# Transforming Ground

## Introduction

## Description

When we stand in an urban setting and look around it is difficult to imagine the open countryside that was once there. Let us summarise a few of the initial steps in changing this countryside into roads and buildings. Prior to the commencement of any construction work, the land’s shape must be transformed. If we are to construct a building then we would normally wish to start off with flat, horizontal land. If we are to construct a road then the land upon which the road is to be laid can be quite a complex shape. The process of transforming the original land to the shape that we would like it to be involves moving earth from one place to another. Earth may also have to be either removed from the site or brought to the site. Estimating how much earth is to be moved and how much of it has to be brought in or taken away is important as its relocation involves a significant cost. Let us look at how we can start to estimate the volumes of earth. We take as our example the construction of a small road.

Step 1: Model the original ground. Perhaps a contour map is available.

Step 2: Draw the outline of the road on the map.

Step 3: On the map, mark the points at which you wish to specify the height of the new road.

The mathematics used to calculate the volumes includes trigonometry, surface fitting, geometry, and integration.

We could use triangulation to model a surface:

This site [1] shows a triangulation taking place.

## Demonstrations

**Props: **Paper, ruler, pencil, eraser

Imagine that you are a land surveyor and that you have recorded the height of the ground at a number of locations on the construction site. *Get your sheet of paper and mark on it a number of crosses, randomly scattered over the paper.* Each cross represents the position at which you have taken the height of the ground above sea level. Overall, therefore, your paper represents a plan view of the ground. Next we need to perform a triangulation. This makes it easy for us to calculate the height of the ground anywhere, not just at the positions of the crosses. Below are two demonstrations for achieving a good triangulation.

1. *Take your pencil and ruler and start connecting the crosses to one another so as to form triangles.* Each vertex of a triangle must only occur at a cross. It is best if you complete this exercise quickly without too much thinking. Let us consider, for a moment, what makes a good triangulation. A good triangle is one where all of the sides are about equal. We must try to avoid long thin triangles. When you have triangulated the whole surface, you can *check that your triangulation is a good one and amend it where necessary*. You can do this by comparing each triangle with each of its neighbours. Two triangles side by side form a quadrilateral. Think to yourself "is the line splitting the quadrilateral in two a good one, or would it be better using the other diagonal?" In this way you will need to erase some lines and insert other lines in order to improve the triangulation. Stop when you have the perfect triangulation.

2. *Take your pencil and ruler and start connecting the crosses on the outside to one another *so that when you are finished you have a polygon enclosing all the of other crosses. Now, look around the edge of the region and try to find the best triangle that you can. (A good triangle is one where all of the sides are about equal. We must try to avoid long thin triangles.) This might be a triangle that has one side on the polygon and a vertex inside the region. Alternatively, it could be a triangle that has two sides on the polygon. *Connect the points to form this new triangle*. Next, look for another good triangle on the edge of the region (which keeps shrinking). *Connect the points to form this new triangle.* Keep adding triangles to the edge of the region until the triangulation is complete. This exercise is rather like having a polygonal shaped pie and biting triangular pieces out of it until it is gone.

You can read more about moving soil (from Wikipedia) or more on the mathematics of calculating volumes [2].

## External links

## Attributions

[1] Lambert, T., "Delauney Triangulation Algorithms", 1998; http://www.cse.unsw.edu.au/~lambert/java/3d/delaunay.html

[2] The Royal Academy of Engineering, "The Mathematics of Earthwork Calculations", http://www.raeng.org.uk/education/diploma/maths/pdf/exemplars_advanced/2_Earthworks.pdf

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