javascript geometric algorithm

JavaScript is a widely used programming language that has many uses, one of which is handling geometric algorithms. In this article, we will introduce the basic content and implementation methods of some JavaScript geometric algorithms.

1. Points and vectors

In geometry, points and vectors are the most basic primitives. In JavaScript, we can use arrays to represent points and vectors. A point is represented by an array containing two elements, where the first element represents the x coordinate and the second element represents the y coordinate. For example, [1,2] represents a point located at (1,2). The vector is also an array containing two elements, but it does not represent the coordinates, but the length and direction. For example, [3,-4] represents a vector with a length of 3 and facing the second quadrant. Through vector subtraction, the vector between two points can be calculated. For example, the vector between point A (1,2) and point B (4,6) is [3,4].

2. Dot product and cross product

Dot product and cross product are the two most commonly used operations in two-dimensional geometry. The dot product is the sum of the products of the corresponding elements of two vectors. For example, the dot product of vectors A[2,3] and B[4,5] is 24 35=23. The dot product can be used to calculate the cosine value of the angle between vectors, which can be obtained through the cosine formula:

cosθ = A·B / |A||B|

where |A| and |B | represents the module length of the vector respectively, |A||B| represents their product. The cross product is the area of ​​the parallelogram formed by two vectors. The calculation formula is:

A × B = |A||B| sinθ

where θ represents the included angle. The result of the cross product is a scalar, and its direction depends on the order of the vectors. The right-hand rule can determine its direction.

In JavaScript, the calculations of dot products and cross products are relatively simple and can be achieved by just using array multiplication, addition and modulo methods.

3. Line lines and line segments

Line lines and line segments are common geometric objects and can also be represented by arrays in JavaScript. A straight line needs to be represented by a point and a vector. For example, straight line L: y=2x 1 can be expressed as [1,1],[2,4], where the first point is an arbitrary point on the straight line, and the second A vector is the direction vector of a straight line. A line segment needs to be represented by two points. The only difference is that they have a beginning and an end. For example, line segment AB can be represented as [1,2],[4,6].

In JavaScript, judging whether a point is on a straight line can calculate the distance between the point and the straight line. To determine whether a point is on a line segment, you need to determine whether it is on the extension of the line segment and between the two endpoints of the line segment.

4. Circles and Rectangles

Circles and rectangles are common two-dimensional geometric objects, and they can also be represented by arrays. A circle can be defined by the coordinates and radius of the center of the circle. For example, a circle O(1,2) with a radius of 3 can be expressed as [1,2,3]. A rectangle can be defined by the coordinates of the upper left corner and lower right corner. For example, the coordinates of the upper left corner of rectangle ABCD are (1,2) and the coordinates of the lower right corner are (3,4), which can be expressed as [1,2,3,4].

In JavaScript, to determine whether a point is within a circle, you can calculate whether its distance from the center of the circle is less than the radius. To determine whether a point is within a rectangle, you can determine whether it is within the area enclosed by the four sides of the rectangle.

5. The closest point pair problem

The closest point pair problem refers to finding the two closest points in a set of points. This problem has applications in computational geometry, computer vision, and machine learning. In JavaScript, you can use brute force algorithm and divide and conquer algorithm to solve the nearest point pair problem. The time complexity of the brute force algorithm is O(n^2), which is not suitable for large-scale data; while the time complexity of the divide-and-conquer algorithm is O(n log n), which is suitable for data of various sizes.

The basic idea of ​​the divide-and-conquer algorithm is to sort all points according to the x coordinate, then divide them into two parts, and deal with the nearest point pair problem of the left and right parts respectively. Then select the smallest distance d among the nearest point pairs of the left and right parts, and then find the shortest distance among the neighbors whose distance is d.

In JavaScript, you can use a sorting algorithm to sort all points, and then recursively handle the closest point pair problem of the left and right parts. For specific implementation, please refer to the examples in the code base.

Summary

In this article, we introduced the basics and implementation methods of processing geometric algorithms in JavaScript. They include the representation of points and vectors, the calculation of dot and cross products, the representation of lines and line segments, the representation of circles and rectangles, and the solution of the nearest point pair problem. By learning these basics, we can better understand and apply geometric algorithms.

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