Mathematics for GIS Professionals

Functions and Graphs

A function is a relationship in which, given the value of one or more variables, the overall value of the function can be calculated. For example, $y = f(x)$ can be read as “$y$ is a function of $x$” with $y$ as the value of the function and $x$ the argument. The function can take any mathematical form such as: $y = ax + b$ (which is a line)

$y = ax^2 + bx + c$(a quadratic or second-degree curve)

$y = ax^3 + bx^2 + cx + d $ (a cubic or third-degree curve), and so on.

A particular group of functions are called polynomials. A polynomial is essen- tially an expression that contains two or more terms such as

$$x^3 + 3x^2y + 3xy^2 + y^3 + x^2 + 3y^2 + xy + x + y + 4$$

More specifically, the term is used to describe a relationship such as:

$$y = a + bx + cx2 + dx3 + ex4 + fx5 + ………….$$

where $a, b, c, d, e, f $ and so on, have fixed values.

There may be a finite number of terms, for example, where

$$y = (1 + x)^4 = 1 + 4x + 6x^2 + 4x^3 + x^4 $$

Alternatively, there may be an infinite number of terms in which case it will either generate an infinite number or else converge onto a specific value. The term con- vergence means that however many terms are added, the function will simply grow closer and closer to a specific value. Thus,

$$e = 1 + \frac{1}{1!} +\frac{1}{2!} + \frac{1}{3!} + …$$

has an infinite number of terms but it never gets bigger than a certain value while the series of terms $1 + 2 + 3 + …$ just gets bigger and bigger and is not convergent.

A series is a sequence of terms that may be finite or infinite. An example of a series is

$$S = a_0 + a_1 + a_2 + a_3 + … + a_(n-1) + a_n$$

where n is a positive integer. It is often written as

$$S = \sum_{i=0}^{i=n}a_i$$

where the symbol $\sum$ (the Greek letter “sigma”) means “the sum of.” In practice, an infinite series will only have a finite sum if the series $(a_0, a_0 + a_1, a_0 + a_1 + a_2, a_0 + a_1 + a_2 + a_3,….$ etc.) converges.

Functions may involve one dependent and one independent variable or may have several independent variables such as: $z = f(x, y)$, meaning that $z$ is a function of $x$ and $y$ where $x$ and $y$ are both independent variables.

If $z$ is dependent on $x$ and $y$ so that we have only one value of $z$ for given values of $x$ and $y$, then we have a surface. If $z$ is linearly dependent on $x$ and $y$ (i.e., $z = f(x,y) = ax^1  + by^1  + c$) then, as we have seen above, this is a plane surface. If $z = ax^2 + by^2 + cx + dy + e$, then we have a second degree or quadratic surface. If $y = f(x)$ so that y is a function of $x$, then there is a unique value for $y$ for every value of $x$. If the relationship between $y$ and $x$ is such that it is also possible to determine $x$ uniquely given the value of $y$, then the relationship is said to have an inverse function, written as $f^{-1}$. Thus, if $y = f(x) = ax + b$, then $x = f^{-1}(y) = \frac{(y-b)}{a}$. This is often not possible; for instance, if $y = x^2$  then $x = +\sqrt{y}$ or $x = -\sqrt{y}$ and there are two possible relationships. Hence, there is no inverse function. In general, functions are mathematical relationships between two (or more) variables that may be in the form of one-to-one or one-to- many or many-to-many.

We have seen that the equation for a circle centered at the origin is $x^2  + y^2  = r^2$ where $r =$ radius. For any value of $x$ that represents a point on the circle there are two possible values of $y$ (one positive and one negative); similarly, for any value of $y$ there are two values of $x$ and thus there is a one-to-two relationship between $x$ and $y$ and between $y$ and $x$.

Relationships between two variables are often better expressed graphically than numerically since many people find a visual image easier to understand than a math- ematical equation. A graph is a drawing showing the relationship between certain sets of quantities by means of points or lines plotted with respect to a set of coordinate axes.

Interpolation – the coordinates of the centroid of a triangle

Let $D$ be the midpoint of $BC$ and let $G$ divide $AD$ in the ratio $2:1$ (so that $GD = \frac{1}{3}^{rd}$ of $AD$). Let the coordinates be $A(x_A, y_A), B (x_B, y_B)$ and $C(x_C, y_C)$. By proportion the coordinates of $D$ are

$(x_D , y_D) =   \frac{x_B  + x_C}{2}$   ,  $\frac{y_B  + y_C}{2}$

The coordinates of

$$G = {\frac{2x_D + x_A}{3}  , \frac{2y_D  + y_A}{3} }$$

$$=    \frac{x_A  + x_B  + x_C}{3} $$  ,  $$\frac{y_A  + y_B  + y_C}{3}$$

Note: Since this is symmetrical it would be the same for BE where E is the midpoint of AC and CF where F is the midpoint of AB. The lines joining the vertices of a triangle with the midpoints of the opposite sides pass through one point G known as the centroid. If the triangle were made of uniform density, then the centroid would be its center of mass or gravity.



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