0) node[right] {$x$};
\draw[->] (0,-1.2) -- (0,1.2) node[above] {$\afFunc{f}{x}$};
\draw[color=blue] plot (\x,{sin(\x r)}) node[below=6mm] {$\afFunc{f}{x} = \sin(x)$};
\draw[color=red, domain=0.7854:3.9270, thick] plot (\x,{1.5*sin(2.0*(\x-0.7854) r)}) node[below=15mm] {$\afFunc{g}{x} = 1.5\sin(2(x-45))$};
\drDim{(0.7854,-1)}{(3.9270,-1)}{-13mm}{New Domain $= 45$ : $225$}{0.5}{3mm}
\drDim{(0,-1)}{(6.2832,-1)}{-23mm}{Domain $= 0$ : $360$}{0.5}{3mm}
\drDim{(6.2832,-1)}{(6.2832,1)}{-12mm}{Range $= -1$ : $1$}{0.5}{5mm}
\drDim{(6.2832,-1.5)}{(6.2832,1.5)}{-22mm}{New Range $= -1.5$ : $1.5$}{0.5}{5mm}
\end{tikzpicture}
The problem for modeling is that determining the new function
is not easy. If you
watch a modeler trying to get such a mapping to "work" you will
see much trial and error. Looking at the function in the figure
above $\afFunc{g}{x} = 1.5\sin(2(x-45))$ gives some indication
why the task is so hard in practice. Which number does what?
Why?
This pattern replaces all of the function-specific changes with
a uniform structure and set of parameters. You never have to
work with a function in any but its simplest
form. Mapping separates where the
model uses a function from where the function is defined.
Consider two rectangles, called the function and the
model respectively. The function rectangle\index{rectangle} is given by
the domain and range of the function represented within it. The
model rectangle can be any size and location you choose. The
goal is to place the original function into the model. The basic
idea is to always use the function rectangle when computing the
function. To find a point on the function in the model, map from
the model domain to the function domain (blue arrow 1 below),
compute the function (red and green arrows in the function box)
and them map from the result back to the model range (blue arrow
2 below).
\begin{drGen}{0.4}{domain=0:6.2832}
\pgfmathsetmacro{\figD}{2.6}
\pgfmathsetmacro{\figSin}{sin(\figD r)} %
\pgfmathsetmacro{\figFnXScale}{0.5}
\pgfmathsetmacro{\figFnYScale}{1} %
\pgfmathsetmacro{\figFnDL}{0}
\pgfmathsetmacro{\figFnDU}{6.2832} %
\pgfmathsetmacro{\figFnRL}{-1}
\pgfmathsetmacro{\figFnRU}{1} %
\pgfmathsetmacro{\figMdDL}{7}
\pgfmathsetmacro{\figMdDU}{14} %
\pgfmathsetmacro{\figMdRL}{1}
\pgfmathsetmacro{\figMdRU}{5} %
\pgfmathsetmacro{\figGapOffset}{0.1}
\pgfmathsetmacro{\figArrowOffset}{2}
\pgfmathsetmacro{\figMdXScale}{(\figMdDU-\figMdDL)/(\figFnDU-\figFnDL)}
\pgfmathsetmacro{\figMdYScale}{(\figMdRU-\figMdRL)/(\figFnRU-\figFnRL)}
{ % function box
\pgftransformxscale{\figFnXScale}
\pgftransformyscale{\figFnYScale}
\draw[color=black] plot (\x,{sin(\x r)}) node[below=4mm] {\textsf{function}};
\draw (\figFnDL,\figFnRL) rectangle (\figFnDU,\figFnRU);
\drLiSeg{(\figD,\figFnRL)}{(\figD,\figSin)}{vector,red}
\drLiSeg{(\figD,\figSin)}{(\figFnDL,\figSin)}{vector,green}
\drPtOpt{(\figD,\figSin)}{emphPointo}
}
{ % connector arrows
% model to function
\drLiSeg{(\figMdDL+\figD*\figMdXScale,\figFnRL-\figGapOffset)}{(\figMdDL+\figD*\figMdXScale,\figFnRL-\figArrowOffset)}{blue}
\drLiSeg{(\figMdDL+\figD*\figMdXScale,\figFnRL-\figArrowOffset)}{(\figD*\figFnXScale,\figFnRL-\figArrowOffset)}{blue}
\drLaLi{(\figMdDL+\figD*\figMdXScale,\figFnRL-\figArrowOffset)}{(\figD*\figFnXScale,\figFnRL-\figArrowOffset)}{\textsf{(1)}}{0.5}{-6mm}
\drLiSeg{(\figD*\figFnXScale,\figFnRL-\figArrowOffset)}{(\figD*\figFnXScale,\figFnRL-\figGapOffset)}{vector,blue}
% function to model
\drLiSeg{(\figFnDL-\figGapOffset,\figSin*\figFnYScale)}{(\figFnDL-\figArrowOffset,\figSin)}{blue}
\drLiSeg{(\figFnDL-\figArrowOffset,\figSin*\figFnYScale)}{(\figFnDL-\figArrowOffset,\figMdRL+\figMdYScale*\figSin)}{blue}
\drLiSeg{(\figFnDL-\figArrowOffset,\figMdRL+\figMdYScale*\figSin)}{(\figMdDL-\figGapOffset,\figMdRL+\figMdYScale*\figSin)}{vector,blue}
\drLaLi{(\figFnDL-\figArrowOffset,\figMdRL+\figMdYScale*\figSin)}{(\figMdDL-\figGapOffset,\figMdRL+\figMdYScale*\figSin)}{\textsf{(2)}}{0.5}{-6mm}
}
{ % model box
\pgftransformxshift{\figMdDL cm}
\pgftransformyshift{\figMdRL cm}
\pgftransformxscale{\figMdXScale}
\pgftransformyscale{\figMdYScale}
\draw[color=orange, thick] plot (\x,{sin(\x r)}) node[below=8mm] {\textsf{model} \hspace{6mm}\quad};
\draw (0,-1.0) rectangle (6.2832,1.0);
\drLiSeg{(\figD,\figSin)}{(\figD,-1)}{vector,red, dashed}
\drLiSeg{(0,\figSin)}{(\figD,\figSin)}{vector,green}
\drPtOpt{(\figD,\figSin)}{emphPointo}
}
\end{drGen}
What follows gives precise definition to the diagram. Variables
relating to a domain start with or have within a ; those over the range start with or have
within an . In one dimension, an affine
map has a single equation, which we introduce first over the
domain interval to . Imagine a function with parameter over this interval. An affine map from the
interval to to
the interval with parameter has the equation
\[\afFn{r}(d)= r_l + d(r_u - r_l), 0 \leq d \leq 1\]
r(d)=Rl+d(Ru-Rl), 0 <= d <= 1
Reversing the map to go from the range () to the domain () gives
\[\afFn{d}(r)=\frac{r-r_l}{r_u-r_l},r_l\leq r \leq r_u \]
d(r)=(r-Rl)/(Ru-Rl), Rl <= r <= Ru
The map has domain with bounds
to and range
with bounds and .
Generalizing, to any domain produces the maps between and as follows:
\[\afFn{d}(r)=\frac{(r-r_l)(d_u-d_l)}{r_u-r_l} + d_l,r_l\leq r \leq r_u\]
d(r)=((r-Rl)*(Du-Dl))/(Ru-Rl) + Dl, Rl <= r <= Ru
\[\afFn{r}(d)=\frac{(d-d_l)(r_u-r_l)}{d_u-d_l} + r_l,d_l\leq d \leq d_u\]
r(d)=((d-Dl)*(Ru-Rl))/(Du-Dl) + Rl, Dl <= d <= Du
Mathematically, these are simple equations, but require attention
every time they are used. This purpose of this pattern is to
abstract these equations so that designers can freely use generic
and simple functions in their models.
WARNING. When using affine maps to apply a function, there are actually two
maps to consider. One goes between the domain of the model and its
domain in the function. The other goes between the range of the function and
its range in the model.
Since the range of the function is determined by the function itself, this means
that mapping between function and model can be described by exactly seven parameters: the
lower and upper bounds of the function's domain ( and
); the lower and upper bounds of the model's domain
( and ); the lower and
upper bounds of the model's range ( and ); and the functional parameter itself.
These seven parameters describe every possible situation in
which you want to use a function in a model. Every possible
situation! Applying them though takes some insight. The
archetypal problem is this: you have a value in the model and
need to find the corresponding value of the function in the
model. Giving more detail to the diagram above, the solution has
three parts: (1) an affine map from the model to the function;
(2) an application of the function; and (3) an affine map from
the result of the function to the result in the model. The
following diagram demonstrates this flow, showing how the
various parameters and bounds relate.
\begin{drGen}{0.4}{domain=0:6.2832}
\pgfmathsetmacro{\figD}{2.6}
\pgfmathsetmacro{\figSin}{sin(\figD r)} %
\pgfmathsetmacro{\figFnDL}{0}
\pgfmathsetmacro{\figFnDU}{6.2832} %
\pgfmathsetmacro{\figFnRL}{-1}
\pgfmathsetmacro{\figFnRU}{1} %
\pgfmathsetmacro{\figMdDL}{14}
\pgfmathsetmacro{\figMdDU}{21} %
\pgfmathsetmacro{\figMdRL}{3}
\pgfmathsetmacro{\figMdRU}{7} %
\pgfmathsetmacro{\figFnXScale}{(\figFnDU-\figFnDL)/(6.2832-0)}
\pgfmathsetmacro{\figFnYScale}{(\figFnRU-\figFnRL)/(1-(-1))} %
\pgfmathsetmacro{\figMdXScale}{(\figMdDU-\figMdDL)/(\figFnDU-\figFnDL)}
\pgfmathsetmacro{\figMdYScale}{(\figMdRU-\figMdRL)/(\figFnRU-\figFnRL)}
\pgfmathsetmacro{\figGapOffset}{0.1}
\pgfmathsetmacro{\figBoxOffset}{1}
\pgfmathsetmacro{\figTextXOffset}{2.5}
\pgfmathsetmacro{\figTextYOffset}{1.5}
\pgfmathsetmacro{\figLabelXOffset}{1.25}
\pgfmathsetmacro{\figLabelYOffset}{0.5}
\pgfmathsetmacro{\figTextVecLen}{0.5}
\pgfmathsetmacro{\figFnBoxXOffset}{2*\figBoxOffset/\figFnXScale}
\pgfmathsetmacro{\figFnBoxYOffset}{\figBoxOffset/\figFnYScale}
\pgfmathsetmacro{\figMdBoxXOffset}{\figBoxOffset/\figMdXScale}
\pgfmathsetmacro{\figMdBoxYOffset}{2*\figBoxOffset/\figMdYScale}
\pgfmathsetmacro{\figArrowOffset}{3}
{ % function box
\pgftransformxshift{0 cm}
\pgftransformyshift{1 cm}
\pgftransformxscale{0.5}
\pgftransformyscale{1.5}
\draw[color=black] plot (\x,{sin(\x r)}) node[below=4mm] {};
% \draw (\figFnDL,\figFnRL) rectangle (\figFnDU,\figFnRU);
\drLiSeg{(\figD,\figFnRL-\figFnBoxYOffset)}{(\figD,\figSin)}{vector,red}
\drLiSeg{(\figD,\figSin)}{(\figFnDL-\figFnBoxXOffset,\figSin)}{vector,green}
\drPtOpt{(\figD,\figSin)}{emphPointo}
\drLa{\textsf{function}}{(\figFnDL-7,\figFnRL-\figFnBoxYOffset)}
\drLiSeg{(\figFnDU+\figFnBoxXOffset,\figFnRL-\figFnBoxYOffset)}{(\figFnDU+\figFnBoxXOffset+\figArrowOffset,\figFnRL-\figFnBoxYOffset)}{blue}
\drLiSeg{(\figFnDU+\figFnBoxXOffset+\figArrowOffset,\figFnRL-\figFnBoxYOffset)}{(\figFnDU+\figFnBoxXOffset+\figArrowOffset,\figSin)}{blue}
\drLiSeg{(\figFnDU+\figFnBoxXOffset+\figArrowOffset,\figSin)}{(\figD+8*\figGapOffset,\figSin)}{vector,blue}
\drLaLi{(\figFnDU+\figFnBoxXOffset+\figArrowOffset,\figFnRL-\figFnBoxYOffset)}{(\figFnDU+\figFnBoxXOffset+\figArrowOffset,\figSin)}{\textsf{(2)}}{0.5}{14mm}
\drLiSeg{(\figFnDL,\figFnRL-\figFnBoxYOffset)}{(\figFnDU,\figFnRL-\figFnBoxYOffset)}{red}
\drPtOpt{(\figD,\figFnRL-\figFnBoxYOffset)}{emphPointo}
\drPtOpt{(\figFnDL,\figFnRL-\figFnBoxYOffset)}{emphPointo}
\drPtOpt{(\figFnDU,\figFnRL-\figFnBoxYOffset)}{emphPointo}
\drLiSeg{(\figFnDL,\figFnRL-\figFnBoxYOffset)}{(\figFnDL,0)}{dashed, thin}
\drLiSeg{(\figFnDU,\figFnRL-\figFnBoxYOffset)}{(\figFnDU,0)}{dashed, thin}
\drLaLi{(\figFnDL,\figFnRL-\figFnBoxYOffset-\figTextYOffset)}{(\figFnDU,\figFnRL-\figFnBoxYOffset-\figTextYOffset)}{$\afWV{fd}_l$}{0.0}{-6mm}
\drLaLi{(\figFnDL,\figFnRL-\figFnBoxYOffset-\figTextYOffset)}{(\figFnDU,\figFnRL-\figFnBoxYOffset-\figTextYOffset)}{$\afWV{fd}$}{0.5}{-6mm}
\drLaLi{(\figFnDL,\figFnRL-\figFnBoxYOffset-\figTextYOffset)}{(\figFnDU,\figFnRL-\figFnBoxYOffset-\figTextYOffset)}{\hspace{1mm}$\afWV{fd}_u$}{0.99}{-6mm}
\drLaLi{(\figFnDL,\figFnRL-\figFnBoxYOffset-\figTextYOffset-0.25*\figLabelYOffset)}{(\figFnDU,\figFnRL-\figFnBoxYOffset-\figTextYOffset-0.5*\figLabelYOffset)}{\footnotesize{\textsf{domain}}}{0.5}{-0.5mm}
\drLiSeg{(\figFnDU-\figTextVecLen,\figFnRL-\figFnBoxYOffset-\figTextYOffset-0.25*\figLabelYOffset)}{(\figFnDU,\figFnRL-\figFnBoxYOffset-\figTextYOffset-0.25*\figLabelYOffset)}{vector, red}
\drLiSeg{(\figFnDL+\figTextVecLen,\figFnRL-\figFnBoxYOffset-\figTextYOffset-0.25*\figLabelYOffset)}{(\figFnDL,\figFnRL-\figFnBoxYOffset-\figTextYOffset-0.25*\figLabelYOffset)}{vector, red}
\drLiSeg{(\figFnDL-\figFnBoxXOffset, \figFnRU)}{(\figFnDL-\figFnBoxXOffset, \figFnRL)}{green}
\drPtOpt{(\figFnDL-\figFnBoxXOffset, \figFnRU)}{emphPointo, green}
\drPtOpt{(\figFnDL-\figFnBoxXOffset, \figSin)}{emphPointo, green}
\drPtOpt{(\figFnDL-\figFnBoxXOffset, \figFnRL)}{emphPointo, green}
\drLiSeg{(\figFnDU/4,\figFnRU)}{(\figFnDL-\figFnBoxXOffset,\figFnRU)}{dashed, thin}
\drLiSeg{(\figFnDU*3/4,\figFnRL)}{(\figFnDL-\figFnBoxXOffset,\figFnRL)}{dashed, thin}
\drLaLi{(\figFnDL-\figFnBoxXOffset-\figTextXOffset, \figFnRU)}{(\figFnDL-\figFnBoxXOffset-\figTextXOffset, \figFnRL)}{$\afWV{fr}_u$}{0.0}{-6mm}
\drLaLi{(\figFnDL-\figFnBoxXOffset-\figTextXOffset, \figFnRU)}{(\figFnDL-\figFnBoxXOffset-\figTextXOffset, \figFnRL)}{$\afWV{fr}$}{0.5}{-6mm}
\drLaLi{(\figFnDL-\figFnBoxXOffset-\figTextXOffset, \figFnRU)}{(\figFnDL-\figFnBoxXOffset-\figTextXOffset, \figFnRL)}{\hspace{1mm}$\afWV{fr}_l$}{0.99}{-6mm}
\drLaLiSl{(\figFnDL-\figFnBoxXOffset-\figTextXOffset-\figLabelXOffset, \figFnRL)}{(\figFnDL-\figFnBoxXOffset-\figTextXOffset-\figLabelXOffset, \figFnRU)}{}{\footnotesize{\textsf{range}}}{0.5}{0.3mm}
\drLiSeg{(\figFnDL-\figFnBoxXOffset-\figTextXOffset-\figLabelXOffset, \figFnRU-0.5*\figTextVecLen)}{(\figFnDL-\figFnBoxXOffset-\figTextXOffset-\figLabelXOffset, \figFnRU)}{vector, green}
\drLiSeg{(\figFnDL-\figFnBoxXOffset-\figTextXOffset-\figLabelXOffset, \figFnRL+0.5*\figTextVecLen)}{(\figFnDL-\figFnBoxXOffset-\figTextXOffset-\figLabelXOffset, \figFnRL)}{vector, green}
}
{ % connector arrows
\pgfmathsetmacro{\figFnRParam}{(\figSin-\figFnRL)/(\figFnRU-\figFnRL)}
\pgfmathsetmacro{\figMdRLoc}{\figMdRL+\figFnRParam*(\figMdRU-\figMdRL)}
% model to function
\drLiSeg{(\figMdDL+\figD*\figMdXScale,\figFnRL-\figGapOffset)}{(\figMdDL+\figD*\figMdXScale,\figFnRL-\figArrowOffset-\figLabelYOffset-\figTextYOffset)}{blue}
\drLiSeg{(\figMdDL+\figD*\figMdXScale,\figFnRL-\figArrowOffset-\figLabelYOffset-\figTextYOffset)}{(0.5*\figD*\figFnXScale,\figFnRL-\figArrowOffset-\figLabelYOffset-\figTextYOffset)}{blue}
\drLaLi{(\figMdDL+\figD*\figMdXScale,\figFnRL-\figArrowOffset-\figLabelYOffset-\figTextYOffset)}{(0.5*\figD*\figFnXScale,\figFnRL-\figArrowOffset-\figLabelYOffset-\figTextYOffset)}{\textsf{(1)}}{0.5}{-6mm}
\drLiSeg{(0.5*\figD*\figFnXScale,\figFnRL-\figArrowOffset-\figLabelYOffset-\figTextYOffset)}{(0.5*\figD*\figFnXScale,\figFnRL-\figArrowOffset-2*\figLabelYOffset)}{vector,blue}
% function to model
\drLiSeg{(\figFnDL-\figBoxOffset-\figTextXOffset,\figSin/\figFnYScale+1)}{(\figFnDL-\figBoxOffset-\figTextXOffset-0.5*\figArrowOffset,\figSin/\figFnYScale+1)}{blue}
\drLiSeg{(\figFnDL-\figBoxOffset-\figTextXOffset-0.5*\figArrowOffset,\figSin/\figFnYScale+1)}{(\figFnDL-\figBoxOffset-\figTextXOffset-0.5*\figArrowOffset,\figMdRLoc)}{blue}
\drLiSeg{(\figFnDL-\figBoxOffset-\figTextXOffset-0.5*\figArrowOffset,\figMdRLoc)}{(\figMdDL-\figMdBoxXOffset-\figTextXOffset-\figLabelXOffset, \figMdRLoc)}{vector,blue}
\drLaLi{(\figFnDL-\figBoxOffset-\figTextXOffset-0.5*\figArrowOffset,\figMdRLoc)}{(\figMdDL-\figMdBoxXOffset-\figTextXOffset-\figLabelXOffset, \figMdRLoc)}{\textsf{(3)}}{0.5}{-6mm}
}
{ % model box
\pgfmathsetmacro{\figMdRangeShift}{\figMdRL-\figFnRL*\figMdYScale}
\pgftransformxshift{\figMdDL cm}
\pgftransformyshift{\figMdRangeShift cm}
\pgftransformxscale{\figMdXScale}
\pgftransformyscale{\figMdYScale}
\draw[color=orange, thick] plot (\x,{sin(\x r)}) node[above=5mm] {\textsf{model} \hspace{6mm}\quad};
% \draw (0,-1.0) rectangle (6.2832,1.0);
\drLiSeg{(\figD,\figSin)}{(\figD,\figFnRL-0.5*\figMdBoxYOffset)}{vector,red, dashed}
\drLiSeg{(-\figMdBoxXOffset,\figSin)}{(\figD,\figSin)}{vector,green}
\drPtOpt{(\figD,\figSin)}{emphPointo}
}
{
\pgfmathsetmacro{\figFnDParam}{\figD/(\figFnDU-\figFnDL)}
\pgfmathsetmacro{\figMdDLoc}{\figMdDL+\figFnDParam*(\figMdDU-\figMdDL)}
\drLiSeg{(\figMdDL,\figMdRL-\figMdBoxYOffset)}{(\figMdDU,\figMdRL-\figMdBoxYOffset)}{red}
\drPtOpt{(\figMdDL,\figMdRL-\figMdBoxYOffset)}{emphPointo}
\drPtOpt{(\figMdDU,\figMdRL-\figMdBoxYOffset)}{emphPointo}
\drPtOpt{(\figMdDLoc,\figMdRL-\figMdBoxYOffset)}{emphPointo}
\pgfmathsetmacro{\figFnMdRC}{\figMdRL+0.5*(\figMdRU-\figMdRL)}
\drLiSeg{(\figMdDL,\figMdRL-\figMdBoxYOffset)}{(\figMdDL,\figFnMdRC)}{dashed, thin}
\drLiSeg{(\figMdDU,\figMdRL-\figMdBoxYOffset)}{(\figMdDU,\figFnMdRC)}{dashed, thin}
\drLaLi{(\figMdDL,\figMdRL-\figMdBoxYOffset-\figTextYOffset)}{(\figMdDU,\figMdRL-\figMdBoxYOffset-\figTextYOffset)}{$\afWV{md}_l$}{0.0}{-6mm}
\drLaLi{(\figMdDL,\figMdRL-\figMdBoxYOffset-\figTextYOffset)}{(\figMdDU,\figMdRL-\figMdBoxYOffset-\figTextYOffset)}{$\afWV{md}$}{0.5}{-6mm}
\drLaLi{(\figMdDL,\figMdRL-\figMdBoxYOffset-\figTextYOffset)}{(\figMdDU,\figMdRL-\figMdBoxYOffset-\figTextYOffset)}{\hspace{1mm}$\afWV{md}_u$}{0.99}{-6mm}
\drLaLi{(\figMdDL,\figMdRL-\figMdBoxYOffset-\figTextYOffset-\figLabelYOffset)}{(\figMdDU,\figMdRL-\figMdBoxYOffset-\figTextYOffset-\figLabelYOffset)}{\footnotesize{\textsf{domain}}}{0.5}{-0.5mm}
\drLiSeg{(\figMdDU-3*\figTextVecLen,\figMdRL-\figMdBoxYOffset-\figTextYOffset-\figLabelYOffset)}{(\figMdDU,\figMdRL-\figMdBoxYOffset-\figTextYOffset-\figLabelYOffset)}{vector, red}
\drLiSeg{(\figMdDL+3*\figTextVecLen,\figMdRL-\figMdBoxYOffset-\figTextYOffset-\figLabelYOffset)}{(\figMdDL,\figMdRL-\figMdBoxYOffset-\figTextYOffset-\figLabelYOffset)}{vector, red}
\pgfmathsetmacro{\figFnRParam}{(\figSin-\figFnRL)/(\figFnRU-\figFnRL)}
\pgfmathsetmacro{\figMdRLoc}{\figMdRL+\figFnRParam*(\figMdRU-\figMdRL)}
\drLiSeg{(\figMdDL-\figMdBoxXOffset, \figMdRU)}{(\figMdDL-\figMdBoxXOffset, \figMdRL)}{green}
\drPtOpt{(\figMdDL-\figMdBoxXOffset, \figMdRU)}{emphPointo, green}
\drPtOpt{(\figMdDL-\figMdBoxXOffset, \figMdRLoc)}{emphPointo, green}
\drPtOpt{(\figMdDL-\figMdBoxXOffset, \figMdRL)}{emphPointo, green}
\pgfmathsetmacro{\figFnMdMax}{\figMdDL+0.25*(\figMdDU-\figMdDL)}
\pgfmathsetmacro{\figFnMdMin}{\figMdDL+0.75*(\figMdDU-\figMdDL)}
\drLiSeg{(\figFnMdMax,\figMdRU)}{(\figMdDL-\figMdBoxXOffset,\figMdRU)}{dashed, thin}
\drLiSeg{(\figFnMdMin,\figMdRL)}{(\figMdDL-\figMdBoxXOffset,\figMdRL)}{dashed, thin}
\drLaLi{(\figMdDL-\figMdBoxXOffset-\figTextXOffset, \figMdRU)}{(\figMdDL-\figMdBoxXOffset-\figTextXOffset, \figMdRL)}{$\afWV{mr}_u$}{0.0}{-6mm}
\drLaLi{(\figMdDL-\figMdBoxXOffset-\figTextXOffset, \figMdRU)}{(\figMdDL-\figMdBoxXOffset-\figTextXOffset, \figMdRL)}{$\afWV{mr}$}{0.5}{-6mm}
\drLaLi{(\figMdDL-\figMdBoxXOffset-\figTextXOffset, \figMdRU)}{(\figMdDL-\figMdBoxXOffset-\figTextXOffset, \figMdRL)}{\hspace{1mm}$\afWV{mr}_l$}{0.99}{-6mm}
\drLaLiSl{(\figMdDL-\figMdBoxXOffset-\figTextXOffset-0.5*\figLabelXOffset, \figMdRL)}{(\figMdDL-\figMdBoxXOffset-\figTextXOffset-0.5*\figLabelXOffset, \figMdRU)}{}{\footnotesize{\textsf{range}}}{0.5}{0.3mm}
\drLiSeg{(\figMdDL-\figMdBoxXOffset-\figTextXOffset-0.5*\figLabelXOffset, \figMdRU-2*\figTextVecLen)}{(\figMdDL-\figMdBoxXOffset-\figTextXOffset-0.5*\figLabelXOffset, \figMdRU)}{vector, green}
\drLiSeg{(\figMdDL-\figMdBoxXOffset-\figTextXOffset-0.5*\figLabelXOffset, \figMdRL+2*\figTextVecLen)}{(\figMdDL-\figMdBoxXOffset-\figTextXOffset-0.5*\figLabelXOffset, \figMdRL)}{vector, green}
}
\end{drGen}
For example, consider a roof whose profile is taken from a
sine curve. The roof spans from a point in the model to another point . A point with parameter gives the location of the roof point along
the line between these two points. The maximum root height is
given by a parameter. Over the span of the roof, it goes through
two complete cycles of the
function. Then the following are the six mapping parameter
settings. The seventh parameter is , which gives
the parametric location of a varying point on the roof.
(defined by the function and thus computed automatically)
(defined by the function and thus computed automatically)
( is parameterized by over the domain to )
(the height is added to the -value of )
(height is chosen in the model)
In summary, the whole process comprises these three steps:
-
Given a model domain value, find the equivalent domain value in the function.
-
Apply the function to get a value in the function's range.
-
Find the equivalent model range value.
Reciprocal
Feather a curve. Make a curve taper gently.
The multiplicative reciprocal is the function . It is seductive as it
provides a simple function that tapers to a non-zero value
(is asymptotic to the -axis), making
it possible to feather effects on a curve or surface. It has
a trap as it increases exponentially as its argument
approaches zero and is undefined at zero. Thus it is
important to set the function domain so that zero and values
close to it are not included in the function. In general,
using this function produces poor results. It is included
here to demonstrate that sometimes the function domain
choice is crucial.
Here are the parameter settings for a useful mapping. The model point is parametric over the domain to .
(defined by the function and thus computed automatically)
(defined by the function and thus computed automatically)
( is parameterized by over the domain to )
(the height is added to the -value of )
(height is chosen in the model)
Click to download MappReverseX.gct
Sine and Cosine
Use sine and cosine functions in complete periodic cycles.
The basic trigonometric functions and provide repeating
periods of and
periods of
between function minima and maxima. This makes such
functions suitable for repetitive design elements. The
functions both have domains from to and ranges from to . Choosing a
finite part of the domain such that the function starts and
ends at a minimum, maximum or zero-crossing yields clean
control over the end-conditions in the model.
Here are sample parameter settings. The model point is assumed to be parametric over
the range to . In this case, is bound to a circle, and vertical lines from each instance of have length given by the function result.
(yields three complete cycles)
(defined by the function and thus computed automatically)
(defined by the function and thus computed automatically)
( is parameterized by over the domain to )
(the minimum length of the lines on )
(the maximum length of the lines on )
When sampling periodic functions and those with minima or
maxima within the chosen domain, the choice of sampling
interval is crucial if the sampled points will be used to
regenerate the mapped function in the model. Samples must be
chosen to include the minima and maxima. Not doing this can
dramatically affect the shape of the reconstructed curve.
Click to download MappCosine.gct
Function Parts
Choosing a small part of a function can yield an intended form.
Simple functions can yield surprisingly rich forms. A
classic example in three dimensions is Foster and Partner's
extensive use of parts of a torus a generator of form
(see Chapter \ref{sec:BradyPeters:brady}).
The function is the sine function: . The trick is that an appropriate choice of
function domain can yield local segments of the sine curve
that do not display the archetypal repetition of the entire
curve.
Here are sample parameter settings. The model point is assumed to be parametric over
the range to . In this case, the model range minimum and
maximum would be chosen by the modeler to suit the
application to hand. The figures on the side bar vary only by
the upper
function domain bound.
\index{function|)}
\index{function!domain|)}
\index{function!range|)}
\index{transformation!mapping|)}
(increases in steps of
(defined by the function and therefore can be computed automatically)
(defined by the function and therefore can be computed automatically)
( is parameterized by over the domain to )
(minimum height is chosen in the model)
(maximum height is chosen in the model)
Click to download MappSinePart.gct
Interactive mapping control
Use a Controller to consistently
express the six mapping parameters.
Complete here an example of mapping function and
frames. Explain the generic mapping function and the mapping
control fames. Debug these.
SineRoof
Create a roof surface controlled by two sine curve mappings.
You have a site in which has certain boundaries and
height. You want to create a sine-like roof that fits into
this site. Create a sine curve function. In order to create
this roof too need to have
Jig as
loft curves for the roof. In this sample, you have two
curves to create the roof, one at the start and another one
at the end point of the site. These curves can be created as
two worlds from one function( you have different dimension
for start and end of the site) first step is to divide
worlds' domain (world one and two) to the number of samples
that you want. Find their equivalents in the function's
domain(see SinePart sample from
Mapping) To have this world's
range you can create new coordinate systems for start and
end point of the site and assign this values to these
coordinate systems. ( Create new coordinate system that
represents the site boundaries in a way that you want) Apply
Sine function to find function range points(FRV's). Find
parametric equivalents for function ranges in each world's
range. Create a surface from these two sine-like curves.
Click to download MappSineRoof.gct