Question Type 6: Assigning values to points using Nearest Neighbor Interpolation Exercises

Four sensors are located at $D(0,0)$ with reading $2$, $E(6,0)$ with reading $8$, $F(6,4)$ with reading $12$, and $G(0,4)$ with reading $6$. Use nearest neighbor interpolation to estimate the readings at: (a) $P_1(3,2)$ (b) $P_2(5,1)$

Consider four sample points $A(0,0)$ with value 1, $B(2,0)$ with value 2, $C(0,2)$ with value 3, and $D(2,2)$ with value 4.

Determine the interpolated values at the following points using nearest neighbor interpolation: (a) $R(0.9, 0.9)$ (b) $S(1.1, 0.8)$ (c) $T(1.2, 0.9)$

Given two sites $A(1,1)$ and $B(4,5)$, derive the equation of the perpendicular bisector that separates their Voronoi regions.

Given three sites $A(0,0)$, $B(5,0)$, and $C(0,5)$, find the vertices of the Voronoi cell of $A$ by computing intersections of the perpendicular bisectors of $AB$ and $AC$ within the first quadrant.

Suppose you have sensors at $A(0,0)$, $B(6,0)$, and $C(0,6)$. A new sensor is added at $D(3,3)$. Describe how the Voronoi cell of $B$ changes qualitatively after adding $D$.

Within the square domain $[0, 10] \times [0, 10]$, there are only two sites: $A(0, 5)$ and $B(10, 5)$. Compute the area of the Voronoi region belonging to $A$.

Given sample points $A(0,0)$ with value $5$, $B(3,0)$ with value $10$, and $C(0,3)$ with value $15$, use nearest neighbor interpolation to find the value at point $P(2,1)$.

Given sites $A(1,1)$, $B(4,1)$, $C(4,4)$, and $D(1,4)$, each with distinct values, write the system of inequalities that defines the region where $B$ is the nearest neighbor.

List the half-plane inequalities that describe the Voronoi cell of $A(2,0)$ given neighboring sites $B(0,2)$, $C(-2,0)$, and $D(0,-2)$.

An air-quality monitoring network has sensors at $P_1(1, 2)$ with a reading of 30, $P_2(4, 2)$ with a reading of 50, and $P_3(2, 5)$ with a reading of 40. Estimate the reading at location $Q(3, 3)$ using nearest neighbor interpolation.