Exercises for Question Type 4: Adding sites to update Voronoi Diagrams

Question 1

Skill question

Given four sites $A = (0, 0)$ , $B = (4, 1)$ , $C = (2, 3)$ , and $D = (- 1, 2)$ , a new site $E = (1, 1)$ is inserted. Determine the equation and the domain of the Voronoi edge between sites $E$ and $C$ .

[6]

Question 2

Skill question

Given four sites $Q_{1} (0, 0)$ , $Q_{2} (5, 0)$ , $Q_{3} (6, 3)$ , and $Q_{4} (1, 4)$ , find the equation of the perpendicular bisector between $Q_{1}$ and $Q_{4}$ . Hence, determine the coordinates of the Voronoi vertex formed by $Q_{1}$ , $Q_{2}$ , and $Q_{4}$ .

[7]

Question 3

Skill question

Using the rectangle sites $R_{1} = (2, 0)$ , $R_{2} = (6, 0)$ , $R_{3} = (6, 4)$ , and $R_{4} = (2, 4)$ , determine the equation and range of the Voronoi edge between $R_{3}$ and $R_{4}$ .

[5]

Question 4

Skill question

Given three sites $S_{1} = (1, 1)$ , $S_{2} = (5, 1)$ , $S_{3} = (3, 5)$ , you add a new site $S_{4} = (4, 2)$ . Determine the equation and domain of the new Voronoi edge between $S_{4}$ and $S_{2}$ .

[7]

Question 5

Skill question

Given two sites at $P_{1} = (0, 0)$ and $P_{2} = (4, 0)$ , the Voronoi diagram is the line $x = 2$ . Now a third site $P_{3} = (2, 0)$ is added. Determine the equations of the boundaries of the Voronoi cell for the new site $P_{3}$ and specify the region it occupies.

[5]

Question 6

Skill question

Five sites are $A (0, 0)$ , $B (4, 1)$ , $C (2, 3)$ , $D (- 1, 2)$ , and $E (3, - 1)$ . Determine the Voronoi cell of $C$ : give the equations of its bounding edges and their intersection points.

[6]

Question 7

Skill question

The following question is based on the application of Voronoi diagrams in a coordinate plane.

Sites $P_{1} (0, 0)$ , $P_{2} (4, 0)$ , $P_{3} (5, 3)$ , $P_{4} (2, 5)$ , and $P_{5} (- 1, 3)$ form a Voronoi diagram. Determine the equation and the domain of the Voronoi edge between $P_{2}$ and $P_{3}$ .

[7]

Question 8

Skill question

For the rectangle with sites $R_{1} = (2, 0)$ , $R_{2} = (6, 0)$ , $R_{3} = (6, 4)$ , $R_{4} = (2, 4)$ , find the Voronoi cell of $R_{1}$ : give the equations of its edges and their intersection points.

[7]

Question 9

Skill question

Sites $P_{1} = (- 2, 0)$ , $P_{2} = (2, 0)$ , and $P_{3} = (0, 3)$ define a Voronoi diagram. Find the equation of the Voronoi edge between $P_{1}$ and $P_{3}$ and specify its domain.

[6]

Question 10

Skill question

For sites $A (0, 0)$ , $B (4, 0)$ , $C (4, 4)$ , and $D (0, 4)$ , the Voronoi diagram partitions the plane into four regions. Determine the equation of the Voronoi edge between $A$ and $D$ and specify its domain.

[4]

Question 11

Skill question

Sites $A = (0, 0)$ , $B = (4, 0)$ , and $C = (2, 4)$ form three regions in their Voronoi diagram. Determine the equation of the edge between $A$ and $C$ and describe its domain.

[6]

Question 12

Skill question

The sites $A (0, 0)$ , $B (6, 0)$ , and $C (3, 4)$ form a triangle. A new site $D (4, 1)$ is added within this triangle to form a new Voronoi diagram.

Determine the equation and the coordinates of the endpoints of the Voronoi edge between sites $D$ and $B$ .

[6]

IB Mathematics Applications & Interpretation Question Type 4: Adding Sites to Update Voronoi Diagrams Exercises