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Two convex bodies are homothetic convex bodies if one is a translation of a positive uniform dilation of the other. Their lengths scale by $s$ and their volumes by $s^n$. Closed disks of positive radii are homothetic convex bodies; axis-aligned positive-sided rectangles are homothetic convex bodies exactly when their aspect ratios agree. The Brunn-Minkowski theorem characterizes this relation by equality of root-volumes in Minkowski addition.

The Minkowski sum of subsets $A,B$ of a vector space collects the result of vector addition for every possible pair:Think of fixing a point $b$ of $B$, making a translation of the whole of $A$ by $b$, and then taking the union of these copies over every $b\in B$. The points of $B$ specify displacement vectors; they need not be the centre of a shape. Minkowski addition applies to nonconvex sets too. It is different from set union: for $A=B=\{0,1\}$ on the real line, $A\cup B=\{0,1\}$ but $A+B=\{0,1,2\}$.

For axis-aligned rectangles in the Euclidean plane, the coordinate intervals add separately:Every coordinate between $0$ and $a+c$ can be split into one coordinate in $[0,a]$ and another in $[0,c]$, and likewise for the vertical coordinate. Thus a $2\times 1$ rectangle plus a $1\times 2$ rectangle gives a $3\times 3$ square of area $9$, although the two input areas total only $4$. The Minkowski sum adds possible displacements, so its area need not be the sum of input areas.

For closed Euclidean disks $D_r=\{x\in {\mathbb{R}}^2:\Vert x\Vert \le r\}$ centred at the origin, the triangle inequality gives $D_r+D_s\subseteq D_{r+s}$. Conversely, if $r+s>0$, split any $x\in D_{r+s}$ as $x=rx/(r+s)+sx/(r+s)$. Therefore $D_r+D_s=D_{r+s}$, including the case $r=s=0$. Disks of radii $1$ and $2$ give a disk of radius $3$, with areas $\pi ,4\pi ,9\pi$. For disks with centres $u,v$, the resulting centre is $u+v$. These two examples lead to the root-of-area comparison in the Brunn-Minkowski theorem.

Minkowski addition is commutative and associative because vector addition is. Its identity is the singleton $\{0\}$, and $A+\varnothing =\varnothing$. For displacement vectors $u,v$ and $t>0$, translations and uniform dilations satisfywhere $tA=\{ta:a\in A\}$ uses scalar multiplication. A Minkowski sum of convex sets is convex: a line segment between $a_1+b_1$ and $a_2+b_2$ is obtained by adding the corresponding points of the line segments in $A$ and $B$. Closure requires care; see sum of two closed sets need not be closed and sum of a compact set and a closed set.

Robot motion planning computes a robot's proposed motion from its modeled state, environment and task constraints. A geometric planning problem asks for a path through configuration space; a timed motion can additionally constrain velocity, acceleration, forces and actuator effort. Hauser's primary text explicitly distinguishes geometric path planning from the broader problem involving dynamics and execution.^[ref]

Let $Q_{\mathrm{free}}$ be the configurations allowed by joint limits and collision detection. A geometric solution is a continuous curveEach point represents a complete robot configuration. A configuration-space obstacle is a forbidden set of configurations, not necessarily the same shape as the physical obstacle in the workspace.

An original planar example uses a point robot starting at $(0,0)$ and aiming for $(1,0)$, in metres. A circular obstacle centered at $(0.5,0)$ with radius $0.10\mathrm{m}$ blocks the straight path. A two-segment detour through $(0.5,0.20)$ has lengthThe perpendicular distance from the obstacle center to either segment isso the detour clears the circular obstacle. Checking only the start and goal would instead miss the collision on the straight segment.

Now give the robot a circular footprint of radius $0.05\mathrm{m}$, with translation as its only degree of freedom. The forbidden center positions form a disk of radius $0.15\mathrm{m}$: the obstacle is expanded by the footprint through Minkowski addition. The same detour still passes, but its center-path margin above the expanded obstacle is only about $0.0357\mathrm{m}$. The geometric reduction depends on the stated footprint and permitted motion.

A path's corner does not supply a dynamically smooth trajectory. Traversing this sharp detour at nonzero constant speed would instantaneously change velocity at its midpoint. A physical controller must stop, smooth the corner or choose a different feasible trajectory, while continuing to respect the obstacle and actuator constraints. A larger path length can be an acceptable cost if it permits the required motion.

A proposed validation compares the continuous planned geometry, sampled collision tests and measured execution on a controlled obstacle course. Report the robot geometry, constraints, computational budget and observed clearance. Failure to find a path in a limited run does not alone prove none exists; success on a numerical path does not alone establish accurate execution. Hauser treats completeness and the distinction between paths and feedback policies separately.^[ref]