Question

In: Statistics and Probability

A polygon is called convex if every line segment from one vertex to another lies entirely...

A polygon is called convex if every line segment from one vertex to another lies entirely within the polygon. To triangulate a polygon, we take some of these line segments, which don’t cross one another, and use them to divide the polygon into triangles. Prove, by strong induction for all naturals n with n ≥ 3, that every convex polygon with n sides has a trian-gulation, and that every triangulation contains exactly n − 2 triangles. (Hint: When you divide an n-gon with a single line segment, you create an i-gon and a j-gon for some naturals i and j. What does your strong inductive hypothesis tell you about triangulations of these polygons?)

Solutions

Expert Solution

orchestra answered 1 year ago
Next > < Previous

Related Solutions

Complete exercise 9.20 from the text. (a) Create a function called polygon that draws a polygon...
Complete exercise 9.20 from the text. (a) Create a function called polygon that draws a polygon in a polar plot. Your function should have a single input parameter – the number of sides. (b) Use a for loop to create a figure with four subplots…You should use the function you created in part (a) to draw each polygon. Use the index parameter from the for loop to specify the subplot in which each polygon is drawn, and in an expression...
Suppose ?⃗ (?,?)=−??⃗ +??⃗ and ? is the line segment from point ?=(2,0) to ?=(0,3). (a)...
Suppose ?⃗ (?,?)=−??⃗ +??⃗ and ? is the line segment from point ?=(2,0) to ?=(0,3). (a) Find a vector parametric equation ?⃗ (?) for the line segment ? so that points ? and ? correspond to ?=0 and ?=1, respectively. ?⃗ (?)= (b) Using the parametrization in part (a), the line integral of ?⃗ along ? is ∫??⃗ ⋅??⃗ =∫???⃗ (?⃗ (?))⋅?⃗ ′(?)??=∫?? ?? with limits of integration ?= and ?= (c) Evaluate the line integral in part (b). (d)...
for every bounded, closed and convex subset K of Y, there exists a retract from Y...
for every bounded, closed and convex subset K of Y, there exists a retract from Y onto K prove the lemma
Evaluate the line integral along the given paths. xy ds (a) C: line segment from (0,0)...
Evaluate the line integral along the given paths. xy ds (a) C: line segment from (0,0) to (5, 4) C: counterclockwise around the triangle with vertices (0, 0), (8, 0), and (0, 2)
Evaluate the line integral along the given paths. xy ds C (a) C: line segment from...
Evaluate the line integral along the given paths. xy ds C (a) C: line segment from (0, 0) to (7, 4) counterclockwise around the triangle with vertices (0, 0), (8, 0), and (0, 4)
Write a C++ function called parse that reads one line of user input from the keyboard...
Write a C++ function called parse that reads one line of user input from the keyboard and creates an array of the strings found in the input.  Your function should be passed the array and a reference variable that is to be assigned the length of the array.  Prompt the user and read the input from within the function. For example:  If the user inputs copy this that, the resulting array would have length 3 and contain the strings “copy”, “this”, and “that”....
At one time or another, almost every retailer claims that their first and only priority is...
At one time or another, almost every retailer claims that their first and only priority is to make the customer happy. Few, however, can duplicate what Nordstrom does. For the Seattle-based upscale retailer, “the customer is always right” is not just a promotional motto but a way of life that guides the organization. The entire company is directed towards one goal––catering to the needs of customers. Sales associates are given incredible freedom to do whatever is needed to make customers...
At one time or another, almost every retailer claims that their first and only priority is...
At one time or another, almost every retailer claims that their first and only priority is to make the customer happy. Few, however, can duplicate what Nordstrom does. For the Seattle-based upscale retailer, “the customer is always right” is not just a promotional motto but a way of life that guides the organization. The entire company is directed towards one goal––catering to the needs of customers. Sales associates are given incredible freedom to do whatever is needed to make customers...
Write parametric equations for each of the following curves. (a) The straight line segment traced from...
Write parametric equations for each of the following curves. (a) The straight line segment traced from (3,2) to (5,8) as t goes from 0 to 1. (b) A circle centered at (3,2), of radius 4, traced out two times, counterclockwise, as t goes from 0 to 2pi.
How do i make a point for line segment from p to infinite in java? in...
How do i make a point for line segment from p to infinite in java? in C++, it's Point p; Point extreme = {INF, p.y}; --------------------------------------------------------------------------------------------- boolean isInside(Point polygon[], int n, Point p) { // There must be at least 3 vertices in polygon[] if (n < 3) return false;    // Create a point for line segment from p to infinite //Point extreme = {INF, p.y};    p= Double.POSITIVE_INFINITY;    Point extreme = p.y; // Count intersections of the...
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT