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

Question 1

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 .

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Question 2

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} .

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Question 3

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} .

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Question 4

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} .

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Question 5

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.

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Question 6

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.

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Question 7

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} .

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Question 8

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.

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Question 9

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.

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Question 10

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.

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Question 11

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.

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Question 12

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 .

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