3.1. Introduction

You know that the paper is a model for a plane surface. When you join a number of points without lifting a pencil from the paper (and without retracing any portion of the drawing other than single points), you get a plane curve.

In this chunk, we learn that a piece of paper represents a flat surface known as a plane. If you take a pencil and connect various points on this paper without lifting the pencil and without going back over any part more than once, the shape you create is a plane curve. Essentially, this is how curves can be drawn using simple techniques.

Think of drawing a continuous line on a piece of paper, like drawing a figure-eight without lifting your pencil. The final picture created can be seen as a plane curve, showing how smooth and continuous lines can form shapes.

A simple closed curve made up of only line segments is called a polygon.

This chunk introduces polygons as shapes that are formed by connecting lines in a closed loop. A polygon must consist solely of straight line segments and should completely enclose a space, not allowing any gaps. This sets the foundation for understanding different types of polygons as we learn more about their properties.

Consider a simple drawing where you connect dots to form a shape, like connecting four dots to create a square, or three for a triangle. Each of these drawings is a polygon because they are formed by straight lines and enclose an area.

Polygons that are convex have no portions of their diagonals in their exteriors or any line segment joining any two different points, in the interior of the polygon, lies wholly in the interior of it.

This chunk explains the difference between convex and concave polygons. A convex polygon is one where no diagonal, which is a line joining two non-adjacent vertices, lies outside the shape. This means that if you were to connect any two points inside the polygon, the line connecting them would stay within the shape. In contrast, a concave polygon would have at least one diagonal that dips outside the shape.

Imagine a convex shape like a regular dining table – if you draw lines between any two points on that table's edge, the line stays above the table. Now think of a concave shape like a star; if you connect certain points, the line might dip below making it 'cave' in. This visual helps distinguish between the properties of these types.

A regular polygon is both ‘equiangular’ and ‘equilateral’. For example, a square has sides of equal length and angles of equal measure.

This chunk defines regular polygons, which have equal-length sides and equal angles, meaning they're symmetrical from every angle. An example is a square, which has four sides of equal length and four right angles. On the other hand, irregular polygons do not have equal lengths or angles, such as a rectangle or a random quadrilateral.

Consider a slice of pizza – if all slices (triangles) are equal in size, you have a regular polygon. If someone takes extra from one slice making it bigger than the others, that’s an example of an irregular polygon.

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