(Geometry: point on line segment) Exercise 3.32 shows how to...

(Geometry: point on line segment) Exercise 3.32 shows how to test whether a point is on an unbounded line. Revise Exercise 3.32 to test whether a point is on a line segment. Write a program that prompts the user to enter the three points for $\mathrm{p} 0, \mathrm{p} 1,$ and $\mathrm{p} 2$ and displays whether $\mathrm{p} 2$ is on the line segment from $\mathrm{p} 0$ to $\mathrm{p} 1 .$ Here are some sample runs: Enter three points for $p 0, p 1,$ and $p 2: 112.52 .51 .51 .5$ (1.5,1.5) is on the line segment from (1.0,1.0) to (2.5,2.5) Enter three points for $p 0, p 1,$ and $p 2: 11223.53 .5$ (3.5,3.5) is not on the line segment from (1.0,1.0) to (2.0,2.0)

(Geometry: point on line segment) Exercise 3.32 shows how to test whether a point is on an unbounded line. Revise Exercise 3.32 to test whether a point is on a line segment. Write a program that prompts the user to enter the three points for $\mathrm{p} 0, \mathrm{p} 1,$ and $\mathrm{p} 2$ and displays whether $\mathrm{p} 2$ is on the line segment from $\mathrm{p} 0$ to $\mathrm{p} 1 .$ Here are some sample runs:
Enter three points for $p 0, p 1,$ and $p 2: 112.52 .51 .51 .5$
(1.5,1.5) is on the line segment from (1.0,1.0) to (2.5,2.5)
Enter three points for $p 0, p 1,$ and $p 2: 11223.53 .5$
(3.5,3.5) is not on the line segment from (1.0,1.0) to (2.0,2.0)

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Key Concepts

Collinearity

In computational geometry, testing for collinearity involves checking whether three points lie on a single straight line. This can be done by using methods such as comparing slopes or computing the cross product of vectors, with a zero result indicating that the points are collinear.

Line Segment

A line segment is a portion of a line bounded by two distinct endpoints. Unlike an unbounded line, a point must not only be collinear with the endpoints but also lie between them to be considered as part of the segment.

Boundary Condition Check

Even when a point is collinear with two endpoints, it must be determined whether it lies between them. This check typically involves ensuring that the point's coordinates fall within the minimum and maximum values of the corresponding coordinates of the endpoints or using vector projection techniques.

Vector Operations

Vector operations such as calculating the dot product and using parametric equations can help to verify both collinearity and the positional relationship of a point relative to the endpoints of a segment. These methods offer a robust approach to determining if a point lies on a specific line segment.

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