Geometric Modeling and Design: Chapter Review

Vocabulary and notation

Geometric model and simplifying assumption

A geometric model represents an object or space with familiar shapes to answer a particular question. A simplifying assumption leaves out or idealizes a detail for that purpose.

Measured and stipulated inputs

A measured dimension comes from a reading. A stipulated dimension is fixed for the calculation.

Measurement interval

The range of unrounded readings that can produce a report under the stated rounding rule.

Prediction interval from these inputs

The range of model outputs allowed by the stated input bounds and assumptions. It carries no probability or confidence level.

Independent observation

A check of the target quantity using separate evidence, such as a grid survey of an area predicted from tape readings.

Area density and mass density

Area density describes a quantity per unit of area, such as residents per square kilometer. Mass density describes mass per unit of volume.

Uniform model and homogeneous material

A uniform model uses the same density throughout the region or material. A homogeneous material has uniform composition and density under the modeled conditions.

Scale and capacity

A scale matches corresponding model and actual lengths. Capacity is available volume inside a container up to the stated limit.

Constraint, feasible design, and objective

A constraint is a requirement. A feasible design meets every requirement. The objective is the quantity to maximize or minimize among feasible designs.

Layout family

The set of layouts that obey specified placement rules, such as one layer and one orientation per row.

Best tested and proven optimum

Best tested means best among the feasible designs compared. A proven optimum is achieved by a feasible design and no design allowed by the model improves on it.

Formulas and theorems

• Readings behind a rounded report

  $$r-\tfrac{\Delta}{2}\le x<r+\tfrac{\Delta}{2}$$

  Text description

  The included segment starts at a filled point at 8.55 centimeters. The report of 8.6 centimeters lies in the middle. A hollow point at 8.65 centimeters excludes that reading. The lower halfway reading rounds up to 8.6 and the upper halfway reading rounds up to 8.7.

  Use when A positive reading $x$ is reported as $r$ to the nearest step $\Delta$, with ties going to the larger report. Keep any physical limits on $x$. The rounding rule bounds the original reading. Instrument or placement error needs separate information.

  e.g. A positive length is reported as 8.6 cm to the nearest tenth, with ties rounded up. Find the possible readings in centimeters. They satisfy $8.55\le x<8.65$.

• Area after a rectangular cutout

  $$A=Lw-bv$$

  Text description

  Two schematic top views compare a cutout that fits inside a rectangle with a cutout that spans the full width. The interior hole has its own width v, measured in the same direction as the outer width w. The second cutout is an end strip that runs from the top to the bottom edge, so it has the same width w as the whole rectangle.

  Use when The cutout of length $b$ and width $v$ fits inside the rectangle of length $L$ and width $w$. With positive fixed lengths, larger $w$ raises $A$ at fixed $v$, and larger $v$ lowers $A$ at fixed $w$. For a full-width strip, set $v=w$ before bounding.

  e.g. Outer and cutout lengths are 9.5 m and 2.5 m. Their independent widths range from 3.2 to 3.4 m and 0.9 to 1.1 m, respectively, excluding each upper endpoint. The cutout always fits. Find the remaining area range in square meters. It is $27.65<A<30.05$.

• Area density and combined regions

  $$\begin{gathered} d=\frac{N}{A}\\ N=dA\\ A=\frac{N}{d}\\ d_{\text{combined}}=\frac{N_1+N_2}{A_1+A_2} \end{gathered}$$

  Use when $N$ is the count or total quantity for the same region as its positive area $A$. Divide by $d$ only if positive. Transfer a rate to another region only if it represents that region. Combine nonoverlapping regions by adding totals, or subtract a known part's count and area to find the remainder.

• Mass, volume, and density

  $$\begin{gathered} \rho=\frac{m}{V}\\ m=\rho V\\ V=\frac{m}{\rho} \end{gathered}$$

  Use when Use positive volume $V$ and the density for that volume. Divide by $\rho$ only if positive. A material density applies to the occupied material volume. The average density of a hollow object using its outer volume divides by the whole volume, including the cavity.

• Model-to-actual length, area, and volume

  $$\begin{gathered} k=\frac{L_{\text{actual}}}{L_{\text{model}}}\\ L_{\text{actual}}=kL_{\text{model}}\\ A_{\text{actual}}=k^2A_{\text{model}}\\ V_{\text{actual}}=k^3V_{\text{model}} \end{gathered}$$

  Use when A ratio $1:k$ puts model first in common units. For area or volume, the same positive length factor must apply in every measured direction. Reverse by dividing by the same power. To recover $k$ from a matching area or volume ratio, take its positive square or cube root. Square or cube the units too if using a conversion such as meters per map centimeter.

• Rectangular capacity and volume units

  $$\begin{gathered} V_{\text{fill}}=A_{\text{inside base}}h_{\text{fill}}\\ 1\text{ L}=1000\text{ cm}^3 \end{gathered}$$

  Use when Use the inside base and depth from the floor to the fill line or rim. The fill fraction equals the depth fraction when horizontal cross-sectional area is constant. Apply cubic scaling to capacity only for similar interiors at corresponding fill levels.

• Fencing cost and planted area

  $$\begin{gathered} C=2r_s s+2r_d d\\ A=sd \end{gathered}$$

  Use when For a rectangle, both parallel sides of length $s$ cost rate $r_s$ per unit length, and both perpendicular sides of length $d$ cost $r_d$. Recount only the lengths bought if an edge is supplied. Price gates separately. Use the full rectangle area when all of it is plantable.

• Box faces, purchased board, and cost

  $$\begin{gathered} S=2(\ell w+\ell h+wh)\\ B=S_{\text{needed}}+W\\ C=rB+F \end{gathered}$$

  Use when $S$ counts six faces of a closed rectangular box with face dimensions $\ell,w,h$. Add material for any divider or overlap to get $S_{\text{needed}}$, and add waste area $W$. With a positive area rate $r$ and fixed fee $F$, charge for purchased area $B$. If sold in whole sheets, price the sheets bought and verify the cutting layout fits.

• Items in one row

  $$na+(n-1)g\le L$$

  Use when For $n\ge1$ whole items of width $a>0$ along a usable row of length $L$, use the minimum gap $g\ge0$ between neighbors and no required end gaps. With no gaps, round $L/a$ down. If one item cannot fit, the maximum is zero.

• Minimum sum at fixed product

  $$\begin{gathered} (a+b)^2=4ab+(a-b)^2\\ a+b\ge 2\sqrt{ab} \end{gathered}$$

  Use when For positive $a,b$ of fixed product, equality requires $a=b$. If the variable material terms are $3w$ and $2\ell$, balance those terms. Verify that the resulting dimensions satisfy every constraint.

• Maximum product at fixed sum

  $$\begin{gathered} x+y=T\\ xy=\left(\tfrac T2\right)^2-\left(x-\tfrac T2\right)^2 \end{gathered}$$

  Use when For positive $x,y$ with fixed sum $T$, the product reaches its upper bound at $x=y=T/2$ if that pair is feasible. With a cap $x+y\le T$, the bound still holds, and attaining it requires both the full sum and equal terms.

Problem types, step by step

Exam traps

• Trap Moving all uncertain inputs in the same direction

  Fix A bigger hole removes more material. Use opposite ends for whole and removed terms only if the needed choices can coexist. Keep a shared dimension at one value.

• Trap Calling an excluded bound a maximum or minimum

  Fix A bound may be approached without being attained. When a prediction decreases as its input rises, the input's high end gives the output's low end.

• Trap Treating precise arithmetic as evidence about the actual shape

  Fix Rounding bounds cover only the uncertainty modeled. An estimate made from the same inputs checks the size of the calculation. Use separate evidence to check the real target.

• Trap Cubing a scale from a top view or from a container's exterior

  Fix A top view supplies two directions. Use a separate depth when needed. Capacity requires the interior and the stated fill level.

• Trap Calling a sampled winner a proven optimum

  Fix State the design set actually covered. An exhaustive comparison or an attained bound proves optimality for that set.