  We consider sketch vectors of geometric objects $J$ through the \mindist
function \[ v_i(J) = \inf_{p \in J} \|p-q_i\| \] for $q_i \in Q$ from a point
set $Q$. Collecting the vector of these sketch values induces a simple,
effective, and powerful distance: the Euclidean distance between these sketched
vectors. This paper shows how large this set $Q$ needs to be under a variety of
shapes and scenarios. For hyperplanes we provide direct connection to the
sensitivity sample framework, so relative error can be preserved in $d$
dimensions using $Q = O(d/\varepsilon^2)$. However, for other shapes, we show
we need to enforce a minimum distance parameter $\rho$, and a domain size $L$.
For $d=2$ the sample size $Q$ then can be $\tilde{O}((L/\rho) \cdot
1/\varepsilon^2)$. For objects (e.g., trajectories) with at most $k$ pieces
this can provide stronger \emph{for all} approximations with
$\tilde{O}((L/\rho)\cdot k^3 / \varepsilon^2)$ points. Moreover, with similar
size bounds and restrictions, such trajectories can be reconstructed exactly
using only these sketch vectors.
