Collinear geometry

Within the field of geometry, collinearity of a specific set of points is the property of their lying on a single line. A set of points with this property is said to be collinear. In a greater visual sense, the term has been implemented for aligned objects which refers to things lying on the same line or sharing a common row.

 Within any kind of geometry, the set of points present on the same line is said to be collinear. In relation to Euclidean geometry, points which are visualized lying on the same line or row refer to as a formation known as the Straight line. However, in most of the existing geometries, a line is typically a primitive object type will not be necessarily appropriate. A specific model for geometry offers an interpretation of how the points, lines or the types of objects which share a relation with one another and the collinearity is maintained within that model of geometry. A mapping of a geometry to itself which sends lines to lines is known as collineation, which preserves the collinearity property. The linear maps of vector spaces are viewed as the geometric maps which perform the job of matching lines with lines. They also perform the job of matching collinear points to a set of collinear points and so are called as collineations. In the field of projective geometry these linear mappings are referred to as the homographies and are just one type of collineation.

Determine collinear points

Collinear points definition include the following mentioned statements,

• Three or more points that belong to the same straight line is called the collinear points.
• Considering a straight line L is above the Cartesian coordinate plane which is formed by the x axis and the y axis respectively.
• This straight line L which passes through the three points A, B and C whose coordinates are (2,4), (4,6) and (6,8) respectively.
• It may also be stated that there is an alternative method that the three respective points A (2, 4), B (4, 6) and C (6, 8) are lying upon the same straight line.
• Three or more points which lie on the same straight line are referred to as the collinear points.

Methods to find if the three given points are collinear refer to two kinds of methods. One of the method refers to the slope formula and the other hand, the second method refers to the area of triangle method. The first method, which is given the name of the slope method is to find that given set of specific points are collinear. The method states that, two or more points are collinear if the slope related to any two pairs of points are the same. With the three given set of points named as A, B and C and the three pairs of points which can be formed are AB, BC and AC. If the slope of AB = slope of BC = slope of AC, then the A, B and C are collinear points.

The second formula was referred to as the Area of triangle which is represented by the formula,

in the above formula, the two vertical bars which are present enclose the variable which are represented using a determinant. Applying the coordinates of the above three points A, B and C in the determinant formula above for the implementation of the triangle formula just to check that the value is zero which comes out as the answer.

Since, the outcome for the area of the triangle is zero, therefore A (2, 4), B (4, 6) and C (6, 8) are referred to as the collinear points.

Graphs and Slopes

Within the field of mathematics, more specifically in graph theory, a graph is a structure amounting to the set of objects in which some pairs of objects are in some sense related to each other. The objects have a correspondence to the mathematical abstractions referred to as vertices, and each of these related pairs of vertices is commonly given a named which is, edge. Typically, a graph is represented in a diagrammatic form as a sets of dots or circles for the vertices and joined by lines or curves which are there to represent the edges. Graphs are one of the major field of study in the field of discrete mathematics. The edges may be either directed or undirected. Graphs refer to the basic portion of study in the subject of graph theory.

A multigraph is a generalization that allows multiple edges which are adjacent to the same pair of existing vertices. In some of the studies, multigraphs are referred to as only graphs. Sometimes, graphs are given allowance to contain loops within them in the diagram with the help of which they are represented. Loops are formed by edges that join the vertex to itself to give rise to the formation of loops.

In the field of mathematics, the slope or the gradient of a line is a number that describes both the direction and the steepness which is related to the line. Slope is often defined by the letter m however, there is no clear answer to the question that why the slope of a line is denoted by the letter m. The calculation of slope is done by finding the ratio between the vertical change and the horizontal change between any two distinct points of a line. In some contexts, the ratio is expressed as a quotient, providing the same number for every two distinct points present upon the same line.

 The steepness, incline or the grade of the line is measured by the absolute value related to the slope.  A slope with a greater absolute value indicates a lot more steeper line.  The direction of a line is either increasing, decreasing, horizontal or vertical. In mathematical language, the slope m of the line is,

The concept of the slope applies directly to the grades or the gradients present within the field of geography and civil engineering as well.