Computational Geometry: Master Geometric Algorithms, Convex Hulls, and Voronoi Diagrams

Question 1

Which of the following accurately defines a convex polygon?

Accepted Answer

A polygon that lacks any concave angles

Answer

A polygon featuring all interior angles measuring more than 180 degrees

Answer

A polygon that possesses at least one concave angle

Answer

A polygon where all interior angles measure less than 180 degrees

Question 2

In the field of robotics, what is a practical application of computational geometry?

Accepted Answer

Path planning for autonomous vehicles

Answer

Natural language processing

Answer

Protein structure prediction

Answer

Database indexing

Question 3

What is the time complexity, using Graham's scan, for computing the convex hull of a set of n points in the plane?

Accepted Answer

O(n log n)

Answer

O(n^2)

Answer

O(n)

Answer

O(log n)

Question 4

Which algorithm efficiently computes the intersection of two convex polygons?

Accepted Answer

Sweep line algorithm

Answer

Graham's scan

Answer

Closest pair algorithm

Answer

Jarvis' march

Question 5

Finding the convex hull of a set of n points has a time complexity of:

Accepted Answer

O(n log n)

Answer

O(n)

Answer

O(n^2)

Answer

O(n^3)

Question 6

Define a Voronoi diagram:

Accepted Answer

A space is partitioned into regions where each region is closer to a specific point than any other point

Answer

A graph showing the shortest distances between a set of points

Answer

A convex hull of a set of points

Answer

A triangulation of a set of points

Question 7

What is the purpose of a sweep-line algorithm?

Accepted Answer

To solve problems where input is ordered along a line

Answer

To compute convex hulls

Answer

To find the closest pair of points

Answer

To triangulate a set of points

Question 8

Which of the following is NOT a widely recognized application of computational geometry?

Accepted Answer

Cryptography

Answer

Computer-Aided Design (CAD)

Answer

Geographical Information Systems (GIS)

Answer

Computational Biology

Question 9

Which of the following is NOT an advantage of using a convex hull algorithm?

Accepted Answer

It can't handle all degenerate cases.

Answer

It can be used to solve certain linear programming problems.

Answer

It finds the minimum bounding box of a set of points.

Answer

It is efficient for large data sets.

Question 10

What is the time complexity of finding the closest pair of points in a set of n points using the brute-force algorithm?

Accepted Answer

O(n^2)

Answer

O(n)

Answer

O(log n)

Answer

O(n log n)

Question 11

Which of the following is a key property of a Voronoi diagram?

Accepted Answer

Each point in the diagram is closer to its corresponding generator than to any other generator.

Answer

It is always a convex polygon.

Answer

It can be used to find the exact shortest path between two points.

Question 12

Which of the following algorithms is commonly used to find the convex hull of a set of points?

Accepted Answer

Graham's Scan

Answer

Merge Sort

Answer

Dijkstra's Algorithm

Answer

Breadth-First Search

Question 13

Explain the fundamental difference between a Delaunay triangulation and a Voronoi diagram.

Accepted Answer

A Delaunay triangulation connects points that are close together, while a Voronoi diagram partitions space based on proximity to generator points.

Answer

A Delaunay triangulation is always convex, while a Voronoi diagram is not.

Question 14

What is the key purpose of the sweep-line algorithm?

Accepted Answer

Finding intersections between a set of line segments.

Answer

Constructing Voronoi diagrams.

Answer

Computing convex hulls.

Question 15

Which data structure is commonly used to efficiently represent a polygon in computational geometry?

Accepted Answer

Doubly-connected edge list

Answer

Array

Answer

Tree

Answer

Linked list

Question 16

Describe the underlying strategy of the nearest neighbor algorithm for point location.

Accepted Answer

Iteratively search within recursively subdivided space to find the nearest point to a query point.

Answer

Use a hash table to store the points and their locations.

Answer

Construct a kd-tree to organize the points for efficient search.

Question 17

Which fundamental geometric algorithm forms the basis of many other algorithms in computational geometry?

Accepted Answer

Convex hull

Answer

Triangulation

Answer

Closest pair of points

Question 18

What is the asymptotic time complexity of computing the convex hull of a set of n points in 2D space?

Accepted Answer

O(n log n)

Answer

O(n^3)

Answer

O(n^4)

Answer

O(n^2)

Question 19

Which geometric data structure partitions the plane into regions of equal distance to a set of points?

Accepted Answer

Voronoi diagram

Answer

Convex hull

Answer

Minimum bounding rectangle

Answer

Delaunay triangulation

Question 20

What is the minimum number of vertices needed to define a convex polygon in 2D space?

Accepted Answer

3

Answer

6

Answer

5

Answer

4

Question 21

What is the significance of the empty circle property in computational geometry?

Accepted Answer

It helps determine whether a set of points forms a convex hull.

Answer

It ensures that every point in a set is contained within a circle.

Answer

It is used to compute the Delaunay triangulation of a set.

Question 22

Which geometric representation is suitable for a finite region in the plane?

Accepted Answer

Polygon

Answer

Matrix

Answer

Curve

Answer

Function

Question 23

What is a primary advantage of employing computational geometry techniques in computer graphics?

Accepted Answer

Improved realism and efficiency in rendering complex scenes.

Answer

Reduced memory requirements

Answer

Faster execution time

Question 24

In computational geometry, which data structure is commonly utilized to efficiently store and manipulate geometric information?

Accepted Answer

Convex hull

Answer

Bloom filter

Answer

Binary search tree

Question 25

Which algorithm is commonly employed to compute the convex hull of a given set of points?

Accepted Answer

Graham scan

Answer

Merge sort

Answer

Prim's algorithm

Question 26

Which of the following is NOT an application of computational geometry in the field of computer graphics?

Accepted Answer

Image compression

Answer

3D modeling

Answer

Ray tracing

Answer

Collision detection

Question 27

What is the name of the theorem that states any point inside a convex polygon can be expressed as a convex combination of its vertices?

Accepted Answer

Carathéodory's theorem

Answer

Descartes' theorem

Answer

Pythagoras' theorem

Answer

Euler's theorem

Question 28

Which of the following is **NOT** a fundamental concept in computational geometry?

Accepted Answer

Binary Search Trees

Answer

Geometric Algorithms

Answer

Voronoi Diagrams

Answer

Convex Hulls

Question 29

In the context of computational geometry, what is the purpose of constructing a convex hull?

Accepted Answer

To identify the smallest convex polygon that encloses all the given points.

Answer

To find the closest pair of points in the given set.

Answer

To determine the maximum area of a circle that can be inscribed within the given points.

Question 30

Which algorithm is commonly used to construct the convex hull of a given set of points in the plane efficiently?

Accepted Answer

Graham Scan

Answer

Merge Sort

Answer

Dijkstra's Algorithm

Answer

Breadth-First Search

Question 31

What is the primary purpose of a Voronoi diagram in computational geometry?

Accepted Answer

To partition the plane into regions where each region is closest to a specific point in a given set of points.

Answer

To determine the convex hull of a given set of points.

Answer

To find the shortest path between specified points on a graph.

Question 32

In computational geometry, which algorithm is suitable for efficiently finding all intersections among a collection of line segments in the plane?

Accepted Answer

Sweep Line Algorithm

Answer

Merge Sort

Answer

Dijkstra's Algorithm

Answer

Depth-First Search

Question 33

What is the time complexity of the Graham Scan algorithm for constructing the convex hull of a set of n points in the plane?

Accepted Answer

O(n log n)

Answer

O(n^2)

Answer

O(log n)

Answer

O(n)

Question 34

Which algorithm is used to efficiently compute the convex hull of a set of points in the plane?

Accepted Answer

Graham's Scan

Answer

Bellman-Ford Algorithm

Answer

Floyd-Warshall Algorithm

Answer

Dijkstra's Algorithm

Question 35

What is the time complexity of computing the Voronoi diagram for a set of n points using Fortune's algorithm?

Accepted Answer

O(n log n)

Answer

O(n^2)

Answer

O(n^3)

Answer

O(2^n)

Question 36

Which of the following is a significant application of computational geometry in the field of computer graphics?

Accepted Answer

Collision detection

Answer

Compiler design

Answer

Database management

Answer

Natural language processing

Question 37

Which of the following is NOT considered a fundamental concept in computational geometry?

Accepted Answer

Graph theory

Answer

Voronoi diagrams

Answer

Triangulation

Answer

Convex hulls

Question 38

What is the primary purpose of triangulation in computational geometry?

Accepted Answer

Decomposing a complex shape into a set of triangles for efficient representation and analysis

Answer

Computing the convex hull of a set of points

Answer

Generating random numbers

Answer

Finding the shortest path between two points in a graph

Question 39

Which data structure is commonly employed to represent a convex hull efficiently?

Accepted Answer

Doubly-connected edge list (DCEL)

Answer

Array

Answer

Hash table

Answer

Binary tree

Question 40

What is the time complexity of computing the convex hull of a set of n points using the Graham's Scan algorithm?

Accepted Answer

O(n log n)

Answer

O(n)

Answer

O(2^n)

Answer

O(n^2)

Question 41

Which algorithm is most efficient for computing the convex hull of a set of points in the plane?

Accepted Answer

Graham's Scan

Answer

Gift Wrapping

Answer

QuickHull

Answer

Jarvis March

Question 42

What is the worst-case time complexity for finding the closest pair of points in a set of n points using brute force?

Accepted Answer

O(n^2)

Answer

O(n)

Answer

O(log n)

Answer

O(n log n)

Question 43

In a Voronoi diagram, what is the region associated with a point called?

Accepted Answer

Voronoi cell

Answer

Bounding box

Answer

Delaunay triangle

Answer

Convex hull

Question 44

Which is NOT a property of a Delaunay triangulation?

Accepted Answer

All triangles are acute

Answer

It maximizes the minimum angle of all triangles

Answer

It is a triangulation of the convex hull of the point set

Answer

The circumcircle of each triangle contains no points in its interior

Question 45

What is the primary application of a line sweep algorithm in computational geometry?

Accepted Answer

Solving geometric intersection problems

Answer

Finding the closest pair of points

Answer

Computing the convex hull

Answer

Constructing Voronoi diagrams

Question 46

Which data structure is commonly used to represent the edges of a Voronoi diagram?

Accepted Answer

Doubly-connected edge list

Answer

Array

Answer

Binary tree

Answer

Hash table

Question 47

What is the fundamental difference between a convex hull and a Voronoi diagram?

Accepted Answer

A convex hull encloses a set of points, while a Voronoi diagram partitions the plane into regions based on proximity to points

Answer

A convex hull is a triangulation, while a Voronoi diagram is not

Question 48

Which of the following algorithms is most efficient for finding the convex hull of a set of points in two dimensions (assuming a simple polygon)?

Accepted Answer

Graham's Scan

Answer

Delaunay Triangulation

Answer

Jarvis March

Answer

Quickhull

Question 49

In a Voronoi diagram, what is the geometric interpretation of the edges?

Accepted Answer

The set of points equidistant to two given points in the point set.

Answer

The lines connecting all points to a central point.

Answer

The boundaries of the convex hull.

Answer

The shortest paths between all pairs of points.

Question 50

Which of the following data structures is commonly used to represent a Voronoi diagram?

Accepted Answer

Delaunay Triangulation

Answer

Linked List

Answer

Hash Table

Answer

Binary Search Tree

Question 51

What is the time complexity of finding the closest pair of points in a set of n points using a divide-and-conquer approach?

Accepted Answer

O(n log n)

Answer

O(n)

Answer

O(n^2)

Answer

O(log n)

Question 52

In a Delaunay triangulation, what is the property that guarantees the optimality of the triangulation?

Accepted Answer

The circumcircle of any triangle does not contain any other points from the point set.

Answer

The sum of the edge lengths is minimized.

Answer

The number of triangles is minimized.

Question 53

Which of the following applications is NOT a typical use case of computational geometry?

Accepted Answer

Compiling a programming language

Answer

Geographic information systems (GIS)

Answer

Robotics and motion planning

Answer

Computer graphics and animation

Question 54

What is the main advantage of using a sweep line algorithm for solving geometric problems?

Accepted Answer

It efficiently handles a large number of geometric objects by processing them in a sequential manner based on their position along a sweep line.

Answer

It is the most intuitive algorithm for solving any geometric problem.

Question 55

Which of the following geometric operations is NOT directly related to the concept of convex hulls?

Accepted Answer

Finding the shortest path between two points

Answer

Finding the smallest enclosing circle for a set of points

Answer

Computing the diameter of a point set

Answer

Determining if a point lies inside a polygon

Question 56

What is the main difference between the Bentley-Ottmann algorithm and the plane-sweep algorithm for line segment intersection?

Accepted Answer

The Bentley-Ottmann algorithm uses a priority queue to efficiently handle events, while the plane-sweep algorithm uses a sorted list to maintain the active segments.

Answer

The Bentley-Ottmann algorithm is faster, while the plane-sweep algorithm is more efficient in memory usage.