The problem of exact observability of the heat equation in an arbitrary space dimension with scanning pointwise sensor is considered in the case when the space for outputs is L(infinity)(e, theta), epsilon > 0. A new method for the construction of observation curves for sensors that are able to ensure L infinity(epsilon,theta)-exact observability at final time is given, based on the maximum principle for the heat equation. An application of the method to the observability problem with discrete-time scanning observations and related approximate controllability results are also discussed.